Lautman

Lautman
MATHEMATICS, IDEAS AND THE PHYSICAL REAL
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MATHEMATICS, IDEAS AND
THE PHYSICAL REAL
Albert Lautman
Translated by Simon B. Duffy
Continuum International Publishing Group
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Originally published in French as Les mathématiques, les idées et le réel physique
© Librairie Philosophique J. VRIN, 2006
This English language edition © the Continuum International
Publishing Group, 2011
All rights reserved. No part of this publication may be reproduced or transmitted in any
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from the publishers.
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
EISBN 978-1-4411-4433-1
Library of Congress Cataloging-in-Publication Data
Lautman, Albert, 1908–1944.
[Mathématiques, les idées et le réel physique. English]
Mathematics, ideas, and the physical real / Albert Lautman;
translated by Simon B. Duffy.
p. cm.
“Originally published in French as Les Mathématiques, les idées et le réel physique.
Librairie Philosophique, J. VRIN, 2006”–T.p. verso.
Includes bibliographical references and index.
ISBN-13: 978-1-4411-4656-4
ISBN-10: 1-4411-4656-3
ISBN-13: 978-1-4411-2344-2 (pbk.)
ISBN-10: 1-4411-2344-X (pbk.)
1. Mathematics–Philosophy. I. Duffy, Simon B. II. Title.
QA8.4.L37613 2010
510.1–dc22
2010041829
Typeset by Newgen Imaging Systems Pvt Ltd, Chennai, India
Printed and bound in Great Britain
Contents
Translator’s Note
ix
Acknowledgements
xi
Introduction, by Jacques Lautman
xiii
Secondary Bibliography on the Work of Albert Lautman
xx
xxiii
xxiv
xxvi
xxviii
xxx
xxxii
xxxiv
Preface to the 1977 Edition, by Jean Dieudonné
xxxix
Albert Lautman and the Creative Dialectic of
Modern Mathematics, by Fernando Zalamea
1. Effective Mathematics
2. Structure and Unity
3. Mixes
4. Notions and Ideas
5. Platonism
6. Category Theory
Considerations on Mathematical Logic
1
Mathematics and Reality
9
International Congress of the Philosophy of Science
13
On the Reality Inherent to Mathematical Theories
27
v
CON TEN TS
The Axiomatic and the Method of Division
1. Equality
2. Multiplication
3. Unity
4. Measure and the Integral
5. The Absolute Value
31
34
35
37
37
39
Book I: Essay on the Unity of the Mathematical Sciences
in their Current Development
Introduction: Two Kinds of Mathematics
45
Chapter 1 The Structure of a Domain of Magnitudes and
the Decomposition of Its Elements: Dimensional
Considerations in Analysis
50
Chapter 2 The Domain and Numbers: Non-Euclidean Metrics
in the Theory of Analytic Functions
60
Chapter 3 The Algebra of Non-Commutative Magnitudes:
Pfaffian Forms and the Theory of Differential Equations
67
Chapter 4 The Continuous and the Discontinuous: Analysis and
the Theory of Numbers
73
Conclusion
80
Book II: Essay on the Notions of Structure and Existence
in Mathematics
87
Section 1: The Schemas of Structure
93
Introduction: On the Nature of the Real in Mathematics
vi
Chapter 1 The Local and the Global
1. Differential Geometry and Topology
2. The Theory of Closed Groups
3. Approximate Representation of Functions
95
102
105
106
Chapter 2 Intrinsic Properties and Induced Properties
1. Parallelism on a Riemann Manifold
2. Structural Properties and Situational Properties
in Algebraic Topology
3. Duality Theorems
4. The Limitations of Reduction
110
113
115
118
124
C ONT E NT S
Chapter 3 The Ascent towards the Absolute
1. Galois’s Theory
2. Class Field Theory
3. The Universal Covering Surface
4. The Uniformization of Algebraic Functions on
a Riemann Surface
125
126
128
130
Section 2: The Schemas of Genesis
139
Chapter 4 Essence and Existence
1. The Problems of Mathematical Logic
2. Existence Theorems in the Theory of Algebraic
Functions
3. Existence Theorems in Class Field Theory
4. The Theory of the Representation of Groups
141
141
Chapter 5 ‘Mixes’
1. Hilbert Space
2. Normal Families of Analytic Functions
157
160
167
Chapter 6 On the Exceptional Character of Existence
1. The Methods of Poincaré
2. The Singularities of Analytic Functions
171
175
178
Conclusion
183
133
148
151
152
Book III New Research on the Dialectical Structure of Mathematics
197
Chapter 1 The Genesis of the Entity from the Idea
1. The Genesis of Mathematics from the Dialectic
199
203
Chapter 2 The Analytic Theory of Numbers
1. The Law of Reciprocity
2. The Distribution of Primes and the Measurement
of the Increase to Infinity
3. Conclusion
207
208
Letter to Mathematician Maurice Fréchet
220
Foreword
213
218
Book IV Symmetry and Dissymmetry in Mathematics and Physics
Chapter 1 Physical Space
229
vii
CON TEN TS
241
242
251
258
Notes
263
Bibliography
281
Index
297
Chapter 2 The Problem of Time
1. Sensible Time and Mathematical Physics
2. The Theory of Partial Differential Equations
3. The Theory of Differential Equations and Topology
viii
Translator’s Note
The collected work of Albert Lautman was first published in 1977 as Essai
sur l’unité des mathématiques et divers écrits by Union Générale d’Éditions.
It was not until 2006 that the re-edited collection upon which this translation is based appeared from Librairie Philosophique J. Vrin. In addition to
permission to include all of the texts by Lautman that appeared in the 2006
volume, Fernando Zalamea has given permission to include his introduction
to Lautman’s work that also appeared in it. This Zalamea article will provide
the main introduction to the work of Lautman in the current volume. The
secondary bibliography of reviews and philosophical commentaries on
Lautman’s work that was compiled by Zalamea for the 2006 volume has
been updated and expanded for inclusion in the current volume.
The four main essays published by Lautman are included in the chronological order of their appearance. The first two that appeared in 1938 were
for the Doctorat D’Etat. ‘Essay on the unity of the mathematical sciences in
their present development’ (1938a) was Lautman’s secondary thesis, and
‘Essay on the notions of structure and existence in mathematics. I. The
schemas of structure. II. The schemas of genesis’ (1938b) was his principal
thesis. Lautman often refers to these two essays in his other work in this
way, that is, as his ‘principal thesis’ and his ‘secondary thesis’. Whenever
he does so, I have included the citations as above.
I have made a few typographic corrections to the equations with reference to the original publications. Where English translations of cited or
referenced material are available, I have provided citations from the English
edition and/or given page references to the English edition in square
brackets after the reference to the original language edition, except where
otherwise indicated.
ix
T RAN S L ATO R’S N OTE
In the French translation of Heidegger’s The Essence of Reason (1969
[1929; 1938]) used by Lautman, Dasein is translated as ‘human reality’.
I have retained this French usage in all quotations from Heidegger 1969,
and indicated this in the text with square brackets around [human reality].
I have also altered the Malik translation to render Heidegger’s distinction
between Sein – as ‘being’, the gerund of ‘to be’ – and Seiend (singular) or
Seiendes (plural) – as ‘an entity’ and ‘entities’. References in the text are
to the English translation edition (Heidegger 1969) followed by page numbers in square brackets to the French translation.
I have translated the term dominer, which Lautman uses to describe the
nature of the relation that exists between dialectical ideas on the one hand,
and mathematical theories on the other, with the term ‘to govern’. I have
translated the French sensible as ‘sensible’ in English, as in sensibility in
contrast to understanding, in keeping with the common English translation of the Kantian term sinnlich.
Lautman refers to combined subdomains of mathematics, such as algebraic topology, differential geometry, algebraic geometry and analytic
number theory as either les mathématiques mixtes, and simply les mixtes or un
mixte, which I have translated, respectively, as ‘mixed mathematics’ and the
‘mixes’ or a ‘mix’.
Additional translator’s notes on particular points have been included in
the Notes following each section. These are indicated by the following:
—Tr.
I would like to thank Daniel W. Smith, who, in consultation with
Continuum, suggested that I undertake this project, and Sandra L. Field,
for her support and encouragement throughout.
Simon B. Duffy
x
Acknowledgements, 2006 Edition
After the new study by Fernando Zalamea and the re-publication of the
preface written in 1977 by Jean Dieudonné, the chronological order has
been retained. In this new edition of the collection it seemed justified to
provide the view that the two articles, ‘Essay on the unity of the mathematical sciences in their present development’ (secondary thesis) and
‘Essay on the notions of structure and existence in mathematics’ (principal
thesis), introduce, among other things, the ideas and the positions formulated by the author after these were first published. Similarly, with the
latest articles, his conviction of a profound affinity between mathematical
structures and the exigency of the external real revealed in the formalization of physical theories is seen to take an increasingly important place in
the course of his short career.
A secondary bibliography listing reviews and philosophical commentaries of the work is also presented. Fernando Zalamea established the second
bibliography and here authorized its use. I thank him warmly.
The bibliography of all cited references has been established for this
edition, thanks to the resources of the Library of Mathematics at the École
Normale Supérieure. Permission was granted to refrain from reproducing
the original bibliographies, which remain rather imperfect, since, at that
time, philosophers other than historians of philosophy had little respect for
the rules which have since become the required standards.
Finally, I would like to express the gratitude of my brother and myself to
the Librairie Philosophique Vrin and Madame Arnaud, its director, and to
Jean-François Courtine for having welcomed these old and difficult texts.
Jacques Lautman
xi
Introduction
by Jacques Lautman
The present book, with preface by Jean Dieudonne from the 1977 edition,
preceded by a recent study by Fernando Zalamea, reproduces the entire
collection, plus a few additions, of the texts of Albert Lautman which had
been reunited in 1977 under the title L’Unité des sciences mathematiques et
autres écrits in one volume of the series 10/18, prepared by Maurice Loi. The
title, taken from the supplementary thesis published in 1938 by Hermann,
indicates one of the many directions of his work, but it was a poor guide to
the hidden metaphysical ambition of the author, which is central, though
it remained underdeveloped. Since Parmenides, the great philosophers
have scrutinized the very complex relations between the opposites: the
finite and the infinite, the continuous and the discrete, the open and the
closed, the local and the global, the same and the other, and movement
and immobility. All are pairs of concepts used to describe the situations, or
processes, of nature. The simplest example is chirality (the impossibility of
superimposing two dissymmetrical objects in the same orientation) and
the importance of dissymmetry is well known in the discovery of crystallography by Pasteur.
Albert Lautman understood the fundamental rupture between mathematics up until Augustin-Louis Cauchy and modern mathematics that
arises with Évariste Galois and Niels Henrik Abel, and progressed rapidly
with Bernhard Riemann, Georg Cantor and then David Hilbert to create
a multitude of developments in which opposites interpenetrate, create
inclusive links, open new domains to the creative imagination and, in the
xiii
IN TRO D UCTIO N
process, produce quantities of mathematical entities whose identity is
established, more or less clearly, at various levels. Essential are the mixes
that appear, unexpectedly or secretly, and that make new improbable passages possible. Mathematics has its particular development but Max Planck
and Albert Einstein have brought to the fore how physics and cosmology
require, ex post, the creations of Riemann and Henri Poincaré, from which
comes Albert Lautman’s conviction that dialectical pairs dominate the
physical real in the same way as the functioning of the mind, and that
mathematics is the most accomplished modality of the development of the
possibilities of operational connections between opposites. In his lectures,
he loved to cite Malebranche: ‘The study of mathematics is the purest
application of the mind to God’.1 Lautman’s last texts, written in 1943–
1944, clearly attest to a shift in his work towards physics.
This work on mathematical philosophy, short and very focused, is
indebted to the entry by Jean Petitot in the Enciclopedia Einaudi (Petitot
1982) for not being completely ignored, even by the specialists. Despite the
two theses defended in December 1937 in front of Leon Brunschvicg and
the mathematician Elie Cartan, both great figures, having been immediately subject to a critical review by Jean Cavaillès in the Revue de Metaphysique et Morale (1938a) and, in 1940, in the Journal of Symbolic Logic, a
penetrating review, highlighting the unresolved difficulties concerning the
status of the existence of mathematical objects, by Paul Bernays (1940), a
colleague of Hilbert. The break in scientific exchanges because of the war
and the death of the author at 36 years of age in 1944 explains in part the
forgetfulness that was only interrupted from time to time, with the 1977
edition by 10/18 (Lautman 1977), and in 1987 by Petitot in the Revue
d’histoire des sciences.2
Three other reasons, each situated at very different levels relative to one
another, are worth mentioning: the uncertainty about a receptive audience, the silence of the logicians and Cavaillès’s shadow. These texts,
whose argument relies heavily on the precise analysis of a number of
mathematical creations during the years 1860 to 1943 are difficult reading
for ordinary philosophers. Mathematicians follow more easily; however,
they are normally devoted to mathematics. It is, in general, only late in
life that some, usually only the great figures, interest themselves in the
ultimate questions, and, as will be found in the preface to the 1977 edition
by Jean Dieudonné, pivotal to the Bourbaki group at its creation and for a
long time thereafter.
xiv
INT R ODUC T ION
Detached from examples of mathematics that are the subject matter on
which work that is technically philosophical is based, the metaphysical
contribution, in the strict sense, which carries the meaning of the work,
has been puzzling: suggestive but too underdeveloped not to be ambiguous, since, even though both the realist and idealist conceptions of science
could retrieve it, it would remain absolutely intolerable to the nominalist.
The absence of interest from mathematical logicians and the frank hostility
of philosophical logicians, with the exception of Ferdinand Gonseth,3 are
better appreciated when it is understood that the first two articles published by Albert Lautman comprise frontal attacks against the heritage of
the Principia Mathematica of Bertrand Russell and Alfred North Whitehead,
and also against Rudolf Carnap and the Vienna Circle.
The few pages from 1939 in which he expressed his interest in Being
and Time (1962 [1927]) have retained much less attention than that his
rationalist reading is far removed from the Martin Heidegger of the French
after the war4, if not from Heidegger himself. Gilles Deleuze (1994 [1968],
chapter 4) and, more recently, Alain Badiou (2005 [1998]) appear to
be the only ones to have made explicit a strictly philosophical use of his
work, outside of a strictly epistemological context or the history of the
discipline.
It is hard to write that the image, quite rightly great, of Jean Cavaillès
cast a shadow over the destiny of the writings of Albert Lautman. Their
trajectories, philosophical as much as during war time, were both too parallel and too unequal in visibility for it to have been otherwise. Cavaillès,
the elder by a few years, was a young substitute professor at the Sorbonne
in 1941. He had a place of national importance in the Resistance, and with
the combination of these two qualities, he has become, post-mortem, the
symbol of a resistant university, in part mythical. For those not looking
carefully, Albert Lautman, whose name has often been quoted next to that
of Cavaillès, could only have been a disciple, at best a double. This is not
how Lautman and Cavaillés saw themselves, as evidenced by their public
exchange, organized by Leon Brunschvicg before the Societe française de philosophie on 4 February 1939 (Cavaillès and Lautman 1946), and also those
few remaining letters from Cavaillès to Lautman (see Benis-Sinaceur
1987). Their friendship, the identity of their ethico–political commitment
and their work community in a domain in which those who might follow
them were scarce did not exclude a fairly large difference in angle of attack
on mathematics and the task of the philosopher.
xv
IN TRO D UCTIO N
The life of Albert Lautman had been deeply inflected twice. In autumn
1923, in the elementary mathematics class at Lycée Condorcet, he met
Jacques Herbrand5 – in the words of his contemporaries, a quite exceptional person – whose influence was absolutely crucial on his orientation
towards mathematical philosophy. Less than 15 years later, the announcement of the triumph of Nazism will transform the philosopher, for whom
the defence of freedom and universal values are a necessity that take precedence over all, in wartime.
Lautman was born in Paris in 1908. His father, Sami Lautman, a Jew of
the Austro–Hungarian Empire, had been excluded from the competitive
examinations to train at the hospitals in Vienna, under the numerus clausus.
Lüger, elected Mayor of Vienna in 1886, is the one who put in place the
first anti-Semitic provisions of contemporary times. So Sami Lautman
arrived in Paris in 1891 and had to start by obtaining a baccalaureate before
resuming his medical studies ab initio. In 1914, he joined the French Army,
but because he was newly naturalized, he could not be an officer and,
therefore, could not be classed a doctor. He was a warrant officer stretcherbearer and, of course, served as a doctor. Critically wounded, he received
the Croix de Chevalier (Knight’s Cross) of the Legion d’Honneur. Albert’s
mother, née Lajeunesse, is from a family of both Avignonians (Papal Jews)
and Alsacians who were settled in Paris for several generations.
In 1926, Albert entered the École Normale Supérieure via the Letters
examination and met Jacques Herbrand again, who had entered in 1925,
first in the Science examination, and who, to join his friend, went to the
literature student rooms. ‘Loving philosophy with a passion, but not seeking in it a rule of life because the practical problem was not interesting’
(Chevalley and Lautman 1931), Herbrand was the individual and daily
instructor for Lautman in mathematics. Together, they became friends
and intellectual companions with Claude Chevalley, then with Charles
Ehresmann, two of the five future founders of the Bourbaki group.6 In
February 1928, Lautman was taken by Celestin Bougie, deputy director of
the school, to Franco–German meetings in Davos, where he met his future
wife, then a student of George Davy in Dijon. He returned convinced that
mathematical creation is done in Germany and made sure to spend a
semester in Berlin in 1929. In 1929, an event of micro-history is indicative
of his politics and morality. A majority of student primary school teachers
from Quimper had decided to refuse to take military service for reserve
officers and were threatened with exclusion from their École Normale
Primaire by the Rector of Rennes. In Rue d’Ulm,7 a petition of support
xvi
INT R ODUC T ION
circulated, borne mainly by former khagneux8 at the Lycée Henri IV converted to pacifism by Alain,9 with Simone Weil at the head.10 Albert Lautman
signed, but, unlike several other signatories, he did not refuse military
service, not wanting the signatories to be excluded or to make them fail.
He promoted freedom of choice, but was elitist and considered that Noblesse
oblige. Qualified in philosophy in 1930, he completed his military service
with fellow mathematicians in the 401st Field Artillery at Metz.
In spring 1931, the Institute of Western Languages in Osaka wrote to
the Director of the École Normale Supérieure, seeking to recruit a young
teacher of that school for two years to teach literature and French philosophy. The offer tempted Jean-Paul Sartre and Lautman, who both decided
to apply. Lautman sent his letter via the USSR and the Trans-Siberian railway, and it arrived before that of Sartre, which was delivered by ship from
Marseilles. Missing, without doubt, other selection criteria, the Japanese
responded positively to the application that arrived first. On his return
from Osaka two years later, welcomed without excessive warmth by the
Inspector General of Philosophy (unique at the time) Dominique Parodi,
who, three years later would open the columns of the Revue de Metaphysique
et Morale, he was assigned to the Lycee de Vesoul. He was there only one
year and earned a one-year scholarship to the Caisse Nationale des Sciences,11
the forerunner of the CNRS.12 In October 1935, he was allocated to the
boys school at Chartres, which allowed him to attend seminars at the Institut Henri Poincaré, notably that of Gaston Julia, as evidenced by many references in his theses. At the invitation of Célestin Bouglé, each year he gave
a small series of lessons to the aggregatifs13 of the Rue d’Ulm. He frequented
the Sunday mornings of Leon Brunschvicg, who received students, colleagues and also members of important radical parties, being the Deputy
Secretary of State in the government of Léon Blum. At the end of 1937, he
defended his theses.14
Politically, Lautman was on the left, but he kept his distance from the
Communists because he did not admit that the end justifies the means. He
had contacts with German colleagues, a number of whom he saw pass
through France on their way to the United States, and not all of them were
Jews. He was convinced, from the beginning of 1938, that war was inevitable, and that the sooner the better. This is why he signed up voluntarily
for fairly heavy training for reserve officers, with training periods in
Suippes and Mourmelon. This earned him, at mobilization, the assignment
of commander of an anti-aircraft artillery battery, and he was quickly promoted to ‘temporary’ captain. His battery had some success: four planes of
xvii
IN TRO D UCTIO N
the Luftwaffe shot down, three hit. As captain he received the Military
Cross. Posted in mid-June 1940 to cover the re-embarkation of British
troops, he was taken prisoner and sent to Oflag IV D at Hoyerswerda,
Silesia. A first hazardous attempt to escape failed before he had even
crossed the last barbed wire fence. Hence, he was sent to the lockup, where
attempted escapees were reunited. He took an active part in the university
of the camp, which had notable lectures on major topics.15
While incarcerated, he made the acquaintance of Jacques Louis, a SaintCyrian16 and captain in a regiment of Tunisian goumiers,17 also committed
to escape for good. The child of a northern city, Louis found himself separated from his parents in 1914, during the first German offensive. Taken in
by British soldiers, he spent the war with an English family and was miraculously found by his parents in 1919. An adventurer but also an organizer,
he soon realized that Lautman the intellectual, who still didn’t know how
to turn an empty tin can into a pickaxe, a shovel or a candle-holder – he’d
eventually learn – presented as a critical asset: he spoke fluent German
and could buy tickets for rail and bread without being noticed. Louis
assembled a group of 28, who spent nine months digging an 80-meter tunnel. On 18 October 1941, they set off. Sixteen arrived safely, after about
ten days; the most delicate passage was that between Alsace–Lorraine,
already regarded as under the control of the Reich, and then France. The
railway workers of the SNCF18 were known intermediaries and the handcar workers at Forbach did a great service, helping the escaped soldiers to
freedom.
Demobilized, Lautman was immediately dismissed as a Jew. After a stay
in Aix for a few months, he moved to Toulouse, where he found Louis,
who devoted himself to recruiting for the Secret Army. Lautman quickly
became one of those responsible for the Haute-Garonne staff headquarters
(see Latapie 1984); however, he also accepted the responsibility for organising passages to Spain for the O’Leary network19 (see Belot 1998, 106–10),
the smugglers were Spanish Republicans in exile. For two years, at the rate
of about 12 to 20 per month, Anglo-Saxon airmen, Resistance fighters on
the run, hounded Jews and young people anxious to join the Free French
forces had crossed the Pyrenees, at the risk then of a few months incarceration in Miranda prison, before being able to make it to Algiers. In April
1944, there was a setback: a summer route over the mountains had been
chosen, yet it snowed late. Tracks were found and the group was stopped.
However, within the scope of his activities in the Secret Army, when, in the
spring of 1943, the message was received announcing that the Normandy
xviii
INT R ODUC T ION
landing would not take place soon, Lautman understood, like many others,
that he must organize the material life and also the activity of these young
people who had become illegal by refusing the STO (Service du Travail Obligatoire, or forced labour, in Germany) and were in the process of swelling
the ranks of the Maquis.20 He was particularly interested in the CorpsFranc near Toulouse21 when, sensing that the landing was imminent, he
decided to join them from 17 May 1944, to ensure their training. But on
Monday, 15 May 1944, when he arrived at the destination of the rendezvous with one of his smugglers, he and the smuggler were arrested by the
German police of the Occupation. Although never proven, the hypothesis
of treason, or at least that there was a weak link in the network, remains
the most likely explanation of his arrest. Lautman was part of the convoy
for deportation, which left Toulouse on 9 July but was turned back and
ended up at Bordeaux (see Nitti 1944). Fifty detainees whose cases were
the most serious were condemned to death. They were taken on 29 July to
the execution posts of Camp de Souges. The required squad of French
mobile police did not appear; it was the same the next day. On 1 August, a
squad of German non-commissioned officers completed the task.
xix
Secondary Bibliography on
the Work of Albert Lautman
Alunni, Charles. 2006. Continental genealogies. Mathematical confrontations
in Albert Lautman and Gaston Bachelard. In Virtual mathematics: the logic of
difference. Edited by S. Duffy. Manchester: Clinamen Press.
Badiou, Alain. 2005. Briefings on Existence: A Short Treatise on Transitory Ontology.
Translated by N. Madarasz. New York: State University of New York Press,
pp. 59ff.
Barot, Emmanuel. 2003. L’objectivité mathématique selon Albert Lautman: entre
Idées dialectiques et réalité physique. Cahiers François Viète 6:3–27.
———. 2009. Lautman. Paris: Belles Lettres.
Benis-Sinaceur, Hourya. 1987. Lettres inédites d’Albert Lautman à Jean Cavaillès;
Lettre inédite de Gaston Bachelard à Albert Lautman. Revue d’histoire des sciences
40 (1):117–129. In Blay 1987.
Bernays, Paul. 1940. Review of Albert Lautman, Essai sur les notions de structure et
d’existence en mathématiques; Essai sur l’unité des sciences mathématiques dans leur
developpement actuel. Journal of Symbolic Logic 5 (1):20–22.
Black, Max. 1947. Review of Jean Cavaillès et Albert Lautman. La pensée mathématique. Journal of Symbolic Logic 12 (1):21–22.
Blay, Michel, ed. 1987. Mathematiques et Philosophie: Jean Cavailles, Albert Lautman,
Revue d’Histoire des sciences. 40 (1).
Buhl, Adolphe. 1938. Review of Albert Lautman, Essai sur les notions de structure
et d’existence en mathématiques. I. Les schémas de structure. II. Les schémas de gense.
L’Enseignement mathematique 37:354–355.
Castellana, Mario. 1978. La philosophie mathématique chez Albert Lautman.
Il Protagora 115:12–24.
Cavaillès, Jean. 1938a. Compte-rendu de Albert Lautman, Essai sur les notions de
structure et d’existence en mathématiques. Essai sur l’Unité des sciences mathématiques.
Revue de Métaphysique et de Morale 45 (Supp. 3):9–11.
xx
S E C O N DA RY B I B L I O G R A P H Y O N T H E W O RK OF ALBERT LAUTMAN
Chevalley, Catherine. 1987. Albert Lautman et le souci logique. Revue d’Histoire
des Sciences 40 (1):49–77. In Blay 1987.
Costa de Beauregard, Olivier. 1977. Avant Propos à Albert Lautman, Symétrie et
dissymétrie en mathématiques et en physique. Lautman 1977, pp. 233–238.
Deleuze, Gilles. 1994. Difference and Repetition. Translated by P. Patton. London:
Athlone Press, pp. 177–183.
Dumoncel, Jean-Claude. 2008. Compte-rendu de Albert Lautman, Les mathématiques, les idées et le réel physique. History and Philosophy of Logic 29 (2):199–205.
Gonseth, Ferdinand. 1950. Philosophie mathématique. Philosophie: Chronique des
années d’après-guerre 1946–1948. Paris: l’Institut international de philosphie.
Republished in Gonseth 1997, pp. 95–189.
———. 1997. Logique et philosophie mathématique. Paris: Hermann.
Granell, Manuel. 1949. Lógica. Madrid: Revista de Occidente, pp. 284–285.
Heinzman, Gerhard. 1984. Lautman, Albert. In Enzyklopedie. Philosophie und
Wissenschaftstheorie. Vol. 2. Mannheim: Bibliographisches Institut, p. 547.
———. 1987. La position de Cavaillès dans le problème des fondments des mathématics, et sa différence avec celle de Lautman. Revue d’Histoire des Sciences 40
(1):31–47. In Bray 1987.
———. 1989. Une revalorisation d’une opposition: sur quelques aspects de la
philosophie des mathématiques de Poincaré, Enriquès, Gonseth, Cavaillès,
Lautman. Fundamenta Scientiae:27–33.
Kerszberg, Pierre. 1987. Albert Lautman ou le Monde des Idées dans la Physique
Relativiste. La Liberté de l’Esprit 16:211–236.
Lichnerowicz, André. 1978. Albert Lautman et la philosophie mathématique.
Revue de Métaphysique et de Morale 85 (1):24–32.
Merker, Joël. 2004. L’Obscur mathématique ou l’Ouvert mathématique. In Le réel
en mathématiques, edited by P. Cartier and N. Charraud. Paris: Agalma, pp. 79–85.
Nicolas, François. 1996. Quelle unité pour l’oeuvre musicale? Une lecture d’Albert
Lautman. Séminaire de travail sur la philosophie. Lyon: Horlieu.
Petitot, Jean. 1982. La filosofia matematica di Albert Lautman. Enciclopedia
Einaudi 15:1034–1041.
———. 1985. La philosophie mathématique d’Albert Lautman. Morphogenèse du
sens Paris: Presses Univésitaire de France, pp. 56–61.
———. 1987. Refaire le Timée. Introduction à la philosophie mathématique
d’Albert Lautman. Revue d’Histoire des Sciences 40 (1):79–115.
———. 2001. La dialectique de la vérité objective et de la valeur historique dans
le rationalisme mathématique d’Albert Lautman. In Sciences et Philosophie en
France et en Italie entre les deux guerres, edited by J. Petitot and L. Scarantino.
Napoli: Vivarium.
Rosser, J. Barkley. 1939. Review of Albert Lautman, Essai sur l’Unité des Sciences
Mathématiques dans leur Développement Actuel. Bulletin of the American
Mathematical Society 45 (7):511–512.
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Sichère, Bernard, ed. 1998. Cinquante ans de philosophie française. Paris: Ministère
des Affaires Étrangères, p. 38.
Thirion, Maurice. 1998. Images, Imaginaires, Imagination. Paris: Ellipses, pp. 332–341.
———. 1999. Les mathématiques et le réel. Paris: Ellipses, pp. 341–402.
Venne, Jacques. 1978. Deux épistémologues français des mathématiques: Albert
Lautman et Jean Cavaillès. Thèse: Université de Montréal.
Zalamea, Fernando. 1994. La filosofia de la matematica de Albert Lautman.
Mathesis 10:273–289.
———. 2006. Albert Lautman et la dialectique créatrice des mathématiques
modernes. In Lautman 2006, pp. 17–33.
xxii
Albert Lautman and the Creative
Dialectic of Modern Mathematics
by Fernando Zalamea*
It is possible today to observe in hindsight the epistemological landscape of
the twentieth century, and the work of Albert Lautman in mathematical
philosophy appears as a profound turning point, opening to a true understanding of creativity in mathematics and its relation with the real. Little
understood in its time or even today, Lautman’s work explores the difficult
but exciting intersection where modern mathematics, advanced mathematical invention, the structural or unitary relations of mathematical
knowledge and, finally, the metaphysical and dialectical tensions underlying mathematical activity converge. Well beyond other better-known
names in philosophy of mathematics – who are focused above all on questions concerning the logical problem of foundations, important but fragmentary studies in the vast panorama of modern mathematics – Lautman
broaches the emergence of inventiveness in the very broad spectrum of the
development of the mathematical real. Group theory, differential geometry, algebraic topology, differential equations, functional analysis, functions
of complex variables and number fields are some of the domains of his
* Departamento de Matematicas, Universidad Nacional de Colombia, mathematician, translator of the work of Albert Lautman into Spanish (Bogotá: Siglo del
Hombre, 2008). He received the Jovellanos Prize in 2004 for his book Ariadna y
Penélope. Redes y mixturas en el mundo contemporáneo, Oviedo: Ediciones Nobel, 2004.
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preferred examples. He detects in them methods of construction, structuration and unification of modern mathematics that he connects to a precise
Platonic interpretation in which powerful pairs of ideas serve to organize
the edifice of effective mathematics.
In what follows, an interpretation of the work of Lautman will be offered
that freely uses some mathematical advances of the second half of the
twentieth century, since, in our view, interest in Lautman consists primarily in what it has to say to us today. His ideas, his method, his wagers are
now more striking than ever. After the particular limits encountered by
analytic philosophy, after the reductionist linguistic dissections of knowledge, after the zigzags of postmodernism, a critical return to the big questions of the history of philosophy and toward a complex appreciation of
reality brings the work of Lautman into close proximity with contemporary
inquiry.1 We will proceed in three stages, according to a triple back-andforth between various levels of the concrete and the abstract: plurality of
effective mathematics and unity of structural methods (Sections 1 and 2);
‘mixed’ mathematics, Lautmanian ‘notions’ and ‘ideas’ (Sections 3, 4: the
adjective Lautmanian deserves to enter into usage); Platonic dialectic and
the dialectic of the mathematical theory of categories (sections 5, 6). The
aim is to pay homage in this way to Lautman’s very method, which permits
rising to the most pure and universal through a dialogical back-and-forth
between complementary notions, following a scale in which the complex and
the concrete are gradually liberated to attain the simple and the abstract.
1. EFFECTIVE MATHEMATICS
With the term ‘effective mathematics’,2 Lautman tackles the theories,
structures and constructions conceived in the very activity of the mathematician. The term refers to the structure of mathematical knowledge, and
what is effective refers to the concrete action of the mathematician to gradually build the mathematical edifice, that such action is constructivist or
existential.3 The mathematical – beyond its ideal set theoretical reconstruction – develops along a hierarchy of real configurations of rather diverse
complexity, in which the concepts and examples are connected through
structural processes of liberation and saturation, resulting in mathematical
creations like mixes between opposite polarities. Lautman detects some
specific features of advanced mathematics that are not given in elementary
mathematics:4 a) the complex hierarchisation of various theories, irreducible
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to systems of intermediate deduction; b) the richness of the models, irreducible to linguistic manipulation; c) the unity of structural methods and of
conceptual polarites, beyond the effective multiplicity of models; d) the
dynamics of the creative activity, in a permanent back-and-forth between
freedom and saturation, open to the Platonic division and the Platonic dialectic; e) the mathematically demonstrable relation between what is multiple
on a given level and what is singular on another, through a sophisticated
lattice of mixed ascents and descents.
The reduction of all mathematical theorems to ‘tautologies’, equivalent
to each other from the point of view of the base Zermelo-Fraenkel axioms,
has levelled (conveniently for many) the broken terrain of mathematics.
Lautman protested against this ‘resignation’ and his work deserves to be
understood as an effort to describe a mathematical topography that is
much more complex and eventful. Attentive to the ascents and descents in
Galois theory, in class field theory, or in the construction of universal covering surfaces, Lautman shows not only how the process of saturation of
an imperfect structure gives rise to mathematical creativity, but also how
these processes combine gradually through scales of construction and very
precise inverse correspondences: between intermediate fields and subgroups of the Galois group, between class fields and groups of ideals,
between covering surfaces and subgroups of Poincaré’s fundamental group.
In these cases, mathematical creativity is responding to a hierarchization of
knowledge, in which the multiple intermediate levels of correspondence
between structures are much more important than a purely logical alternation between particularization and generalization. According to Lautman,
mathematical creation emerges with the division and definition of differences (hence the influence of Lautman on Deleuze)5, coupled with the
utilization of mixes which permit the liberation of simple notions. It happens to be a process that is opposed to the arbitrary search for generalizations, in which ‘a whole conception of mathematical intelligence’ is put
in play:
It is, in effect, extremely important for the philosopher to prevent the
analysis of ideas and the research of notions that are the most simple and
separable from each other, from appearing as research of the broadest
kind.
In fact, a closer look at the gradations and the back-and-forth of concepts
allows the emergence of mathematical thought to be observed. The hatching
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and the genesis of mathematical structures, that the analytic set-theoretical
approach hides, can be studied with greater insight by the Lautmanian
approach. When he detects that the metrization of topologies is not always
possible, or that the Hilbert space serves as a ‘remarkable synthesis of continuous and discontinuous notions’, when he discovers the precise moment
when algebra and analysis are related in the work of Hilbert on integral
equations, when he recalls the ‘affinity of the underlying dialectical structure’ to the Weil conjectures, when he shows the emergence of Herbrand’s
domains as gradual mixes which allow the construction of proofs of consistency, when he marvels at lattices that appear throughout mathematics,
Lautman is always on the lookout for creative movements in mathematics:
movements in which a problem, a concept or a construction is transformed
through the sheaf of partial solutions to the problem, the definitional
delimitations of the concept, and the saturations and divisions of the
construction. Mathematics, a space of thought that is always alive and in
constant evolution, can in this way appear as truly iconic in the work of
Lautman, one of whose principal merits (as Bernays remarks) consists precisely in the capacity to think the mathematical world faithfully.
At the base of his work, when he contemplates the dynamic spectrum of
the technique of mathematics, Lautman discovers a number of tectonic
shifts in the new mathematics that will shake it right up to the late twentieth
century: ‘the primacy of geometric synthesis over that of the “numerical”
analysis’ (see the combinatorial geometries of Boris Zilber, Poizat 2000), a
full perception of the intrinsic richness of arithmetic systems: ‘we may be
led to believe that it is wrong to claim to consider arithmetic as fundamentally more simple than analysis’ (see the reverse mathematics of Friedman
and Simpson, Simpson 1999), a linkage of properties of reflection and
properties of closure in the back-and-forth between the local and the
global (see the theorems of representation in Peter Freyd’s theory of allegories, Freyd et al. 1990). The agile receptivity of Lautman allows the
opening up of a number of major currents in the mathematics of his time,
that still have much to teach us.
2. STRUCTURE AND UNITY
According Lautman, mathematical theories are constructed following
‘a whole series of precisions, limitations, exceptions’, through which the
Ideas and concepts acquire their effective life. In fact, ‘restrictions and
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delimitations [. . .] should not be conceived as an impoverishment, but
rather as an enrichment of knowledge, due to the increased precision and
certainty provided.’ Nevertheless, the exact, concrete and differentiated
impetus consigned in this way has as a constant complement an inverse
alternating impetus, oriented towards integrality and unity. In the science
of the 1930s, the ineluctable emergence of the notion of mathematical
structure gave Lautman a supple instrument to order the diversity. Still
nascent when he was writing, and not yet defined in all their generality
(the labour of Bourbaki), mathematical structures included a wide network
that underlies the work of the German school (Hilbert, Emmy Noether,
Emil Artin), studied closely by Lautman during his final years at the École
Normale, with Herbrand, Chevalley and Ehresmann. He was capable of
grasping the emergence of various levels of mathematical structure: structured concrete objects (groups, algebras, Hilbert spaces, etc.), intermediate
structural correspondences (duality theorems, Galois correspondences,
conformal representations, etc.), generic structures (lattices, etc.). Going
even further, he began to see ‘structural schemas’ in which – at an ‘upper’
logical level: more simple and universal – a number of free oppositions
are assembled (local/global, intrinsic/extrinsic, discrete/continuous, etc.)
throughout collections of possible structures.6
Lautman emphasises a synthetic perception, ready to determine the
value of complex networks of mathematical interaction, beyond a stifling
search for ‘primary’ notions. The unity of mathematics is expressed, not in
a common base to rebuild the whole, but in the convergence of its methods
and in the passage of ideas between its various networks: logical, arithmetic, algebraic, analytic, topological, geometric, etc. The penetration of
the methods of algebra in analysis, analysis subordinated to topology, the
ubiquitous appearance of the geometric idea of domain, the agreement of
the complex variable in arithmetic – these are all examples studied in detail
by Lautman, in which, in the local fragment, the global unity of mathematics is reflected. It happens to be a real unity, at the interior of the synthetic universe of effective mathematics, that disappears when the plurality
of mathematical knowledge is reduced to its fictional analytic reconstruction. After all, the set theoretical presentation provides convenient layers
of relative consistency, but, in practice, it is increasingly evident that mathematics develops far from its so-called fundamentals.7 An epistemological
inversion shows how, contrary to what one might think in the first
instance, a practical observation of diversity can then reinstate the multiplicity in the unity. In fact, a full awareness of diversity does not lead to a
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lack of connectivity, but returns to the unity, whether in Charles Peirce’s
pragmatism, in Walter Benjamin’s montage, in the figurative relay of Pierre
Francastel, or in the difference of Deleuze. The metaphor of ‘rising again’
(remonter) – present in several of Lautman’s references: return, ascend,
sources, etc. – exposes once again the integral behind the differential.
Through logical gradations in which the structure and dynamics are in
dialogue, the ascent towards Ideas is one of the preferred Lautmanian
displacements. There is perhaps no more audacious and magnificent an
ascension than his penultimate text, when he assures us – and then convinces us in the demonstrative context of statistical mechanics – that ‘the
materials of which the universe is formed are not so much the atoms and
molecules of the physical theory as these great pairs of ideal opposites such
as the Same and the Other, the Symmetrical and Dissymmetrical, related
to one another according to the laws of a harmonious mixture.’ Similarly,
in a mathematical sheaf, the sections are complemented with projections,
in a back-and-forth that unites reality and knowledge, ‘the conceptual
analysis necessarily succeeds in projecting, as an anticipation of the concept, the concrete notions in which it is realized or historicized’.
3. MIXES
One of the characteristic traits of modern mathematics is its ongoing effort
to put an end to impermeable enclosures, its constant transport of examples, theories and ideas between different subdomains of mathematics,
with the consequent transformation of objects when they are read from
multiple variable contexts. If one of the strengths of mathematics has
always been its ability to realize the passages between the possible (models), the necessary (theorems) and the current (applications), this facility to
control and mould the mediations becomes a veritable plastic arsenal
throughout the twentieth century. Mixed mathematics are legion. Among
the most studied by Lautman, one finds algebraic topology, differential
geometry, algebraic geometry and analytic number theory. The connections of the noun (central subdiscipline) and the adjective (infiltrating
subdiscipline) only signal very weakly the real osmosis of modem mathematical thinking. The rigid and outdated delimitations disappear, and flexible frontiers arise in a new classification (MSC 2000), which, rather than
a Porphyrian tree, seems more like a liquid surface in which the information flows between mobile nuclei of knowledge.8
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Albert Lautman is the only modern philosopher of mathematics to have
remarked upon and studied, on the one hand, the emergence of mixed
mathematics in action, and on the other, the ‘ideas’ and ‘notions’ that
allow an understanding of the increasing potential of the mixes. Given the
central importance of mixed structures in contemporary mathematics, it is
natural to emphasize the immense value that Lautmanian philosophy of
mathematics can (and should) acquire today. The failure of a supposed
mathematical tautology (purist invention of analytic philosophy) and the
opening towards a contaminated mathematics, much closer to reality, are
the order of the day. Contamination by physics and metaphysics is indispensable for mathematical creativity.
Clearly opposed to the efforts of purification advanced in the foundations of arithmetic, Lautman exalts the richness of ‘transcendent’ analytical methods in number theory, and explains why mediations and mixes
tend to be required in the creative act in mathematics. In fact, the primary
role of mixes consists in their joint reflective action: a reflective back-andforth between partial properties, which are located in a neighbourhood
mediate between extrema and which act as a precise relay9 in the transmission of information. Whether in a given structure (Hilbert space), a collection of structures (Herbrand’s ascending domains) or a family of functions
(Montel’s normal families), mixes, on the one hand, imitate the structure
of underlying domains, and on the other hand, act as partial blocks to construct the higher domains. Without the desired form of contamination,
without these premeditated alloys, contemporary mathematics would simply be incomprehensible. A technical success like the proof of Fermat’s
theorem (1994),10 the symbol of the full deployment of twentieth-century
mathematical invention, is only possible as a final effort after a very complex
back-and-forth of mixes: a problem on elliptic curves and modular forms
resolved through deep connections between algebraic geometry and complex variables, developed around g -functions, their Galois representations
and their deformations and associated rings. The implicit scope of Lautman’s
work, registered in such a deployment of mixes, opens on surprising perspectives that appear to exceed the present philosophy of mathematics. The
‘mixes’ appear in Lautman’s first known writing, when he brilliantly
describes the construction of Herbrand’s ‘domains’ and shows how ‘the
Hilbertians knew to interpose an intermediate schematic, that of individuals and domains considered not so much for themselves as for the infinite
consequences that allow finite calculations to operate through them’.
Comparing this intermediate schematisation with Russell’s hierarchy of
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types, Lautman indicates that ‘we are in the presence, in each case, of a
structure whose elements are neither entirely arbitrary, nor really constructed, but composed as a mixed form that derives its fertility from its dual
nature’ (emphasis added). Understanding at the outset the significance of
mixed forms of logic in the 1930s, when logic was viewed on the contrary
as a pure form, demonstrates well the independence and the acuity of the
young philosopher. When it has today become evident that it is precisely
these mixed forms of logic that led to the real explosion in mathematical
logic (happily infiltrated by algebraic, topological and geometric methods),
and when it has became clear that a logical system should not be understood a priori, but rather as closely related to a collection of mathematical
structures a posteriori, it is all the more remarkable that these contemporary
mathematical facts are able to be found with such fidelity in the profound
philosophical insights of Lautman.
In his article on the ‘method of division’, Lautman connects the reference to ‘mixes’ to the Platonic tradition (Sophist, Philebus, Plato 1997),
emphasises their dynamic interest, and situates mathematical creativity in
a dialectic of liberation and composition. With terms that are not found
in his work, but that summarize his position on better known terrain, his
work shows how mathematics, on the one hand, divides the contents of a
concept through definitions (syntax) and derivations (grammar), thereby
releasing simple components, and how, on the other hand, it constructs
intermediate relays through models (semantic) and transport theory (pragmatics), thus rekindling the existence of simple filaments and allowing
for their reorganization into new concepts. When one of these mixes
succeeded in uniting a clear simplicity and a high powered reflector – as in
Riemann surfaces or in Hilbert spaces, admired and exemplarily studied by
Lautman in his two theses – the mathematical creativity reaches perhaps
its greatest heights.
4. NOTIONS AND IDEAS
In Lautman’s theses, the whole dynamics of his philosophy is driven by an
alternating contrast between complementary concepts – global/local,
whole/part, extrinsic/intrinsic, continuous/discrete, etc. – but it is in his
New Research on the Dialectical Structure of Mathematics that Lautman introduced the terms that govern these dialectical connections. He defines a
notion as one of the poles of a conceptual tension, and an idea as a partial
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resolution of this polarity. Thus, the concepts of finitude, infinitude, localization, globalization, computation, modeling, continuity or discontinuity are
‘notions’ (examples given by Lautman), and the propositions that express
state that infinity is obtained as the non-finite (cardinal skeleton), the
global as the patching together of the local (compactness), the modelizable
as the realisation of the deductively calculable (set-theoretical semantics),
or the continuous as the completion of the discrete (Cantorian right) are
some ‘ideas’ (examples given by us).
The interest in notions and ideas is threefold: they allow the filtering
(liberation) of some unnecessary ornaments and in this way allow the
mathematical framework to become clear; the unification of various constructions through a ‘higher’ dialectic level; and the opening of the spectrum
of mathematics to alternatives. Whether by filtering or unifying the mathematical landscape – duality theorems in algebraic topology and in the
‘general theory of structures’ (i.e. lattices) (Lautman’s examples) – whether
by opening it to a new framework of possibilities – non-standard ideas
that resolve in another way the oppositions between notions: the infinite
as unbounded (Robinson), the discrete as a sequence of demarcations on a
primitive continuous (L. E. J. Brouwer), the deductively calculable as a system of coordinates for the modelizable (Per Lindström) (examples proposed
by us) – in all cases, the Lautmanian notions and ideas cover the universe of
mathematics transversely, and explain the amplitude of this universe as much
as its surprising harmonic agreement between the one and the many.
For Lautman, notions and ideas are situated at a ‘higher’ level, in which
the intellect can imagine the possibility of a problematic independently of
mathematics, but whose real meaning can only be obtained when it is
embodied in effective mathematics. In the tension between a universal (or
generic) problematic and its partial concrete (or effective) resolutions,
according Lautman, much of the structural and unitary back-and-forth of
mathematics is to be found. As we will see later, it is a very precise question
of the paradigm promoted by the mathematical theory of categories.
Lautman is aware that he seems to introduce a delicate a priori in the
philosophy of mathematics, but he explains it as a simple ‘exigency of problems, anterior to the discovery of their solutions’: ‘in a purely relative sense,
and in relation to mathematics, [this a priori] is exclusively the possibility of
experiencing the concern of a mode of connection between two ideas and
to describe this concern phenomenologically’. In fact, the anteriority of
the problem should be understood solely at a purely conceptual level, since
the elements of a solution are often given first in practice and incorporated
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only after in a problem (which does not prevent, in a conceptual rearrangement, the problem in the end from preceding the solution). So:
Mathematical philosophy, as we conceive it, therefore consists not so
much in retrieving a logical problem of classical metaphysics within a
mathematical theory, than in grasping the structure of this theory globally
in order to identify the logical problem that happens to be both defined
and resolved by the very existence of this theory. (Lautman 1938b)
It is distressing to observe that the huge effort made by Lautman to grasp
globally a few mathematical theories of his time, and extract valuable ideas
from them, is no longer repeated in philosophy.11
In parallel to his strategy to apprehend the global structure of a theory
before defining its logical status, Lautman situates the place of logic at the
very interior of mathematics, as a discipline that does not precede mathematics, but should instead be situated at the same level as the other mathematical theories. He prefigures – as the later Peirce – our conception of
logic as it arises from model theory, in which a ‘logic’ is not only determined, but even defined (à la Lindström) by an adequate collection of structures. According to Lautman, ‘for logic to exist, a mathematics is necessary’,
and it is in the back-and-forth of mixed logical schemas with their effective
realizations that the strength of mathematical thinking lies.
The reciprocal enrichment between effective Mathematics and the
Dialectic (Lautman’s capitals) is reflected in an ascent and descent between
Lautmanian notions and ideas, on the one hand, and mixes, on the other.
In fact, if from the mixes one ascends, notions and ideas are ‘liberated’
which allows situating these mixes in a more ample dialectic; and if, conversely, from the notions one descends, new mixes are constructed to
clarify and incarnate the content of the ideas in play. One of the most
salient features of Lautman’s work is to have shown in detail how these
processes of ascent and descent must be inextricably linked in the philosophy of Mathematics in extenso, in the same way that they are in a Galois
correspondence in nuce.
5. PLATONISM
As Jean Cavaillès remarks in the first sentence of his report on Lautman’s
theses, his work constitutes ‘a new attempt to define the inherent reality
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of mathematical theories: the most recent works are used and the result
invokes Plato’ (1938a, 9). In effect, according Lautman, mathematical
reality – far from being reduced to the observation of natural objects, independent of the human eye – extends to encompass an entire hierarchical
lattice of ideal constructions, which cease to be arbitrary when they are
integrated generically in stable scientific networks. Mathematical constructions then become general reals, and mathematical reality is organised
around a complex imbrication of levels, situated below a schematic echelon formed by ideas:
We do not understand by Ideas the models whose mathematical entities
would merely be copies, but in the true Platonic sense of the term,
the structural schemas according to which the effective theories are
organized. (p. 199)
The invoked Platonism is, therefore, far from a simplistic idealism that he
denounces. Following Léon Robin, Julius Stenzel, and Oscar Becker,12
Lautman reads a dynamic Plato, far from pure contemplation, and much
closer to a full becoming of qualitative distinctions, which allows the
incarnation of the schematic tensions of ideas in effective mathematical
theories.
Even if Lautman’s first references to a rereading of Platonism can be
found in the conclusion of his main thesis (1938b), from his first articles he
establishes a dialogue with Plato, to underline a ‘participation of the sensible in the intelligible’ – which serves to support understanding the osmosis
of mathematical activity, its processes of transport and its creative mixes –
and to begin to explore, with striking examples in modern mathematics,
‘the Platonic distinction between the Same and the Other that is found in
the unity of Being’. This dialogue with Plato takes shape in New research on
the dialectical structure of mathematics, when he indicates that ‘the Ideas of
this Dialectic are certainly transcendent (in the usual sense) with respect
to mathematics’, Lautman is not thinking of any temporal, methodological
or even logical priority, he assumes no a priori, but he suggests simply the
possibility of the existence of a connection of the ‘why’ with the ‘how’,
where the question can come to be situated before the answer. What is
proper and specific to mathematical activity is then to transit between a
world of ideas of possible, free relations, and a world much richer in determinations, full of precisions and delimitations. The back-and-forth is inevitable, and it opens up new possibilities for a Platonism better adapted to the
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complexities of mathematical reality: a structural and dynamic Platonism,
not reified, immobile or eternal.
In this Platonic manifold – according to Lautman, closer to the true
Plato – mathematics and physics converge. Whether in the forms of
mathematical discovery that neighbor physical dissociations, or, in a conception of mathematical theories as middle terms between ideas and
experience:
Ideas / Theories
Theories
Experience
or, in ‘the hypothesis of a similar importance of the dissymmetrical
symmetry in the sensible universe and the anti-symmetric duality in the
mathematical world’, the structural base that, with powerful arguments,
renews support for the classical correlation:
Physics
/ Mathematics
Sensible world
Ideal world
Lautman always manages to compare forms of physical knowledge and
those of mathematical knowledge by projecting them onto situational structural
networks whose relative coordinates can be put in perfect correspondence,
even if the objects studied in each network are quite different. Thus, a
reticular commensurability of physics and mathematics acquires the right
of existence, and is clearly situated in opposition to its reduced (and too
proclaimed) grammatical incommensurability.13
6. CATEGORY THEORY
A first idea of the objectives and methods of the mathematical theory of
categories can be obtained when the ontological content of an already
quoted sentence of Heidegger’s is eliminated: ‘renewing the relation
between structures each time to a unity, so that from this unity the whole
history of an entity can be followed up to its concreteness’. In fact, the
dialectics of the one and the multiple, of the structural and the concrete,
reached one of its most fortunate expressions in category theory, because
an object defined through the universal properties in abstract categories –
one – becomes in turn a multiple throughout the plurality of concrete categories in which it is ‘incarnated’. The technical power of the theory resides
xxxiv
A L B E RT L AU T M A N A N D T H E C R E AT I V E D I A L E C T I C O F MODERN MATHEMATICS
in the permanent back-and-forth of functors, natural transformations and
adjunctions, which serve to constitute a very ductile network of exchanges
and blockages in mathematics. It is remarkable that Lautman’s conceptions
are able to take shape fully (that is, theorematically, with their corresponding
‘procession of precisions’) through the fundamental concepts of category
theory.14
Lautman often describes the conceptual resources implied in certain
techniques of category theory: functors in algebraic topology (the description of Poincaré’s and James Alexander’s duality theorems), functors
representable in manifolds (description of the ascent to a universal covering surface and the hierarchy of intermediate isomorphisms connected to
subgroups of the fundamental group), logical adjunctions (the description
of an inversion between Kurt Gödel’s completeness theorem and Herbrand’s
theorem), free allegories (the description of a ‘structure is like a first
drawing of the temporal form of sensible phenomena’). Basically, when
he maintains that it is necessary to accept ‘the legitimacy of a theory of
abstract structures, independent of the objects connected together by these
structures’, Lautman is very close to a mathematical theory oriented
towards the structural relations beyond objects: the mathematical theory of
categories.
The Lautmanian language, with its idiosyncratic ‘notions’, ‘ideas’ and
dialectical hierarchies, acquires much delimited technical support in category theory. The ‘notions’ may be defined through universal categorical
constructions (diagrams, limits, free objects), the ‘ideas’ through the elevation of classes of free objects to pairs of adjoint functors, the dialectical
hierarchies through scales of levels in the natural transformations. Thus,
for example, the Yoneda lemma15 shows technically the inevitable presence
of ideal constructions in any full consideration of mathematical reality –
one of Lautman’s strong positions – when the lemma proves that any small
category can be submerged in a category of functors, in which, in addition
to the representable functors that form a copy of the initial category, other
ideal functors (presheafs) also appear that complement the universe. It is a
matter of a forced apparition of the ideal at same moment as grasping the
real, a permanent and penetrating osmosis in all forms of mathematical
creativity. The irreducible back-and-forth between ideality and reality (very
much present elsewhere in Hilbert’s article ‘On the Infinite’, 1926, known
to Lautman) is one of the major strengths of modern mathematics.
Most of the structural schemas and schemas of genesis studied by
Lautman in his main thesis (1938b) can be explained and, above all,
xxxv
AL B E RT L AU T M A N A N D T H E C R E AT I V E D I A L E C T I C O F M O D ERN MATHEMATICS
extended, through the aide of category theory. For example, the study of
the duality of the local and the global is extended to a complex duality
of functorial localizations and global reintegrations (Freyd’s representation
theorem); the duality of the extrinsic point of view and of the intrinsic
point of view is extended to the internal logic of an elementary topos
(F. William Lawvere’s geometric logic); the logical schemas of Galois’s theory are extended to a general theory of residuality (Galois’s abstract connections, à la George Janelidze). When we see how, according to Lautman,
the affinities of logical structure allow different mathematical theories to
be brought together, because of the fact that they each provide an outline
of different solutions to one and the same dialectical problem; how one
can talk about the participation of mathematical theories distinct from a
common dialectic that governs them; how the dialectic is purely problematic, outline of the schemas; or how the indetermination of the dialectic at
the same time ensures its exteriority (See p. 126–130), it is impossible not
to situate these ideas in the environment of category theory. Whether in
the back-and-forth between abstract categories (‘common dialectic’) and
concrete categories (‘distinct mathematical theories’), in free objects (‘indetermination of the dialectic’) whose external applicability on the whole
mathematical spectrum is precisely the consequence of their indetermination, or in diagrams, sketches and limits that allow the grand schemas of
mathematical exchanges to be to outlined.
A lot of the apparent distance between modern mathematics, classical
philosophy and contemporary thought can be reduced. In this sense – one
of the senses of Lautman’s militant work, open at once to Plato and Hilbert,
to the ancient dialectic and to the advanced mathematics of his time – we
suggest in the following table some translations between Lautman’s
thought, category theory and the trimodal thought of universals according
to Avicenna:
xxxvi
Lautman
Categories
Avicenna
notions and ideas
universal constructions
(abstract categories)
universals
ante multitudinem
effective
mathematics
given
structures
universals
in multiplicitate
mixes
kinds of structures
(concrete categories)
universals
post multiplicitatem
A L B E RT L AU T M A N A N D T H E C R E AT I V E D I A L E C T I C O F MODERN MATHEMATICS
The universals ante multitudinem of Avicenna (mixed products in the
Neoplatonic understanding of Aristotle’s categories) are intrinsic concepts,
without referentials; the universals in multiplicitate are collections of
individuals, or are referentials; the universals post multiplicitatem are forms
conceived by the understanding beyond collections.16 Between Avicenna
and the mathematical theory of categories – simultaneously near and yet
so far – the work of Lautman is situated in a fascinating mixed terrain,
which serves as a bridge between some of the major ideas of philosophy
and some of the great achievements of contemporary mathematics. Full of
ardor and exigency,17 Lautman’s work opens up original perspectives for a
vigilant, engaged mathematical philosophy, by taking hold of both the real
activity of the discipline and the major problems – still alive despite their
alleged death – in the history of philosophy.
xxxvii
Preface to the 1977 Edition
by Jean Dieudonné
MEMBER OF THE ACADEMY OF SCIENCES, FRANCE
Contemporary philosophers who are interested in mathematics are in most
cases occupied with its origins, its relations with logic, or its problems with
foundations; quite a natural attitude for that matter, since these are the
questions that themselves call for philosophical reflection. Few are those
who seek to get an idea of the broad tendencies of the mathematics of their
time, and of what guides more or less consciously present mathematicians
in their work.
Albert Lautman, on the contrary, seems to have always been fascinated
by these questions. Like Jean Cavaillès, he made the effort to become initiated in the basic mathematical techniques, which enabled him to find
out about the latest research without risking being drowned in a flood of
abstract notions that are difficult to assimilate for a layman. Also, in contact with his comrades and friends Jacques Herbrand and Claude Chevalley
(two of the most original minds of the century), he had acquired of mathematics in the years 1920–1930 views far more extensive and precise than
had most mathematicians of his generation, often narrowly focused. I can
vouch for what concerns me personally.
It is these views that he presents in his two theses, and with 40 years of
distance, one can’t but be struck by their prophetic bearing. Because from
the title of these works, one finds, as if highlighted, the two key ideas–
forces that have dominated all subsequent developments: the concept of
mathematical structure, and the profound sense of the essential unity underlying the apparent multiplicity of the diverse mathematical disciplines.
xxxix
PR EFACE TO TH E 197 7 E D I T I O N
The word ‘structure’ is one of those that has been the most overused
during the last decades, but for mathematicians, it has acquired a very precise meaning. In 1935 this meaning had not yet been completely explicit,
but many mathematicians were quite aware of the reality that he recovers,
notably among those who were inspired by Hilbert’s ideas on the axiomatic
conception of mathematics. The essential point in this conception is that a
mathematical theory is concerned primarily with the relations between the
objects that it considers, rather than the nature of these objects. For example, in group theory, it is most often secondary to know that the elements
of the group are numbers, functions or points of a space. What is important
is to know whether the group is commutative, or finite, or simple, etc. This
view has so permeated the development of mathematics since 1940 that it
has become quite banal, but this was not yet the case at the time when
Lautman was writing, and he repeatedly insisted on it, as for example
when he stressed the fundamental identity of structure between the Hilbert
space, composed of functions, and the usual Euclidean space. Even more
remarkable is the long passage devoted to what is now called the notion of
universal covering (revêtement) of a manifold (at the time it was referred to
as a ‘universal manifold of covering (recouvrement)’). The ‘ascent towards
the absolute’ that he discerned, and in which he saw a general tendency,
took in effect, through the language of categories, a form applicable to all
parts of mathematics: it is the notion of ‘representable functor’ that today
plays a significant role both in the discovery and in the structuration of
a theory.
The theme of the unity of mathematics seems more central again in the
thought of Lautman. We know that, since antiquity, philosophers have
been pleased to highlight the opposition of points of view and of methods
in all domains of intellectual activity. Many mathematicians have long
been impressed by these antagonistic pairs that philosophers had taught
them to discern in their own science:
Finite versus Infinite
Discrete versus Continuous
Local versus Global
Algebra versus Analysis
Commutative versus Non commutative, etc.
A significant portion of Lautman’s theses are devoted to examination of
these oppositions, on which his position is similar to that of Hilbert, in
xl
P R E FA C E TO T HE 1 9 7 7 E DIT ION
whom one finds the strongly expressed conviction that they only happen
to be superficial appearances masking much more profound relationships.
The entire development of mathematics since 1940 has only confirmed
the soundness of this position. It has thus been well recognized that these
supposed oppositions are actually poles of tension within a same structure,
and that it is from these tensions that the most remarkable progress
follows.
As regards the ‘local–global’ antagonism, Lautman had known to appeal
to the work of E. Cartan, the value of which very few appreciated before
1935, and whose central place in all mathematics is now universally recognized. But this fertile polarity now extends far beyond its initial geometrico–topological framework. One of the grand axes of the work of Algebra
and Number Theory, since Hensel, Krull, Hasse and Chevalley, has been
to analyze the problems by ‘localizing’ them to begin with, the notion of
prime ideal (or of ‘valuation’) replacing the ‘points’ of geometry; while in
the opposite sense, the technique of sheafs and their cohomology, created
by J. Leray, provided adequate instruments in the most diverse theories to
‘globalize’ the local results and measure, in some way, the obstructions
to this globalization.
If this evolution was possible, it is because it overcame at the same time
an opposition that seemed much more radical (since it goes back to Greek
mathematics, and has caused much ink to flow over the centuries): that of
the ‘discrete’ and the ‘continuous’. The time has passed when the ideas of
approximation and completion seemed foreign to Algebra for which they
have become particularly valuable tools. And the extraordinary intuition
of A. Weil, foreseeing the possibility of structures recovered from Topology
(later discovered by A. Grothendieck) at the very heart of Number theory,
the science of the ‘discreet’ par excellence, has opened vast horizons.
A whole new science was created, Homological Algebra, which borrows
its methods from algebraic Topology to enrich itself, for example, group
theory or ‘abstract’ rings, and inversely allows Topology to be considered
as a peculiar application of an essentially ‘combinatorial’ theory: ‘simplicial’
Algebra.
The no less venerable opposition of the finite and the infinite has suffered the same fate. To give one striking example, one of the most important advances in finite group theory has been the discovery, by Chevalley,
of a method allowing a whole new family of finite simple groups to be
deduced from simple complex Lie groups (formerly called ‘continuous
groups’). To illustrate the interpenetration of the ‘commutative’ and the
xli
PR EFACE TO TH E 197 7 E D I T I O N
‘noncommutative’, Lautman again had recourse to the work of E. Cartan,
making Grassmann’s exterior algebra (noncommutative) the essential tool
in Differential Geometry and Pfaffian systems theory. He could also have
mentioned, from that era, Brauer’s group theory, which plays a central
role in the theory of commutative fields, despite having the noncommutative algebras constructed on these fields as its object. But today there are
many other examples of this type: in the theory of spherical functions (a
generalization of Laplace’s ‘spherical harmonics’, which essentially relates
to the theory of certain noncommutative Lie groups), the key result is
the fact (discovered by Gelfand) that a certain algebra of functions is commutative; while in the inverse sense, it was found that the theory of
commutative algebraic groups is closely related to the structure of a certain
noncommutative ring.
As for the old classification of mathematics into Arithmetic, Algebra,
Geometry and Analysis, it has become as out-of-date as the divisions of the
‘animal kingdom’ by the early naturalists into species grouped according to
fortuitous and superficial resemblances. Modern mathematical objects
appear as centers where surprising combinations of many diverse structures come to converge. A typical modern mathematical paper1 will unfold
as follows: The aim is to demonstrate that a certain group, occurring in
number theory (and, more precisely, in the arithmetic theory of roots of
unity) is finite. One begins by interpreting this group using homological
Algebra, which reduces the theorem to be proved to a result concerning
the cohomology of certain discrete subgroups of a Lie group; and finally
this last result is obtained by calling upon E. Cartan’s theory of symmetric
spaces and Hodge’s theory of harmonic forms. It started with ‘Arithmetic’,
and then passed via ‘Algebra’ to end up ultimately in ‘Geometry’ and
‘Analysis’! One could give many other analogous examples, showing
conclusively that the old conceptions can only be intolerable fetters in the
understanding of mathematics today.
This shows that Lautman had foreseen this extraordinary development
of mathematics, which fate prevented him from participating in. He had
filled it with enthusiasm, as much for the unparalleled harvest of new
theories and solutions to old problems,2 as for the eminently aesthetic
character that the central parts of this vast edifice now offer (to those who,
like Lautman, seek to understand them). It is hoped that the new edition
of his works gives rise among philosophers to young emulators, capable
like him to appreciate and interpret one of the most amazing monuments
of the human spirit.
xlii
Considerations on Mathematical Logic*
It is impossible to speak of mathematical logic without first of all mentioning certain recent work. There is a broad historical and critical exposé of
the question in the magnificant Treatise of Formal Logic by Professor
Jorgen Jürgensen (Jürgensen 1931). Moreover, the problems posed by the
philosophy of mathematics have been the subject of a special philosophy of
science congress held at Koenigsberg in 1930; the report of the congress
sessions has been published in Volume II of the new German review of
philosophy Erkenntnis;1 and finally Arnold Reymond, Professor at the
University of Lausanne, has had a book appear last year entitled: Les principes de la logique et la critique contemporaine (1933), which is the most
comprehensive and most precise of books on mathematical logic, the only
one that is also written in French.
It is known that the interest of mathematical logic is the reunification
of two independent series of research. Logicians along with Boole and
Schröder had wanted to share with logical deduction the benefits of the
algebraic calculus. They had reached a calculus of classes, propositions and
relations, and thus opened the ‘calculus ratiocinator’ that Leibniz had
required, a domain that was extending itself to understand the mathematics.
Independently of this work, the mathematicians were led to investigate
the logical foundations of mathematics whose certainty seemed shaken by
the discovery of the famous paradoxes of set theory. But set theory proved
* This text, easily datable from 1933, appears in the 10/18 edition of 1977 without
reference to an original publication, of which neither Fernando Zalamea nor I have
succeeded in finding a trace (Jacques Lautman).
1
CON S ID ERATIO N S O N M AT H E M AT I C A L L O G I C
so successful for the study of number theory that it was able to keep the
benefit of transfinite calculus while introducing numbers and sets in a
fairly rigorous way so that the contradictions, like the set of all sets, are
eliminated in advance. It then went on to try to deduce all of mathematics
from a small number of logical notions and primitive propositions. And, in
addition, as the paradoxes raised by Cantorian set theory were in nature
more grammatical than mathematical, it undertook to transcribe all mathematical arguments in a symbolic language, the idea of which also goes
back to Leibniz and to his ‘lingua characteristica’ project. This was the work
first of Frege, then of Peano and above all of Russell and Whitehead in
Principia Mathematica. Mathematics and logic were in this way conflated
in a general theory of deduction: logicism.
The philosophical problems posed by the existence of a theory of deduction are twofold. It is a matter of knowing, on the one hand, whether the
initial claims relative to the deduction of mathematics from logical notions
and primitive propositions have been justified, that is, that there has been
no appeal to a principle other than the primitive principles, which would
thereby be found to be necessary and sufficient. It is then a matter of
assessing the validity of the same theory, that is, to obtain the certainty
that it can be applied without ever encountering a contradiction. We will
see that independent of any preconceived metaphysics and for technical
reasons of calculation, the answer to these two questions is possible in
Russell’s theory only at the cost of asserting a certain reality of the outside
world. A metaphysical structure of the world is entailed by the demands of
the theory, and in such a way that the stages of the deduction are always
compatible with ordinary experience.
Let us examine first of all the logical construction of mathematics. The
frame of this construction is constituted by the famous theory of types.
Russell, having noted that the antinomies of analysis and set theory had
been made possible by the consideration of sets whose elements were
defined through the consideration of the set itself or through functions
that could take as arguments values defined by the totality of possible
argument values, wanted to make such definitions illegitimate. To do so,
he defined the hierarchy of individuals as type 0, properties of individuals
as type 1, properties of properties as type 2, and so on, in such a way that
elements of a class or the arguments of a function are always of the type
inferior to the class or function in question. The theory of types thus eliminates the antinomies like the set of all sets that contain itself as an element,
but it does not exclude the possibility of vicious circles within the same
2
C O N S I D E R AT I O N S O N M AT HE M AT IC A L LOGIC
type of functions or properties. When the attribution of a property to
an individual is based on the totality of properties of that individual, the
consideration of that totality is illegitimate. Take, for example, the theory
of cardinal numbers. This theory is based on primitive notions of ‘0’ and
the ‘recurrence of n to n + 1’. A whole number n + 1 is a number that has
all the recurrent properties of n starting from 0. However, the property of
being a whole number defined by recurrence is also among the properties
of the number n. The notion of whole number would therefore be illegitimately implicated in the elements of its definition (see Carnap 1931).
That’s why Russell was led to establish a new ramification at the interior of
the set of properties of the same type. The properties involved in the definition by recurrence will be of a certain order and the property of being
a cardinal number will be a property of the same type and higher order.
The notion of cardinal number was thereby saved, however a very large
number of theorems of the theory of real numbers, in which the consideration of all real numbers irrespective of order played a part, were rendered void. Their preservation was only possible through the introduction
of the ‘axiom of reducibility’ which asserts that there exists for all predicative functions of any order, a function equivalent to the same type and of
order 1. The introduction of this axiom is the recognition that mathematics
does not form a set of tautological propositions. It is indeed impossible to
indicate a procedure for constructing the function of order 1 equivalent
to a function of any order. Russell and Whitehead rely only on the realist
certainty that certain beings possess certain properties even though we
could not attribute these to them by only appealing to rigorous logical
operations. Ramsey, a disciple of Russell who unfortunately died in 1930,
considered for example the property of being the tallest man in the room.
The attribution of this property to a specific man requires consideration of
all the men in the room, and it may be that it is impossible to find the man
in question even though we are confident that there is a man who has this
property. The axiom of reducibility is the translation into symbolic language of sometimes impossible operations by the strict application of the
theory of ramified types, but only by the necessity with which we possess
an empirical intuition. This axiom of existence was not the only theory of
deduction in the Principia Mathematica. In the same way as we defined
whole numbers by the fact that they are different from each of their predecessors, we had been obliged by this to admit an axiom of infinity, posing the
existence of an infinity of objects. If the number of objects in the universe
were limited to 10 for example, 10 + 1, 10 + 2, etc., would be identical to
3
CON S ID ERATIO N S O N M AT H E M AT I C A L L O G I C
the null class and would therefore all be equal to one another, unlike the
general property of whole numbers. The notion of cardinal number is
inseparable from the real existence of things to count; it relies on the existence of classes. There exist for example dualities necessary to the definition
of a law of correspondence between these dualities, which is the cardinal
number 2. It is through this constant appeal to experience in the choice of
primary notions and in the introduction of different logical operations
that Russell and Whitehead are certain to have eliminated in advance the
contradictions and paradoxes. One doesn’t say that the King of France is
bald, not that he isn’t, but that he doesn’t exist. Suppose we wanted to
introduce the use of the article ‘the’ (in a sentence like: W. Scott is the
author of Waverley, for example), we will only introduce beings defined
by ‘the’ with all necessary precautions to be in agreement with experience.
If Waverley exists and if there is only one man who wrote Waverley, that
man is Sir Walter Scott. Whereas, and we shall see later with the example
of Hilbert’s theory, a deductive theory is generally obliged to give proofs
of consistency for it to be admitted in all rigor. Russell’s theory linking
mathematical existence to sensible reality, escapes into the intention of its
author at this requirement. It is nevertheless true that the admission of
the axiom of infinity, the axiom of choice (taken from set theory), the
axiom of reducibility, lack logical justification. Also the followers of Russell
are trying hard to eliminate the axioms of logicism. It also seems to them
that this point of support in experience that is at the base of the Principia
harms the purely symbolic character of mathematics. It seems nevertheless
that the theory of types is indispensable to all rational construction and
mathematical philosophy can retain as the fundamental contribution of
logicism this notion of an order common to the generation of mathematical functions and to the description of any logical properties.
There is one category of mathematician for whom the most skilful efforts
of rigorous deduction seem to be unreal exercises, it is those who, as a result
of Brouwer and Weyl, see in mathematical objects the product of an activity of the mind exercising itself freely and independently of the possibilities
of any logical transcription. The only law to which this activity is subject
also governs all operational activity whether manual or intellectual. It is
the law of only being possible in a finite and discontinuous number of
steps. The intuitionists thus prohibit the consideration of elements distinguished by certain properties within infinite sets. Classical logic, like logicism, admitted that the rule of the excluded middle could be applied to
the disjunction formed by a universal proposition and its contradictory
4
C O N S I D E R AT I O N S O N M AT HE M AT IC A L LOGIC
particular as in the following reasoning: either all elements of this set have
such-and-such a property, or there is one that doesn’t. The Brouwerians
assert that it makes no sense to deny a universal proposition as long as the
object which is the manifest contradiction of it has not been effectively constructed. In a very large number of cases, the mind would therefore not be
entitled to conclude from the absence of certain properties, the assertion of
the contrary property. The intuitionist is related by this to the phenomenologist disciples of Husserl, Heidegger and Oscar Becker (see Heyting 1931).
There exists for them a positivity of non-being that is not conceived as a
simple negation, but as the object of a sui generis intention of thought. The
concern to see that the reasoning never goes beyond these actually effected
operations leads the intuitionists to only attribute to mathematical notions
a provisional always revisable existence, at the mercy of the first particular
determination that will come to change the meaning of any outlined
edifice. The definition of mathematical entities is secured the moment in
which the mind stops in the series of operations undertaken. While the
mathematician has thereby perhaps gained in certainty, mathematics has
been diminished in scope.
It is nevertheless possible to reconstruct mathematics by avoiding the
antinomies while retaining Cantor’s results, but without relying on the
realism of the notion of class like the logicists. This is the result of the work
of Hilbert and his followers, principally von Neumann in Berlin and Herbrand in France, in the elaboration of the axiomatic method. Mathematics
is conceived in a set of signs devoid of any meaning and of which the
calculation is as follows: Take a certain number of letters, some of which
are always called constants, others variables, the others properties or functions, functions of functions, etc. Also take a certain number of signs
corresponding to the logical operations of disjunction and negation. Then
write certain sets of letters and signs that constitute the primitive axioms.
Take also a certain number of rules of passage that allow certain sets of
letters in the axioms to be replaced by others, and say that a proposition is
demonstrated in the system of axioms when a process exists that allows it
to be transformed by the use of rules of passage into an identity derived
from the axioms.2 The result is thus a symbolism somewhat similar to that
of Russell, but whose meaning is profoundly different. The characteristic
of such a mathematics is in effect to give the definitions in understanding
and no longer in extension to avoid the vicious circles in the definitions.
The whole number is no longer Russell’s inductive number, but the simple
property common to a set of signs, to be one of the arguments in the
5
CON S ID ERATIO N S O N M AT H E M AT I C A L L O G I C
axioms that define the use of the letter Z (first letter of the German word
Zahl). Several groups of axioms can be distinguished, each corresponding
to an extension of the domain defined by the set of preceding groups.
Analysis is thus reconstructed with the axioms of arithmetic, the set of
axioms defining the total induction, those that introduce the notion of
function, and those that define the expressions: all and some, that is, the
axioms of set theory. The strictly mathematical operations being each time
finite in number and bearing upon signs devoid of meaning, it is necessary
to involve other arguments than mathematical reasoning to guarantee the
consistency of the system of axioms and of their transfinite consequences.
The Hilbertians’ intention would also be to prove that the systems of
axioms introduced are at complete determination, that is, that it is possible
to recognize in a finite number of steps if a given proposition is or is not
demonstrable in the system of axioms. Unlike mathematical reasoning,
the proofs of consistency and research on general solvability (or complete
determination) are concerned therefore with a real object, namely the formalist mathematical theory. Hilbert gave the name of metamathematics to
this study of mathematical theories.
Up until Hilbert, there were four methods of proving consistency (see
Poirier 1932, 150). The first is to develop the theory and not to find contradictions, but this method does not provide the certainty that one will
never be found. The second is to see clearly via intuition the simplicity and
compatibility of the primitive axioms; this is a little the certainty of Russell
at the start of the Principia. The third is to find an empirical interpretation
that justifies the invention of such an entity, for example a physical interpretation: the existence of diagonals justifies that of the irrationals; a problem of mechanics has a solution if it corresponds to a physical phenomenon.
This assimilation of consistency with the construction carried out is furthermore the sole definition of compatibility admitted by the intuitionists.
The fourth is to reduce one system of axioms to another, which only postpones but does not solve the problem. Hilbert and his followers have in
truth invented a new method that I will try to characterize according to the
work of Herbrand (Herbrand 1930; 1931).3 Consider all the elementary
propositions of a theory. Attribute to each of them a logical value, that is,
that each has a sign T (true) or F (false) associated with it; then set the rules
of attribution of the signs T and F to the propositions formed with the signs
of disjunction and negation, then reduce all propositions of the theory to a
conjunctive or disjunctive canonical form. If this canonical form contains n
different elementary propositions, one can, after 2n tries, find out if the
6
C O N S I D E R AT I O N S O N M AT HE M AT IC A L LOGIC
canonical form has or doesn’t have the logical value T, whatever the logical
values given to elementary propositions. Suppose that variables or functions have not yet been introduced, there is still only the calculus of logical
propositions and it is proved that in this theory, we can determine whether
a given proposition is or is not demonstrable. There is therefore, in this
very simple case, consistency and complete determination (Entscheidung).
Let us now consider expressions containing variables. It is possible to
reduce these expressions to a canonical form by involving the notion of
domain. A domain is a set of n letters such that if certain of them denote
variables, others will serve by convention of ‘value’ the functions having as
arguments the values that replace the variables and so on until the exhaustion of all individuals and functions of the expression. For a proposition
containing a finite number of variables and functions, it is possible to consider all expressions obtained by substituting the variables with letters in
any way and the functions by their ‘value’. The logical value of each of the
expressions obtained can be calculated for all cases and if this logical value
is always T, the proposition is said to be true in the considered domain of
order n. In these circumstances, if it is possible to construct for all h a
domain of order such that all the hypotheses of the theory y are true, then
the theory is not contradictory. Herbrand died before he could apply his
conception of consistency to a theory broader than ordinary arithmetic.
Further research seems to show that it does not apply to analysis. We will
mention here in any case only the essential idea guiding Herbrand. It is
impossible to carry out all the calculations implied by a theory, because
obviously they are infinite in number, and the intuitionists are right to say
that by proceeding in this way there would never be certainty of not finding contradictions. But it is possible to replace, as concerns the study of
logical value, the consideration of an infinity of particular values with a
letter or a function ‘of choice’ so that the results obtained in the finite
domain of these values of choice have transfinitely valid consequences for
all particular mathematical entities whose values are symbolized by this
value of choice. This is to take up in another form Hilbert’s former logical
function which Hilbert himself gave a vivid interpretation with the example of Aristide: if Aristide the unbribable has also let himself be bribed, it is
certain that a fortiori all men will be bribable (Hilbert 1923; 1936, 183
[1996, 1141]). The conclusions valid for an individual of choice are also
valid for an infinite class of individuals of which he is the representative.
Between the intuitionists’ demands of construction and the pure introduction of notions by axioms, the Hilbertians knew to interpose an intermediate
7
CON S ID ERATIO N S O N M AT H E M AT I C A L L O G I C
schematic, that of individuals and domains considered not so much for
themselves as for the infinite consequences that allow finite calculations to
operate through them. There remains a strong analogy between the domains
of mathematics and Russell’s hierarchy of types and orders. We are in the
presence, in each case, of a structure whose elements are neither entirely
arbitrary, nor really constructed, but composed as a mixed form that
derives its fertility from its dual nature.
None of these three theories, no more that of the logicists than that of
the intuitionists or the formalists, has yet been presented in a manner
deemed satisfactory by their authors themselves. But for the philosopher,
more than anyone else, attempted and roughly outlined theories are just
as fruitful as definitive results.
8
Mathematics and Reality
PRESENTATION TO THE INTERNATIONAL CONGRESS
ON SCIENTIFIC PHILOSOPHY, PARIS 1935*
The logicians of the Vienna Circle claim that the formal study of scientific
language should be the sole object of the philosophy of the sciences. This
is a thesis that is difficult to accept for those philosophers who consider
establishing a coherent theory of the relations of logic and the real as their
essential task. There is a physical real and the miracle that is to be explained
is that there is a need of the most developed mathematical theories to
interpret it. There is also a mathematical real and it is similarly an object of
admiration to see domains resisting exploration until they can be tackled
with new methods. It is in this way that analysis was introduced into
arithmetic or topology into the theory of functions. A philosophy of the
sciences that isn’t entirely concerned with the study of this solidarity
between domains of reality and methods of investigation would be singularly devoid of interest. The philosopher is not in effect a mathematician by
nature. If the logico–mathematical rigor can seduce him, it is certainly not
because it allows a system of tautological propositions to be established,
but because it sheds excellent light on the connection between the rules
and their domain. It even brings up the curious fact that what is for the
logicist an obstacle to be eliminated becomes for the philosopher the highest
* Lautman 1936.
9
M ATH EMATICS AN D R E A L I T Y
object of his interest. These are all the ‘material’ or ‘realist’ implications
that logicism is obligated to admit: they are Russell’s well-known axioms,
the axiom of infinity and the axiom of reducibility. It is, particularly in
Wittgenstein, the assertion that to all true propositions there corresponds
an event in the world, which entails a whole procession of restrictions and
precautions for logic. In particular all propositions relating to the set of
propositions, all logical syntax, as defined by Carnap, is impossible since
this would require being able to consider correlatively the world as a totality, which is illegitimate.
The logicists of the Vienna Circle always assert their full agreement with
Hilbert’s school. Nothing is however more debatable. The logicism school,
that follows from Russell, tries hard to find the atomic constituents of every
mathematical proposition. The operations of arithmetic are defined from
the primary notions of element and class, and the concepts of analysis
are defined by extension from arithmetic. The notion of number therefore
plays a crucial role here, and this role is augmented again by the arithmetization of logic that follows from the work of Gödel and Carnap. This
primacy of the notion of number seems however not to be confirmed by
the development of modern mathematics. Poincaré had already indicated
in respect of the theories of dimension that the arithmetization of mathematics does not always correspond to the true nature of things. Hermann
Weyl, in the introduction to his book The Theory of Groups and Quantum
Mechanics (1928), established a distinction that seems fundamental to us
and that will be taken into account by all future philosophy of mathematics. He distinguishes two currents in mathematics. One from India and the
Arabs highlights the notion of number and leads to the theory of functions
of a complex variable. The other is the Greek point of view which claims
that each domain carries with it a characteristic system of numbers. It is the
primacy of the geometric idea of domain over that of whole number.
The axiomatic of Hilbert and his students, far from claiming to reduce
all of mathematics to being only a promotion of arithmetic, aims on the
contrary to identify for each domain studied a system of axioms such that
from the reunion of the conditions implicated by the axioms arises both a
domain and valid operations in this domain. It is in this way that group
theory, the theory of ideals, and of systems of hypercomplex numbers, etc.,
are constituted axiomatically in modern algebra.
The consideration of a purely formal mathematics must therefore give
way to the dualism of a topological structure and of the functional properties
10
M AT HE M AT IC S A ND R E A LIT Y
in relation to that structure. The formalist presentation of similarly axiomatized theories is only a question of greater rigor. The object studied is not
the set of propositions derived from axioms, but the organized, structured,
complete entities, having an anatomy and physiology of their own. As
an example, the Hilbert space defined by axioms that give it a structure
appropriate to the resolution of integral equations. The point of view that
prevails here is that of the synthesis of necessary conditions and not that of
the analysis of primitive notions.
This same synthesis of the domain and the operation is found in physics
from a slightly different point of view. Carnap sometimes seems to consider the relation between mathematics and physics as that between form
and matter. Mathematics would provide the coordinate system in which
the physical data is inscribed. This conception seems hardly defendable
since modern physics, far from maintaining the distinction between geometric form and physical matter, unites on the contrary spatio–temporal
data and material data in the common framework of a mode of synthetic
representation of phenomena; whether through the tensorial representation
of the theory of relativity or by the Hamiltonian equations of mechanics.
There is thus for each system a simultaneous and reciprocal determination
of the container and the contents. This is once more a determination
unique to each domain at the interior of which a distinction between
matter and form no longer subsists. Carnap seems, it is true, to also have
another theory of the relation between mathematics and physics, much
more in accordance with his logicism tendency. He regards physics, not as
the science of real facts, but as a language in which experimentally verifiable statements are expressed. This language is subject to the rules of
syntax, of a mathematical nature, when uniformly valid throughout their
domain of definition, of a physical nature, when their determination varies
with experience. There is here once more that assertion of the universality
of mathematical laws as opposed to the variation of physical data. It seems
to us that this conception does not account for the fact that this variation
of physical data only makes sense in relation to the prior choice of magnitudes that are likely to vary, and this choice is physical. Carnap’s example
in his book The Logical Syntax of Language (1934, 131 [1937, 178]) is characteristic. When the components of the metrical fundamental tensor are
constants, he argues, it is a mathematical law; when they vary, they obey
a law of physics. The real philosophical problem is rather to know how a
differential geometry can become a theory of gravitation. This accord
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M ATH EMATICS AN D R E A L I T Y
between geometry and physics is proof of the intelligibility of the universe.
It results from the clarification by the mind of a way of structuring the
universe in profound harmony with the nature of this universe. It is conceivable that this penetration of the real by human intelligence has no
meaning for certain extreme formalists. They would in effect rather see in
the pretensions of the mind to know nature an approach drawn from
the studies of Levy-Bruhl (1931). Understanding for them would be a
mystical belief analogous to the participation of the subject in the object in
the primitive soul. The term participation has nevertheless in philosophy
another more noble origin, and Brunschvicg has rightly denounced the
confusion of the two meanings. The participation of the sensible in the
intelligible in Plato permits the identification, behind the changing appearances, of the intelligible relations of ideas. If the first contacts with the
sensible are only sensations and emotions, the constitution of mathematical physics gives us access to the real through the knowledge of the structure with which it is endowed. It is even impossible to talk about the real
independently of the modes of thought by which it can be apprehended,
and far from denigrating mathematics to being only a language indifferent
to the reality that it would disparage, the philosopher is commited to it as
though in an attitude of meditation in which the secrets of nature are
bound to appear to him. There is therefore no reason to maintain the
distinction made by the Vienna Circle between rational knowledge and
intuitive experience, between Erkennen and Erleben. In wanting to suppress the connections between thought and reality, as in refusing to give to
science the value of a spiritual experience, the risk is to have only a shadow
of science, and to push the mind in search of the real back towards the
violent attitudes in which reason has no part. This is a resignation that the
philosophy of science must not accept.
12
International Congress of the Philosophy
of Science: Paris, 15–23 September 1935*
The eighth International Congress of Philosophy held in Prague in September
1934 was preceded by a sort of ‘small Congress’ of the Philosophy of Science in which the Vienna Circle in particular took part, and whose work
Cavaillès summed up in an article that was presented here in January
1935. With the intention of marking the autonomy of the philosophy of
science with respect to general philosophy, this ‘Preparatory Conference’
had agreed to hold regular Congresses of the philosophy of science, and
a committee, including Carnap, Frank, Jörgensen, Lukasiewicz, Morris,
Neurath, Reichenbach, Schlick and Rougier, were responsible for preparing the first of these Congresses whose meetings were held in Paris from 15
to 23 September 1935. Despite some resistance, Rougier, who took almost
all the organizational work upon himself alone, was concerned that all
trends in contemporary philosophy of science would be presented and discussed, also it was agreed that the papers would focus on general problems
over technical questions being in principle reserved for future Congresses.
The popularity of the neo-positivist Vienna Circle in Central Europe
and America, and above all the influence of the work and personality of
Carnap naturally lead the speakers to be divided into two classes: there
were those who placed themselves on the terrain of the Vienna Circle, and
the others. The first studied the same problems and spoke the same language, their adversaries presented isolated theses in relation to personal
conceptions of science or philosophy, less likely to be condensed into formulas and erected into a common doctrine.
* Lautman 1936b.
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INTERNATIONAL CONGRESS OF THE PHILOSOPHY OF SCIENCE
The presentation by Carnap was concerned with the relations between
science and philosophy as understood by the Vienna Circle, the most complete presentation of which lies in the important work of Carnap: The
Logical Syntax of Language (1934). Following Wittgenstein, Carnap considers an experimental science not as the study of a certain domain of reality,
but as a coherent set of propositions that involve certain words and certain
attributes corresponding to objects of experience and their observable
properties. These are the statements of physics, psychology, sociology, etc.
These statements must always correspond to a determined experience, the
Viennese denote them by the term ‘protocols’ to give them the character
of simple accounts of experiences. The protocols form the only synthetic
propositions of science, and neopositivism is essentially an empiricism. To
obtain from these ‘protocol–statements’ other statements, it is necessary to
submit the protocol to a purely formal ‘calculus’ of logic and mathematics.
A logicism is therefore going to be grafted onto the empiricism, and
thus neopositivism realises the synthesis of Mach’s phenomenalism and
Russell’s logicism. The translation of experimental statements into a formalized scientific language is thus the essential condition for scientific
reasoning. The scientific language understands then, regardless of the science in question, two kinds of signs: descriptive signs corresponding to the
objects and empirical properties, and formal signs borrowed from logic
and mathematics. The only problems that arise in such a formalism are
problems relative to these signs, independent of the meaning of the signs:
what are the rules that allow the recognition that an assemblage of signs
constitutes a proposition of the science studied? (This is the problem of
formal determinations, Formbestimmungen). What are the rules that allow
the deduction, from certain admissible premises, to other propositions?
(This is the problem of formal transformations, Umformungbestimmungen).
All of these rules constitute what Carnap calls, by analogy with the grammar that is the study of the syntax of ordinary language, the syntax of
scientific language. There is up to this point nothing that is characteristic
of the Viennese logicism, since these two syntactic problems have already
been quite clearly formulated by Herbrand in the preface to his thesis on
proof theory (Herbrand 1930).
What characterizes the logicist neopositivism of Wittgenstein and Carnap,
on the contrary, is the reduction of philosophy to the syntactic study of
scientific statements. The role of philosophy is thus a role of clarification
of the propositions involved in what is generally called the theory of
knowledge. Some of these propositions are concerned with questions that
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INTERNATIONAL CONGRESS OF THE PHILOSOPHY OF SCIENCE
fall within the sciences proper, as, for example, all the propositions relative
to space and time. There are also other propositions that are concerned
with the logical relations between concepts or scientific propositions. The
role of the ‘logic of science’ is to submit these propositions to the critique
from the syntactic point of view. Some will be amenable to a correct formulation, others, which might have a meaning in common language,
cannot be correctly formulated in scientific language in the least and will
thus be eliminated as concerned with pseudo-problems. Rougier has, in
this direction, shown how many philosophical problems of Aristotelianism
had been made possible by the logical scandal of the dual meaning of the
verb to be in Greek, which serves at the same time to link the attribute to
the subject and to assert the substantial existence of this same subject.
Cavaillès has indicated in his article the principal problem that Carnap
tackled in his book: can the rules of syntax be formulated in a formal
language governed by this syntax? Carnap resolved the question in the
affirmative for a simple enough language no. 1, only employing the signs
of logical quantification (‘there is an x such that’ . . . and ‘for all x’) in
the case of finite collections of objects. The rules of syntax are intended
to establish the conditions under which properties can be attributed to a
proposition that comes from the possibilities of its admission in the deductive system being studied, such as, for example, the following properties: to
be demonstrable, to be refutable, to be compatible (meaning several propositions), to be complete (meaning a system of axioms). Carnap shows how
these properties, which result from logical connections existing between
all the propositions of a system, can still be expressed formally as the properties of one or several propositions taken in isolation, and he can do so
through a process of arithmetization due to Gödel. By making the numbers
selected in a certain way correspond to all the signs involved in a proposition, a characteristic number of all the propositions can be identified such
that the syntactic properties of the proposition are related to arithmetic
properties of this number. The determination of the rules of syntax of
language no. 1 is thus reduced to the determination of the arithmetic statements expressible integrally by means of the signs of this same language.
Carnap also defines a language no. 2 in which he introduces the operators
(for all x) where the general variable can take an infinity of values. In this
language no. 2, Carnap appeals to a much broader mode of consecution of
the propositions than proof in a finite number of steps, namely, to sequential determinations (Folgebestimmungen) capable of including an infinite
number of steps, and that can thus give correct definitions in this system of
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INTERNATIONAL CONGRESS OF THE PHILOSOPHY OF SCIENCE
syntactic properties analogous to those whose formal conditions of attribution are set moreover in the simpler system of language no. 1.
It is not certain that all the notions introduced by Carnap are already
formulated in his eyes in a definitive way. The syntactic study of scientific
language is still in its infancy and Carnap’s Logical Syntax (1934) is presented instead in the form of logical tests rather than a dogmatic treatise.
Carnap was concerned, indeed, in a remarkable effort of intellectual sincerity, to mark the points where he is in disagreement with the other currents of contemporary logicism. For Wittgenstein, there cannot be syntactic
studies, since the properties of propositions, like their compatibility, for
example, appear intuitively in the examination of the propositions themselves, and cannot be formulated. This prohibition of all reflexive consideration of the statements of science comes from the author of the Tractatus
Logico-Philosophicus (Wittgenstein 1922), from his realist attitude. Wittgenstein
considered, in effect, that all propositions must correspond to a real situation in the world of facts, so that a proposition true for all sentences of the
language, as would be a rule of syntax, would have the sense of a proposition relative to the totality of the universe. But the consideration of this
totality is illegitimate, because it cannot be given in the experience of
any situation. That is why the validity of scientific language cannot be discussed; this validity is manifested in the act by which we feel that our
words correspond to things. The Vienna Circle has abandoned the realistic
point of view of Russell and Wittgenstein, so that the restrictions that are
going to be imposed on logicism no longer arise from consideration of
real, but solely from the necessities of the formal calculus. The most
famous restriction in this sense is that which was discovered by Gödel.
Gödel established that we can never demonstrate, in a formal theory containing arithmetic, the consistency of this theory. He indicates, in effect, for
all formalisms containing arithmetic, a process that leads to the effective
construction of unprovable propositions, that is, propositions that it is
equally contradictory to suppose either their truth or the truth of their
negation.
Now, the following proposition: ‘The theory studied is not contradictory’
is justifiably an unprovable proposition in such a formalism. There is a limit
here to the symbolism that arises from symbolism itself, just as in quantum
mechanics, Heisenberg’s uncertainty relations can be proved from the formal properties of mathematical operators corresponding to the physical
magnitudes studied.
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INTERNATIONAL CONGRESS OF THE PHILOSOPHY OF SCIENCE
Carnap’s syntax is also different from Hilbert’s metamathematics,
although it deals with the same problems. For the Hilbertians, metamathematics cannot be part of the formalized mathematics, and this is explained
by Herbrand as follows: a metamathematical reasoning is necessarily discursive and relies by the same token upon a recurrence in the finite whose
validity comes from being able, as Descartes claimed, to make a revision of
reasoning so rapid that it becomes intuitive (Herbrand 1930). This metamathematical recurrence is thus the necessary condition for the proof of
the consistency of reasoning by ordinary recurrence in arithmetic. But by
the same token it belongs to another language, of a higher type, as defined
by Russell’s hierarchy of types. There were at least two Hilbertian mathematicians at the Congress, Bernays and Chevalley, but it seems they were
not concerned to defend the point of view of mathematical formalism in
front of the logicians of the Vienna Circle. Bernays, who, with Hilbert, was
the author of Grundlagen der Mathematik (Hilbert and Bernays 1934; 1939),
preferred, to the delight of philosophers and the surprise of logicists, to
show that there were things in metaphysics other than the famous pseudoproblems, and Chevalley, who resumed Hilbertian class field theory from
new foundations in algebra, tried to rediscover in mathematical thinking
the effort of the human person in order to insert everything that had ceased
to be about life and real needs into the automatisms.
Carnap’s opposition to logic is only manifested on the technical terrain
of logical syntax with the Polish metalogicians and Tarski. Tarski seems to
be more concerned than Carnap with effective proofs in metamathematics.
Carnap does not reason, in effect, on the true axioms of mathematically
determined theories, but on schematic models of possible systems of axioms. He can thus define a predicate as the ‘provable’ predicate and attribute
it to a given proposition, which assures a logical sense to the following
proposition: ‘Such a proposition is provable’, but there is still here no outline, however vague, of the effective proof of the proposition in question.
It can therefore only be said that the fate of Mathematics as engaged in
syntactic research is detached from real mathematical facts. Tarski seems to
follow the lead of Herbrand in trying to define metamathematical or syntactical notions in a manner that is in conjunction with effective proofs of
consistency, of the independence of notions with respect to each other, of
compatibility, etc. Abandoning the pure point of view of comprehension
which is that of Carnap, he reintroduced into metamathematics the consideration in extension of domains of individuals whose construction is
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INTERNATIONAL CONGRESS OF THE PHILOSOPHY OF SCIENCE
necessary in order to study the propositions. Let us, for example, define the
property for a system of axioms to be ‘complete’ (vollständig), that is, to
be such that a new axiom cannot be added to it without introducing contradictions in the consequences. Tarski has shown, in his Prague lecture
(Tarski 1935), that any definition is purely nominal if it doesn’t tie the
research of the ‘Vollständigkeit’ [completeness] to the consideration of the
‘interpretations’ of this system of axioms, that is, of the classes of individuals likely to present among themselves these relations implicitly defined
by the axioms. A system of axioms being said to be ‘monotransformable’ if
all its interpretations are isomorphic, Tarski proves that any system of
monotransformable axioms is ‘complete’ as defined by ‘Vollständigkeit’. The
converse of this theorem is yet to be proved, but the example helps to
understand the thought of Tarski. He seeks to attribute the predicate
‘complete’ to a system of axioms only if certain mathematical relations can
actually be found between the individuals of the domains attached to this
system of axioms. This notion is therefore not part of mathematics, because
it is of a type superior to the propositions that play a part in the system
studied. The proposition: ‘this system of axioms is complete’, or, what
amounts to the same, ‘this x is complete’, in effect only makes sense in a
‘metalogic’ in which the variable subject x, as well as the properties likely
to be attached to it, are of a type superior to variables and properties that
are related to the individuals of the domains attached to the propositions
of the system studied. Tarski not only makes Hilbertian notions of metamathematics enter into his metalogic, he tried to make a ‘semantic’ of it, or a
general theory of correspondence between signs and things signified, and
began the study of notions such as that of truth or of definition that question the very essence of formalism and its philosophical value. In an orthodox formalism, the object of the calculus is the graphic sign independent of
any reference to a reality designated by the sign. To assert the truth of
a proposition signifies uniquely that the proposition is provable in the
formalism. If one adopts then, like Wittgenstein and Carnap, a tautological
conception of logico-mathematical formalism, ‘true’ is conflated with
‘analytic’ and ‘false’ with ‘contradictory’ (the problem of the verification or
refutation of protocol statements by experience is not posed within the
formalism itself). Tarski, on the contrary, restores to formalism its nature
of language serving to express a reality and tries to give in this formalism a
definition of truth that ensures the correspondence between the results of
the calculus and the real. Consider a proposition P obtained in the formal
language. This proposition is of a certain type determined by the type of
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I N T E R N AT I O N A L C O N G R E S S O F T H E P HILOSOP HY OF SC IE NC E
variables involved in it. As previously, let x be a new variable of a higher
kind introduced in metalogic to designate a proposition of the formal calculus. In these conditions, one has the right to write ‘x is true if P is true’.
This is tantamount to saying, as Carnap notes with some irony that doesn’t
at all exclude the sympathy, that ‘it is true that snow is white’ if we already
have this proposition: ‘snow is white’. Tarski’s semantics is no less a very
serious effort to justify the name of the science of significations and has the
great merit of showing that within the formalism there exists, thanks to
Russell’s wonderful theory of types, a means of exiting the pure calculus
and regaining contact with physics.
This necessity of adapting the formalism to the physics of physicists,
Carnap cannot escape from, and his book reveals in many places how he
undertook to solve the question of the relation between logic and reality.
The problem is posed from the initial definition of descriptive signs when
Carnap says simply that they correspond to the objects and to the properties involved in the protocols. This definition seems to imply that Carnap
conceives a protocol statement as necessarily attributing a property to an
object. Gonseth has rightly noted that the task of logic would be greatly
facilitated if it found itself facing a set constituted by judgments of attribution or of existence that would require nothing further than to be codified,
whereas in fact the least description of experience implies that the mind
has to impose on things the order of a certain relation. Carnap tries nevertheless to avoid the objection by giving a purely formal criteria to the distinction of signs into logical signs and descriptive signs: let, for example, a
mathematical sign play a part in a physical theory like the components
g 0 v of the metrical fundamental tensor of the theory of relativity. When
the value of these components is determined by a general law, as in homogeneous spaces of constant curvature, they are mathematical signs. When
these components vary in a non-homogeneous space with the distribution
of matter in this space, they are physical signs.
The basic problem that arises in full amplitude with this definition of
descriptive signs is that of the introduction of new signs within a formal
theory. It is a problem of logic that is encountered both in mathematics
and physics and that can receive two solutions: either the new signs can
be defined by an effective combination of previously established signs, as
in any reconstruction of mathematical logic analogous to the Principia
Mathematica (1910) of Russell and Whitehead; or the sign can be implicitly
defined by a new axiom, as in Hilbert’s axiomatic. As concerns numbers,
the problem was handled in masterly fashion in front of the Congress
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INTERNATIONAL CONGRESS OF THE PHILOSOPHY OF SCIENCE
by Padoa. Padoa, taking up the problem of the introduction into mathematics of negative numbers, fractions, and real and imaginary numbers,
showed that these categories of numbers cannot be considered as extensions of the notion of whole number. Let us, for example, define the fractional numbers by the sign A/B. If the exact division of A by B is possible,
the sign B only defines a whole number C that is already known, if it is not
possible, the sign A/B only defines the existence of a fractional number if
we admit in advance that only whole numbers exist. Padoa, recalling his
work on the axiomatic definition of numbers, argued that it is necessary to
abandon the logicist point of view, which only considers the precondition
of the set of whole numbers, and presupposes in advance the existence of
a ‘maximal’ set within which the different categories of numbers are
defined by the different groups of axioms.
The opposition of methods is no less strong in physics. The Vienna Circle
originally only admitted signs, in fact descriptive signs, connectable to an
object of experience or definable from the same signs. This is the attitude
of Carnap in The Logical Structure of the World (1928). This integral positivism can be generalized into a system of philosophy exactly like Comtism,
and it is physicalism in its primitive rigor. Under the influence of Karl
Popper (1935), Carnap admitted that the laws of physics were not protocol
statements, that there are, in effect, notions that play a part which are
neither more nor less relatable to any experience. Thus, for example, the
vector of an electric field or a magnetic field in Maxwell’s theory should be
considered as defined implicitly by Maxwell’s equations and is not an
object of experience. It is mostly English-speaking philosophers who are
attached to the Congress to define the meaning of the notions involved
in the laws of physics independently of all experience. The problem was
furthermore treated differently by Benjamin (Chicago) and Braithwaite
(Cambridge).
Benjamin defined with great discernment the range of the two methods.
In the so-called method of construction, the aim is to deduce the unknown
(to which corresponds what Benjamin calls ‘suppositional’ symbols) solely
from known elements, as if it was implicated by them. The hypothetical (or
hypothetico–deductive) method on the contrary gives new signs axiomatically and tries to make with them the antecedent relations of implication
whose consequents would be experimentally verifiable. The use of both
methods is absolutely necessary and Benjamin reintroduced certain indispensible metaphysical hypotheses in the philosophy of science to legitimize
the hypothetical method, as, for example, the hypothesis that the known
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I N T E R N AT I O N A L C O N G R E S S O F T H E P HILOSOP HY OF SC IE NC E
and the unknown are part of a same nature. Braithwaite, trying to define
the meaning of a word, like the word electron, also admits, under Ramsey’s
influence, the impossibility of defining the new terms of physics by means
of the terms already known. Such an attitude would, in effect, render the
passage from an old theory to a new theory very difficult, which is most
often anything but a simple generalization of the former theory. These are
ideas that have been presented in France by Bachelard in his latest book,
The New Scientific Spirit (1934). Bachelard spoke out against the easy conception that claims to see in the new mechanics a generalization of the old
mechanics, although his discovery corresponds to a veritable ‘mutation’ of
the scientific mind. Braithwaite, like Benjamin, is also not satisfied to ask
of the axioms that introduce non protocol signs to permit the deduction of
experimental consequences, he gives these abstract symbols the meaning of
an ideal reality analogous to the reality of a fairy tale. There is here an attitude that is not without analogy with Husserl’s phenomenology, in which
the real is characterized by a disposition of the mind to accept it as such.
These metaphysical attitudes that are found in certain theorists of
knowledge seem to be necessary to prevent the philosophy of science from
ending up as a radical nominalism toward which the Vienna Circle tends,
as did the scholasticism of Occam in the past. The word scholasticism was
mentioned by Enriques with mildly critical intent; the memory of Occam
had been recalled with the very clear intention of being linked to the
Vienna Circle by Morris, of Chicago, who had already contributed to the
Congress in Prague to establish by his general theory of signs (semiotic) a
rapprochement between American pragmatism and the logicist nominalism of Vienna. When one considers in effect that a statement is meaningful
only in the language in which the signs involved in the statement are
defined, even though one admits that these statements can be rendered
false by a specific experience translated into another statement of the language, but one does not admit that a correctly verified statement provides
any knowledge whatsoever of the reel. One adopts the radical conventionalism of Le Roy against which Poincaré spoke out so strongly in the final
pages of The Value of Science (1905) and which seems nevertheless to have
been taken up by Ajdukiewicz. A philosophical development was necessary and it was Schlick who provided one in a presentation sent to the
Congress which he unfortunately could not attend. Schlick established a
crucial distinction between a physical law and the statement of the law.
Such a statement has meaning only in a particular language, it can be true
in one system and false in another. But something subsists beyond the
21
INTERNATIONAL CONGRESS OF THE PHILOSOPHY OF SCIENCE
diversity of expression; it is the truth of the law itself. This truth seems to
lie for Schlick in the invariance of the relation that expresses the law with
respect to its different possible methods of verification. It is seized by an act
of intellectual intuition beyond the discourse, in a moment of contact
between the mind and reality.
Similar consequences were already discernible in Schlick’s article on the
foundations of knowledge published in the journal Erkenntnis in 1934
(Schlick 1934). There Schlick established an equally crucial distinction
between the protocol statements that describe an experiment in a scientific
language, and the expected intuitive observation of facts, at the moment
when the experience that they constitute is witnessed. The joy in which
consciousness is then enfolded is the very guarantee for consciousness of
contact with reality. But these moments of contact imply a difficult effort
of intellectual tension, the mind falls back as soon as it has raised itself up
to the level of things and must then follow its course again into pure logic
to further recognize the real. The definitive ban that the Viennese logicians
had believed to pronounce for all reference to an unthinkable world to
which the statements of science would apply is in any case removed, and
Schlick finds again, in the activity of the intelligence, a mode of intuitive
knowledge in which the statements assert their own truth value. Schlick
distances himself, perhaps intentionally, from this school founded by him,
but surely draws closer to the position of Brunschvicg or Enriques.
Enriques has already had the opportunity to introduce to the French
philosophical public his conception of the philosophy of science, in particular at the time of the discussion that was held on his ideas at the Societé
française de Philosophie (Enriques 1934). In his communication to the
Congress he showed how the history of science was an instrument at least
as necessary for the study of scientific truth as logicist formalism. It is
known that Enriques rejects the purely phenomenalist attitude of Mach
and stresses the importance of a priori rational demands in the progress
of science. The true object of the philosophy of science lies for him much
less in the study of the constitution of the statements of science than in
the fertile role of hypotheses about the nature or the simplicity of things.
It is in this regard regretable that a discussion about phenomenalism in
contemporary physics had not been established before the Congress. It is
known, in effect, that it is in specific reference to the ideas of Mach that
Heisenberg, renouncing any representative hypothesis in atomic physics,
oriented quantum mechanics in the direction of a calculation based solely
on measurable magnitudes. The philosophical problems posed by theoretical
22
I N T E R N AT I O N A L C O N G R E S S O F T H E P HILOSOP HY OF SC IE NC E
physics were only discussed at the Congress in their relation with the calculation of probabilities, and this thanks to the presentations by Reichenbach.
Cavaillès has already given detailed guidance, in his article last year
(Cavaillès 1935), on the junction that Reichenbach has made in his masterful treaty on probabilities (Reichenbach 1935) between the calculation
of probabilities and logic with an infinity of values. Cavaillès pointed out
that, in addition to classical logic or logic of two values (true and false),
three valued logics, simultaneously developed by Post and Lukasiewicz,
already existed (by adding an intermediate value corresponding to the
probable), but they don’t give the certainty of finding the mathematical
rules of total and compound probabilities. Reichenbach defines the probability of a proposition asserting a fact as a frequency attached to this probability when it is considered within a series of propositions asserting or
denying that fact. A logical value between 0 and 1 is attributed to this
proposition; and the determination of the logical value of the sum, product, equivalence and implication of propositions is made by the same rules
as the calculation of probabilities.
Reichenbach was able to establish on this basis a purely logical theory of
induction, in the ordinary sense of the anticipation of a statement in physics. The hypothesis of induction comes back in effect to a wager, in a series
of frequencies, on the limit towards which it is believed these frequencies
tend. A proposition relative to this measure of the probability of the limit
is neither true nor false, it is equally a probable proposition, whose probability is measured according to the rules of combination of elementary
probabilities. Each observation adding a new term to the series of frequencies comes to change and correct the successive wagers, that is, the hypotheses made about the limit of the sequence, and whether there really is a
limit. It is necessary that from a certain point these hypotheses are verified.
Induction thus refers to probabilities of probabilities, which, as Reichenbach demonstrates, converge much faster than the first order probabilities
in the case of the existence of a limit of these probabilities. From the
moment we admit that it is better to know rather than not know, we
are thus engaged in this successive formulation of wagers each correcting
the other. The only way to get a true statement related to the future is just
to continue to wager. Thus all pragmatist or finalist hypotheses on the basis
of induction are eliminated, and the result is obtained by the sole means of
a rich enough logic.
Probability theory to date provides the clearest case in which one can
see a logic obtained from Russell fruitfully contribute to the formulation or
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INTERNATIONAL CONGRESS OF THE PHILOSOPHY OF SCIENCE
resolution of a mathematical problem. However there is in the development of contemporary mathematical philosophy another current that is
often conflated with logicism and which is no less profoundly different in
its methods and its goals: it is the axiomatic gained from Hilbert. Instead of
trying to recompose all mathematical statements from the same set of
primitive logical notions, as in the Principia Mathematica (1910), the axiomatic in its recent development is rather an inverse effort to characterize
mathematical theories in their irreducible specificity with respect to one
another.
Chevalley has already shown, in his article on style in mathematics
(1935), how the axiomatic definition of notions such as ‘limit’, by their
characteristic properties, respond to an attitude quite different from that
seen in the generalization of constructive methods in analysis, the tool
par excellence of rigor in mathematics. When we consider the systems of
axioms that are the basis of modern theories of arithmetic, algebra, functional calculus: axioms of group theory, ideals, Hilbert spaces, hypercomplex numbers, etc., we envisage simultaneously a domain and operations
carried out in this domain. So what matters in the establishment of a
system of axioms is not the logical elements of which the statements are
composed, but the structural concern to appropriate exactly the chosen
axioms to the domain that we want to define by its properties. The solidarity which is thus manifested between the domain and the possible operations on this domain places the connection between abstract operations
and concrete domains at the forefront of mathematical research.
It is mainly the Swiss mathematicians who have made every effort to
describe this obvious connection in their communications to the Congress.
Swiss Romandy mathematicians were in fact in the habit of meeting philosophers in the Philosophical Society of Switzerland Romandy chaired by
Arnold Reymond, Professor at the University of Lausanne. Their relative
conceptions of the philosophy of mathematics provide extremely precise
suggestions to the proccupations of the philosophers. Gonseth, in his
lecture on logic considered as a science of any object, resumed the ideas
he had already presented in his lecture on the law in Mathematics that
appeared in the publication of the Centre de Synthèse (Gonseth 1934). He
considered logical axioms as the last stage of a process of progressive
abstraction and objectification from concrete experience to physical objects.
At each level of abstraction, the mind studies more and more general properties. Logic is merely the last chapter of physics, which studies the properties of concrete objects that come from the simple fact of existence, as, for
24
I N T E R N AT I O N A L C O N G R E S S O F T H E P HILOSOP HY OF SC IE NC E
example, the following properties: two things being given, either they are
simultaneously present or simultaneously absent, or else there is one without the other. The problems of logic are thus attached not only to a real
mathematics, but also a real physics. Juvet, focusing on problems of group
theory, showed how the study of the structure of a group allowed the
formal point of view and the practical point of view to be joined in all
branches of mathematics. The abstract structure of a group of transformations, for example, reflects the characteristic properties of space in which
the transformations of the algebraically defined group operate. In his book
La Structure des Nouvelles Theories Physiques (1933), Juvet gives a magnificent
comparison that allows one to understanding what profound harmony can
exist between a schematic structure and a material realization:
Placed at a great distance from a window, the eye can first distinguish
two axes of symmetry. On approaching, it recognizes in each quarter of
the structure two new symmetries; certain motifs are repeated five times
around a center. From closer yet again, more subtle ornamentations are
seen in these motifs. It is the same with physical reality and the mind
that examines it. The symmetries of phenomena, their alternations
observe certain invariants at a given scale. The description that we give
actually preserves these invariants and mimics these symmetries and
these alternations in a game that reflects the structure of a group. Can it
not be said that, in its way, physical reality at this scale mimics the structure of the group, or as Plato said, participates in this group? (Juvet
1933, 173)
The reference to Plato is particularly significant and beneficial. To study
only the signs, we can in effect come to believe that science deals only with
those signs and excludes any consideration of a reality that the symbolism
would have as its function to describe. The rational idea that the mind
penetrates the becoming of things by knowledge of the mathematical connections in which they participate appears to some to be as obscure as the
mystical beliefs in the participation of the subject in the object for the primitives spoken of by Levy-Bruhl (1931). Philosophers are then entitled to
ask themselves whether or not the philosophy of science lacks the essential
mission of every philosophy when it ceases to search for methods that give
man access to the real. Placed in front of a purely tautological conception
of mathematics, the philosopher should stop linking the discovery of truth
in science to the spiritual progress of a consciousness in search of a real to
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INTERNATIONAL CONGRESS OF THE PHILOSOPHY OF SCIENCE
know and dominate. Scientific philosophy, by its formalism, would thus
have contributed to the rejection of philosophy as belonging to the exclusive cult of irrational attitudes. One may however wish for the philosophy
of science a higher ambition.
26
On the Reality Inherent to
Mathematical Theories
PRESENTATION TO THE NINTH INTERNATIONAL
CONGRESS OF PHILOSOPHY*
Descartes Congress
Paris, 1–16 August 1937
SUMMARY – I try to show that the reality inherent to mathematical
theories comes to them from what is incarnated in their own movement as
the schema of connections that between them support certain abstract
ideas that are dominating with respect to mathematics. I show in particular
how the problem of the relations of essence and existence in effective
mathematics receives quite a different solution to those of intuitionism and
formalism.
Mathematical philosophy tends often actually to be mistaken for the study
of different logical formalisms. This attitude generally entails as a consequence the assertion of the tautological character of mathematics. The
mathematical edifices that appear to the philosopher so hard to explore, so
rich in results and so harmonious in their structures contain in fact no
more reality than is contained in the principle of identity. We claim to
* Lautman 1937a.
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ON T H E R E A L I T Y I N H E R E N T TO M AT H E M AT I C A L T H E O R I E S
show how it is possible for the philosopher to disregard such flawed conceptions and find within mathematics a reality that fully satisfies the
expectation they have of it.
This work conveys all the more to the philosopher that it is the reality
inherent to mathematics, as all reality, in which the mind encounters
an objectivity that is imposed on it. It is necessary to know to relate the
modalities of the spiritual experience to the intrinsic nature of this reality
in which it can be apprehended. The reality of mathematics is not made in
the act of the intellect that creates or understands, but it is in this act that
it appears to us and it cannot be fully characterized independently of the
mathematics that is it’s indispensible support. In other words, we think
that the proper movement of a mathematical theory lays out the schema
of connections that support certain abstract ideas that are dominating with
respect to mathematics. The problem of connections that these ideas are
likely to support can arise outside of any mathematics, but the effectuation
of these connections is immediately mathematical theory. Mathematical
logic does not enjoy in this respect any special privilege. It is only one
theory among others and the problems that it raises or that it solves are
found almost identically elsewhere.
We will show on a precise point how we think we can justify this
presentation of things, and to do this let us study the problem of the relation of essence and existence. This problem, which is connected furthermore to the problem of the finite and the infinite, is essentially philosophical.
Classical metaphysics, with the dialectical means of which it is disposed,
has always tried to carry out the passage, for a single entity, from the
essence of this entity to its existence. This problem is met with again in
mathematics in the discussions concerning the transfinite and the axiom of
choice, and the intuitionist or formalist mathematicians have in general
placed the debate on the terrain of traditional philosophy.
For supporters of the actual infinite, the non-contradictory definition
of a mathematical entity entails its existence. For the nominalists, there is
only effectively constructed existence. It seems that what these two attitudes have in common is that they still conceive the problem of the relation of essence and existence as arising with regard to a same entity. If we
now abandon the idea that a schema of solution for such a problem could
even be conceivable independently of mathematics and try on the contrary
to derive from the movement of mathematical theories the framework that
underlies them, we arrive at very different conclusions. When the passage
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O N T H E R E A L I T Y I N H E R E N T TO M ATHEMATICAL THEORIES
from essence to existence is possible, it has always taken place from one
kind of entity to another kind of entity, and likewise in logic and in the rest
of mathematics.
The point of view of ‘essence’ in logic and that of the non-contradictory
structure of a system of axioms is the structural point of view, or ‘beweistheoretisch’, that Bernays opposes to the extensive point of view, or ‘mengentheoretisch’. The extensive point of view is that of the existence of
interpretations of a system of axioms, of domains of individuals that realize
it, and almost all the metamathematical proofs try to establish a link
between the structural properties of the propositions of a system and the
existence of domains in which these propositions are verified. The passage
from essence to existence thus results from the structure or essence of the
system of axioms being apt to give rise to interpretations of the system.
We’ll find analogous schemas of genesis in the most different mathematical theories.
The structure of a Riemann surface is expressed by its genus. The genus
p makes it possible to know the maximum number of 2p closed curves
that can be drawn on this surface without dividing it into two separate
regions. Now that number 2p, which is thus connected to the ‘canonical
cutting’ surface, is immediately interpretable in terms of existence for
entities other than the base surface, since it also measures the maximum
number of elements of a real base of Abelian integrals, everywhere finite,
definable on this surface. The overall structure of a group is reflected in the
number of ‘classes of elements’ of the group, and this number, in the case
of a finite group, is equally that of the irreducible and non-equivalent
representations definable on the space of the group. The structure of an
algebraic field is manifested, like that of a group, by a decomposition into
‘classes’ of elements of the field, and the properties of this structural
number depends on the existence in the base field k of a number such that
k a is a quadratic class field over k. Again, let an operator be defined over
the Hilbert space. The bringing the eigenvectors and eigenvalues of this
operator to the fore results in a structural decomposition of the Hilbert
space into eigenspaces, each defined by the eigenvectors of the operator
in question. In all these purely mathematical examples, we always see a
mode of structuration of a basic domain interpretable in terms of existence
for certain new entities, functions, transformations, numbers, that the
structure of the domain thus appears to preform. The problem of the geneses in which the passage between essence and existence is carried out is
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ON T H E R E A L I T Y I N H E R E N T TO M AT H E M AT I C A L T H E O R I E S
perhaps formulable abstractly but it is only in the proper movement of
mathematical theories that the distinctions necessary for its solution thus
take shape.
It would likewise be possible to extract from the mathematical theories
the schema of connections that support other logical, or more precisely
dialectical, ideas: those of whole and part, complete and incomplete, intrinsic and extrinsic, existence and choice. We only wish to indicate here the
Platonic conclusion that these researches seem to us to impose: the reality
inherent to mathematical theories comes to them from their participation
in an ideal reality that is dominating with respect to mathematics, but that
is only knowable through it.
These ideas are quite distinct from pure arrangements of signs, but they
have no less need of them as mathematical matter that lends them a field
in which the layout of their connections can be provided.
30
The Axiomatic and the Method
of Division*
The development of abstract theories in modern mathematics and the
research into axiomatic definitions are often accompanied by a restoration
of the idea of generalization. Axioms are considered both, in comprehension, as a system of conceivable conditions independently of the mathematical entities that they realize, and, in extension, as defining the most
extended class of entities likely to realize them. This point of view of extension is sometimes so well allied with the classification into genera and
species that the possibility of fitting the least extensive types in the most
extensive is met with again even in the constitution of ‘abstract’ theories.
Thus, for example, in his book Les espaces abstraits (1928), Frechet envisages
the axiomatic constitution of abstract spaces from the point of view of successive generalizations of the notion of space. He first defines the D spaces
or distances, that is, the spaces in which to any pair of points a number can
be attached that is called the distance between these two points, and which
satisfies the axioms of distance. He then considered L spaces or spaces in
which a convergence of sequences of elements can be defined without it
being necessary to define a distance in advance. Any space D is a space L,
but the reciprocal is not true; there are L spaces which are not D spaces.
Frechet then considers a more general category, V spaces, whose definition
does not even appeal to the notion of convergence and relies solely on
the notions of neighborhood and point of accumulation. It is possible to
show that the class of L spaces is entirely contained in the class of V spaces.
* Lautman 1937b.
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TH E A X I O M AT I C A N D T H E M E T H O D O F D I V I S I O N
These examples are sufficient to show that the axiomatic study of abstract
spaces is able to be interpreted as a generalization. We shall see however
that it is possible to give to axiomatic thought a completely different bearing the idea of which moreover is equally found in the work of Frechet.
Consider again the importance of the idea of generality in the conceptions of Bouligand relative to causality in mathematics. Bouligand calls a
proof ‘causal’ if the proof of a relation between hypotheses and a conclusion is such that any reduction carried out in the statement of the hypotheses is likely to compromise the conclusion (Cf. Bouligand 1934; 1935,
175). This relation can then be ‘realized’ in an extensive enough domain of
mathematical facts that constitute the domain of causality of the relation in
question. The idea of group then necessarily introduces the fact that the
different realizations of the same relation obey the laws of composition
of the elements of a group. With respect to its domain of causality, the
causal statement thus plays the role of an invariant with respect to a group.
Bouligand expressly links the concern of causality to the revision of the
initial notions implicated by the axiomatic method and to the search for
greater generality. The link between these three ideas, for example, is
asserted in this sentence from ‘Geometry and Causality’:
The search for the most general conditions of validity of a determined
statement, if it is ready to reveal its causal proof, doesn’t succeed without
a constant reworking of the implemented notions. (Bouligand 1935)
Thus, for example, the classic statement of Pythagoras’s Theorem is too
restrictive and obscures the causal fact that the area of any figure constructed on the hypotenuse is the sum of the areas of figures similar to the
first and constructed on the other two sides. These areas can be of squares
but also of any figures whatsoever. It can be seen that the consideration of
the most general statement is destined to shed light on a necessary connection that proves insufficient in particular cases. The search for generality,
in Bouligand, is therefore in no way due to a concern with generalization,
it presents itself rather as a consequence of the search for the necessary
connection and we would like to further study the true logical nature of
this approach.
Whether it’s a matter of the constitution of abstract notions, or the
search for necessary connections, it seems to us that mathematical discovery does not at all consist in subsuming the particular under the general
but in carrying out the dissociations comparable to those that condition the
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T H E A X I O M AT I C A N D T HE METHOD OF DIVISION
progress of physical knowledge. Physical experimental discovery very often
results from being able to carry out, within a phenomenon, a dissociation
by which the complexity of facts, which had beforehand seemed to be
simple, is revealed. This experimental dissociation is often preceded or
followed by a theoretical dissociation established within the system of
notions corresponding to the experience. In atomic physics, it is the dissociation of spectral lines into doublets or multiplets, the dissociation of
molecules in ortho and para molecules, the discovery of the positive electron, de Broglie’s hypothesis on the complementarity of two half-photons
that constitute the photon. It is the same in the theory of relativity: the
plurality of time or the duality of masses. When it takes place in theoretical
physics, this dissociation can sometimes result from an activity totally
abstracted from the critical postulates implicitly admitted in the common
notions. The dissociation of the unicity of time results in a critique of the
notion of coincidence just as the criticism of the notion of measurement
can cause the dissociation of the point of view of the observed and of the
observing. It thus seems that the concrete experiences present themselves
to the intelligence as resulting from the exceptional encounter of certain
notions whose separation can be carried out abstractly. This separation
does not in the least have the character of a generalization since the consideration of other particular facts of an order of higher precision is substituted for a particular fact of a certain order of precision.
If one tries to characterize the role of the method of division in mathematics, one finds first of all two fairly simple and well known cases: 1)
when two properties are wrongly identified, the discovery of one case in
which one is realized without the other shows their difference (Weierstrass’s
discovery: continuous functions without derivatives; Borel’s discovery:
functions of a complex variable that are ‘monogenic’, in Cauchy’s sense,
without being analytic, in Weierstrass’s sense); 2) another form of dissociation is that which, by an appropriate treatment, establishes the differences
between certain elements having use of a common property, the singular
points of an analytic function are those in the neighborhood of which the
function cannot be expanded into a series but a dissociation into essential
singular points and non-essential singular points (or poles) is introduced
immediately when instead of studying the series expansion of f(z) in the
neighborhood of these points, the development of
1
f (z)
33
TH E A X I O M AT I C A N D T H E M E T H O D O F D I V I S I O N
is studied. A similar development exists for the poles and not for the essential singular points. We now wish to describe, in a few examples of axiomatized notions, a third form of dissociation the philosophical importance of
which seems to us to be considerable because it shows, in mathematics at
least, the close connection of critical reflection and effective creation.
1. EQUALITY
In the Grundlagen der Mathematik (1939), by Hilbert and Bernays, arithmetic equality is defined by two axioms:
a=a
a = b " [A (a) " (b)] .
The first axiom establishes the reflexivity of equality and it can easily be
proven that transitivity and symmetry are implicated by the second. The
second axiom states that for two numbers to be equal, it is necessary that
any arithmetic property that applies to one applies to the other. It is necessary, in sum, that two equal numbers are indiscernible at all possible points
of view, as regards at least the properties defined by the signs of predicates
A introduced in the theory in question. Conversely, if two numbers are
discernable in comprehension, they measure classes of distinct elements in
extension. Thus we see that the notion of indiscernibility is closely linked
to the axioms that define the number of elements of a set. Bernays introduced, in addition, these axioms of countability immediately after the axioms
of equality by indiscernibility. Here, for example, is the axiom that defines
the fact that a domain only contains one individual: (x)(y)(x = y), there is
only one individual in the given domain if given any object x and any
object y, we have x = y. Similarly, the formula (x) (y) (z) (x = y 0 y = z 0 x = z)
means that there are at most two individuals in the domain considered,
and the formula (7x) (7y) (x ! y) (there is an x and y such that x is different
from y) means that there are at least two of them. If we then agree to
arrange the elements between which there is a relation of equality into
the same class, we see that there is in any set of numbers as many elements
as classes of equal individuals. It is perfectly possible to dissociate the
number of classes of elements of a domain, from the number of elements
of this domain by no longer envisaging the relation of equality (=), but an
34
T H E A X I O M AT I C A N D T HE METHOD OF DIVISION
equivalence relation ( / ) which is reflexive, symmetric, transitive like
equality, and amenable to being defined in many ways. Suppose, for example, that the domain in question was any group. Two elements of the group
a and al are said to be equivalent, or belong to the same class, if b is any
fixed element, such that al = bab - 1 . The result for the elements of a same
class is no longer a complete indiscernability, but a certain partial indiscernibility. If a number | (a) , such that | (ab) = | (a) | (b) , is made to correspond, in effect, with any element a of a group, then the functions |
(which are the characters of the group) take the same value, for all the elements of a same class: | (a) = | (bab - 1) . The consequence of this partial
character of the indiscernibility is that the formulas of countability corresponding to a relation of equivalence can no longer measure the number
of elements of the group, but only the number of their classes. A formula
asserting that there is at most n nonequivalent elements in a group would
signify that there are at most n classes of equivalent elements, without any
indication of the number of elements of the group. This shows how the
study of equivalence relations allows the dissociation of the point of view
of countability of classes from that of the countability of individuals, which
are conflated in the equality of arithmetic.
2. MULTIPLICATION
Consider a domain of individuals x, y, whose nature isn’t specified for the
moment, and write the equation y = ax , in which a is any whole number.
If x equally traverses the domain of whole numbers we have the ordinary
multiplication of arithmetic and y is equally a whole number. Multiplication can even be defined when the elements a and the elements x traverse
different domains. For example, consider the case of a vector space. We
know that in such a space, multiplying a vector x by a number a has a
meaning and gives, as a result, a new vector y = ax . The whole numbers
and vectors constitute respectively distinct domains which obey distinct
laws: the vectors form a module, that is, a domain in which the sum of two
elements is uniquely defined, and the whole numbers form a ring, that is,
a domain in which the sum and the product of two elements are defined.
The following axioms are obtained, the first of which concerns the addition
of vectors, the second and third the addition and the multiplication of
numerical multipliers
35
TH E A X I O M AT I C A N D T H E M E T H O D O F D I V I S I O N
a (x + y) = ax + ay
(a + b) x = ax + bx
a (bx) = (ab) x.
The ordinary notion of multiplication is seen here as split into two distinct
ideas: 1) the idea that we can make the elements of a domain of operators
act on the elements of a basic domain to recover other elements of the
basic domain; 2) the idea that this action of operators on the elements of a
domain is, in some cases, reducible to the formation of arithmetic products.
In the case of numbers that multiply vectors, the formation of products is
still found, because the vector x is defined by the coordinates (x2 . . . xn)
the vector ax is defined by coordinates (ax 2 f ax n) , but the operators acting
on a domain to restore an element of the domain can be considered without the formation of arithmetic products playing a part in any way. These
operators are not always composed according to the axioms defined above,
but they nevertheless constitute a domain of elements as characterized as
the elements of the basic domain on which they act.
If the basic domain is constituted by the points of a space, transformations of a group can be defined as operators acting on these points, which
are composed among themselves following the law of group composition.
The transformation S acting on the point p, transforms it into a point
p´ = Sp and the transformation T transforms the point p´ = Sp into a point
p۞ = TSp such that there is a transformation R of the group directly transforming the point p into p۞ = Rp. If the basic domain is constituted by the
functions y = (fx1 . . . fxn) the differential operators
2
2x n
can be considered to be acting to the left on functions y to restore another
function:
2 y = 2y .
2x o
2x o
These operators are likely to be composed following the known laws of the
differential calculus:
2 c 2y m = 22y
2 x n 2x o
2x n 2x o .
36
T H E A X I O M AT I C A N D T HE METHOD OF DIVISION
This shows how the axioms defining the action of operators on a basic
domain result from what has been dissociated in ordinary multiplication,
the operational function, which is independent of the specific nature of the
envisaged operators, from the formation of products, which is linked to the
particular nature of arithmetic operators. This duality between the action
and the nature of the operators is particularly significant in the study of
unity.
3. UNITY
When writing x1 = x (or 1x = x), the term 1 plays a dual role: it is the unit
element of the domain of operators acting on the domain of x, and it is also
the identical operator that transforms the elements x of the basic domain.
Here’s what these distinctions mean: suppose that the domain of operations is a ring of numbers. The unit element 1 of this ring is such that for
every element a of the ring, there would be a1 = a and 1a = a . This is an
internal definition that only envisages the nature of the unit element
within the domain to which it belongs. If, furthermore, this unit element
acting on the element x of the basic domain transforms this element, in
addition to the fact of being a unit element of the ring of operators, it is the
identical operator, and this concerns not so much its own nature as its
outward actions. This distinction of the specific nature of certain elements
and the action they exercise on the elements of other domains, is for us
essential. We shall see later that it permits the method of division in mathematics to be characterized as a whole.
4. MEASURE AND THE INTEGRAL
In the classical theory of the integral, the expression
#
a
f (x) dx
b
is defined by means of the sum of an infinite number of products. Each
product represents the multiplication of an ordinate by the length dx of an
infinitesimally small increase of the abscissa. The lengths dx therefore play
a dual role: being defined as measures of magnitude, they are practically
37
TH E A X I O M AT I C A N D T H E M E T H O D O F D I V I S I O N
conflated with those geometric segments to which they are connected; and
secondly, their function in the determination of the integral is to be the
numbers that play a part as the factors in a product of numbers. Here again
the Lebesgue generalization can be interpreted in terms of dissociation.
The fact of being given in a product as factors, like the contribution to a
basic domain, is not in the least connected to the geometric magnitude of
this domain. Rather, dx can be considered to be simply a number attached
to this domain, satisfying certain determined conditions.
This is necessarily so when the basic domain is no longer a segment but
a family E of sets of points of any kind. The determination of numbers
attached to these sets can be made as follows (Cf. Possel 1935): let us call
set function a function that makes a number n correspond to each set of the
family E whose value is real, finite or equal to + 3 , but never to - 3 . This
function is said to be completely additive if, E1, E2 . . . being disjoint sets of E
whose sum comes to E:
l) n (E 1) + n (E 2) has a finite sum or is equal to + 3 , independent of the
order of the terms;
2) this sum of values is equal to the value of the sum of the sets.
It proves that only those set functions that are never negative can be
considered among the completely additive set functions. Such a function is
called a measure if:
3) any set E of E such that n (E) =+ 3 can be covered by a countable
infinity of sets for which n is finite;
4) the family of sets E contain every intersection and every reunion of a
finite or countable number of sets of E;
5) when nE = 0 , any portion E´ of E belongs to E, and we have: nEl = 0 .
Here’s how such a measure is suitable for the formation of the Lebesgue
integral: let a point function f(p), defined over a set of points E, take the
non-negative values (and if necessary + 3 ) and let n be a measure defined
in E. It is then possible, in certain conditions the statement of which is
irrelevant here, to define on the set E an integral
#
E
f (p) dn,
through the formation of Lebesgue sums, each of which contains an infinity of products of type l i .nE i in which li measures an interval (in ordinates)
of the set of real numbers and in which nE i is the measure attached to a
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T H E A X I O M AT I C A N D T HE METHOD OF DIVISION
subset the fundamental set E. The measure nicely plays here a role analogous to that of the length of infinitely small increments of the abscissa in
the classical theory of the integral. It brings the contribution of space as
defined by the function to an infinity of products. But this new conception
of the integral could only be constituted by the work of Lebesgue and his
successors through the dissociation of the idea of measure attached by convention to a space, from the consideration of magnitudes that were originally conflated with the numbers that measured them.
5. THE ABSOLUTE VALUE
Consider a number field K, that is, a set of numbers such that it is always
possible to form the sum, product and quotient of any two numbers. This
field is said to be ordered if, for any element a , the property of being positive
( 2 0 ) can be defined by the following axioms (Cf. van der Waerden 1931,
ch. 10):
(I) 1) For any element a of k, either a 2 0, a = 0 , or - a 2 0 ;
2) if a 2 0 and b 2 0 , then a + b 2 0 and ab 2 0 ;
3) if - a 2 0 , then a is negative.
This having been posed, we call the absolute value ; a ; of an element a of an
ordered field that of the two terms a and - a that are not negative. The
absolute values satisfy the following axioms
(II)
; ab ; = ; a ; . ; b ;
; a + b ; # ; a ; + ; b ;.
The notion of absolute value, being linked to the definition of the positivity
of the elements of a field, is thus closely linked to the order of the elements
of the field. It is also equally linked to another quite different property, the
closure of the field question. This is what should be understood by that: a
field is said to be closed (or complete) if every sequence of terms a 1, a 2, f ,
such that from a certain rank the absolute value of the difference between
two terms is less than any given positive number f , converges in absolute
value towards a limit a . It proves that given an ordered field K, an extension field X of K which is both ordered and closed can always be constructed. When K is the field of rational numbers, X is the field of real
numbers.
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TH E A X I O M AT I C A N D T H E M E T H O D O F D I V I S I O N
The two notions of order and closure are thus closely associated in the
notion of absolute value, but it has been possible to dissociate them by noting that the construction of the closure X of K doesn’t use the order of
elements a of K but the order of absolute values ; a ; . In the case of ordered
fields, we make an element a correspond to an absolute value ; a ; by choosing, of the two terms a and - a , the one that is not negative, because by
virtue of axioms (I), there is always one. But what matters in the construction of closure is not that the ‘absolute value’ of a is ; a ; , but rather that this
absolute value obeys axioms (II). There is therefore a ‘generalization’ of
the notion of absolute value, when, to each element a of a field still not
amenable to order, an ‘evaluation’ (Bewertung) { (a) can be made to correspond that satisfies the axioms analogous to axioms (II):
(III) 1)
2)
3)
4)
{ (a) is an element of an ordered field
{ (a) 2 0 for a ! 0, { (o) = 0
{ (a) .{ (b) = { (ab)
{ (a + b) # { (a) + { (b)
It can be shown by emphasising this time the properties of the evaluation,
that given a valued field K, we can always construct an extension field X
whose evaluation extends that of K and such that any fundamental sequence
y is convergent (in the sense of the evaluation defined by axioms III).
The absolute values (II) do indeed constitute a particular case of evaluations (III), but the passage from axioms (II) to axioms (III) consisted essentially in detaching the properties defined in (II) from their too close
connection with the properties of order defined in (I).
The few examples that have just been analyzed will allow us to be more
specific about the philosophical importance of the activity of dissociation
in mathematics. We have seen in all the cases certain notions of elementary arithmetic and algebra, which seemed simple and primitive, envelop a
plurality of logical or mathematical notions, delicate to specify but in all
cases clearly distinguishable from one another. It is in this way that arithmetic equality is the only equivalence relation such that the countability
of the individuals of a set is conflated with the countability of classes of
equivalent individuals as defined by this relation. Likewise, the idea of
multiplication contains both the formation of arithmetic products and the
action of operators on a domain of elements distinct from these operators.
The idea of unity can be considered either from the point of view of the
unit element of a ring of numbers, or from the point of view of the operator identical to a domain of operators. The length of a segment is connected
40
T H E A X I O M AT I C A N D T HE METHOD OF DIVISION
to the magnitude that it measures but it is only a number attached by convention to this magnitude. Finally, the absolute value of classical algebra
envelopes both the idea of order and the construction of the closure of a
number field. The passage from notions said to be ‘elementary’ to abstract
notions doesn’t present itself as a subsumption of the particular under the
general but as the division or analysis of a ‘mix’ which tends to release
the simple notions with which this mix participates. It is therefore not
Aristotelian logic, that of genera and species, that plays a part here, but the
Platonic method of division, as taught in the Sophist and the Philebus, for
which the unity of Being is a unit of composition and a starting point for
the search for principles that are united in the Ideas.
Another rapprochement with the Platonic dialectic is necessary. We have
seen that distinct notions revealed by the dissociation of a mathematical
idea are related most often, on the one hand, to the intrinsic nature of
certain entities, and on the other, to the action of these entities on other
entities. Let us take a few examples. The fact that the unequal elements of
a set of numbers are discernible, from a certain point of view, is a property
concerning the nature of these numbers. The fact that the relation of equality, like any relation of equivalence, determines within the set in question
a division into classes (each of which only contains one element), relates to
the structure what the relation of equality imposes on the set. The fact that
numbers can be multiplied with each other relates to the intrinsic properties of the ring of whole numbers, whereas, on the other hand, the fact that
the multiplication of a number by a vector gives a vector is linked to the fact
that a vector space admits the action of numerical operations on it. Likewise, finally, the positivity of a number is rightly an intrinsic notion, but the
closure of a field that can be carried out with the notion of absolute value
or evaluation relates rightly to the constructive fertility of the notion of
evaluation. The axiomatic definitions ‘by abstraction’ of equivalence, measure, operators, evaluation, etc., thus characterize not a ‘type’ in extension
but the possibilities of structuration, integration, operations, of closure conceived in a dynamic and organizational way. The distinction that is thereby
established within a same notion between the intrinsic properties of an entity
or notion and its possibilities of action seems to be similar to the Platonic
distinction between the Same and the Other that is found in the unity of
Being. The Same would be that by which a notion is intrinsic, the Other
that by which it can enter in relation with other notions and act on them.
We said at the beginning of this study that certain mathematicians like
Frechet or Bouligand sometimes associate the effort of axiomatic abstraction
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TH E A X I O M AT I C A N D T H E M E T H O D O F D I V I S I O N
and the idea of generalization. Generalization is however, with them too,
only the consequence of more essential preoccupations. With Bouligand,
as we have seen, it is the search for the necessary connection, with Frechet,
a concern for analysis that often relegates the point of view of generalization to second place, as in these lines intended for philosophers, where he
writes:
It is extremely curious to see a notion such as that of distance which
appears, primarily, to be a primary notion, an irreducible notion, able to
be dissociated into notions of nature very different from each other.
(Frechet 1928, 158)
It is, in effect, extremely important for the philosopher to prevent the
analysis of Ideas and the search for notions that are the most simple and
separable from each other, from appearing like the search for the most
extended types. A whole conception of mathematical intelligence, issuing
from Platonism and Cartesianism is, in effect, at stake in this distinction.
42
BOOK I
Essay on the Unity of the Mathematical Sciences
in their Current Development
Introduction:
Two Kinds of Mathematics
In 1928, in the preface to The Theory of Groups and Quantum Mechanics,
Hermann Weyl wrote:
There exists, in my opinion, a plainly discernible parallelism between
the more recent developments of mathematics and physics. Occidental
mathematics has in past centuries broken away from the Greek view
and followed a course which seems to have originated in India and
which has been transmitted, with additions, to us by the Arabs; in it the
concept of number appears as logically prior to the concepts of geometry.
The result of this has been that we have applied this systematically
developed number concept to all branches, irrespective of whether it is
most appropriate for these particular applications. But the present trend
in mathematics is clearly in the direction of a return to the Greek standpoint; we now look upon each branch of mathematics as determining its
own characteristic domain of magnitudes. The algebraist of the present
day considers the continuum of real or complex numbers as merely one
‘field’ [or ‘antisymmetric field’ among an infinity of others that have the
right to the same consideration. (The property of being antisymmetric
results from not conserving the commutative law of multiplication).]
The recent axiomatic foundation of projective geometry may be considered as the geometric counterpart of this view. This newer mathematics,
including the modern theory of groups and ‘abstract algebra’, is clearly
motivated by a spirit different from that of ‘classical mathematics’, which
found its highest expression in the theory of functions of a complex variable. The continuum of real numbers has retained its ancient prerogative
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
in physics for the expression of physical measurements, but it can justly
be maintained that the essence of the new Heisenberg-SchrödingerDirac quantum mechanics is to be found in the fact that there is associated with each physical system a set of magnitudes, constituting a
non-commutative algebra in the technical mathematical sense, the elements of which are the physical magnitudes themselves. (Weyl 1928, vi
[1931, p. viii]) (Comments inserted by Lautman)
This text asserts in the clearest way the existence of an essential division in
contemporary mathematics. It would be necessary in effect to distinguish
‘classic’ mathematics, which starting from the notion of whole number
leads to analysis, from ‘modern’ mathematics, which, opposed to the mathematics of number, asserts on the contrary the primacy of the notion of
domain with respect to numbers attached to this domain.
There is no doubt about the existence of a new mathematics, animated
by a very different spirit to that of the mathematics of last century. Analysis
of the nineteenth century, as its name suggests, analyses the infinitesimally small. In the theory of functions, it is the convergence of series, the
passage to the limit, continuity, derivation and integration. In the theory of
differential equations, it is the search for local integration in the neighborhood of the origin. It is the differential geometry that emerges from Gauss’s
General Investigations of Curved Surfaces (1827) and Monge’s Applications de
l’Analyse à la Géométrie (1807): the theory of contacts, curvature, envelopes,
etc. However, in spite of the radiance with which the analysis of the infinitely small shined throughout the course of the nineteenth century, it did
not stop the emergence of ideas whose development would lead to the
synthetic mathematics of the modern epoch. Some of these new theories,
such as set theory, are due to the very difficulties of analysis and the need
to find a way to overcome them. Others, such as group theory, have their
origins in algebraic problems completely foreign to analysis. We will not
stop at these questions of origin, because our goal here is solely to characterize in their common resemblances the diverse theories that, in opposition to
analysis, have as their object the study of the global structure of a ‘whole’.
Let us envisage in the first instance topology. Currently, two kinds of
topology are sometimes distinguished: the topology based on the methods
of the set theory of points; and combinatorial topology, or algebraic topology, which characterizes the properties of invariant figures arithmetically or
algebraically by a continuous transformation. The global character of these
two theories is obvious. They generalize in effect intuitive considerations of
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I N T R O D U C T I O N : T W O KINDS OF MATHEMATICS
synthetic geometry like the fact for a surface to be opened or closed, to
have holes or to cover itself with places. The considerable development in
contemporary topology derives from it being currently impossible to conceive a theory of analysis that is not based on a prior topological study of
the domain of definition of the envisaged functions. Consider, for example,
the expression y = f(x). This expression asserts the existence of a selective
correspondence between a domain of the x plane and a domain of the y
plane. The function f which ensures this correspondence can be perfectly
determined uniquely by the topology of the two domains in question,
without needing to know the algebraic or transcendental expression,1
which gives the value of y from that of x. This necessity to upport analysis
with topology derives from Riemann as regards the functions of a complex
variable, and the topology in question is that which has since been called
combinatorial topology. As for the theory of functions of real variables,
since the work of Cantor, Borel and Lebesgue, it is intimately linked to the
topology of point sets and the global ‘measure’ of the domains they represent. The distinction established by Weyl between classical mathematics
and modern geometrical mathematics would therefore not tend to oppose
the methods of the theory of functions and those of topology. They are the
same, and nothing more is needed to be convinced of this than to read the
book by Weyl, The Concept of a Riemann Surface (1913), that is often cited in
our principal thesis (Lautman 1938b).
By opposing the methods of modern mathematics to those of classical
mathematics, Weyl thought, and the indications that he gives prove it, of
the theories of modern algebra as they are found presented for example in
the two volumes of the work of van der Waerden (1930; 1931). We would
like briefly to emphasize the differences that separate the spirit of classical
arithmetic and analysis from the spirit of modern arithmetic and algebra:
group theory, rings, fields, ideals, hypercomplex systems, etc. Classical
mathematics is constructivist, first of all as regards the definition of the
operations of analysis beginning with the operations of elementary arithmetic, but also and above all as regards the individual generation of real or
complex numbers from whole numbers. Modern algebra is on the contrary
axiomatic, and the result is that by giving the axioms which the elements
of a group or field obey, the often infinite totality of elements of the group
or field are also given, at the same time. These elements are in general no
longer amenable to being individually constructed from the arithmetic of
whole numbers. They can in fact be of any kind, in addition to numbers
there are also vectors, operators, transformations, matrices etc.
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
This axiomatization of modern algebraic theories brings with it a number
of consequences that for Weyl seem essential: the first relates to the priority of the global structure of a domain with respect to numbers attached to
this domain. This can be understood in two different but complementary
ways. The domain described by Weyl can be the first ‘domain of magnitudes’ (or in the narrow sense of the word: algebra) that constitutes the
studied magnitudes themselves, whose individual properties are then governed by the laws of organization of the ‘whole’ of which they are the
elements. What in effect characterizes an algebra are not the elements of
which it is the set, but the fact that, whatever elements are considered,
they maintain relations among themselves which affirm their membership
of this algebra.
The priority of the notion of domain can then be asserted with respect to
the notion of number, by considering the appropriation of a system of
numbers or magnitudes of a domain previously defined in a geometrical,
topological or even physical way. We have already alluded above to the
connections that unite the topology of certain domains to the existence of
functions defined on these domains. The connection of topology and algebra is infinitely closer because the algebraic invariants attached to a domain
are not only defined on it, but serve moreover to explore and recognize it.
There is another aspect in the text of Weyl that distinguishes modern
algebra from classical mathematics, that which is relative to the non-commutativity of multiplication. The non-commutative algebra, in which a
product ab is different to a product ba , also differs as radically from ordinary algebra as the algebra of logic in which a # a = a . The algebra of logic
does not however constitute, in all truth, a new mathematics because its
field of application is too narrow, while the non-commutative algebras
have an increasingly appreciable importance, both in mathematics and
physics.
Finally, a fourth characteristic of the theories of modern algebra can be
described, which, though not formulated in Weyl’s text, is nonetheless an
extension of the previous characteristics: in contrast to the analysis of the
continuous and the infinite, algebraic structures have a clearly finite and
discontinuous aspect. Whatever the infinity of elements that constitute a
group, a body, an algebra (in the narrow sense of the word), the methods
of modern algebra consist most often in imposing on these elements a division into classes of equivalent elements, and in thus substituting for an
infinite set, the consideration of a number of classes which, in application,
is most often finite.
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I N T R O D U C T I O N : T W O KINDS OF MATHEMATICS
The methods and the fundamental conceptions of modern algebra being
thus distinguished from those of analysis, an interpretation of the meaning
of this duality, which appears to be deeply installed within contemporary
mathematics, can be sought. Is it an essential duality between theories
irreducible to one another, or is it rather a duality of methods that could
one day be reconciled? The text of Weyl quote above does not contain a
clear answer to this question; however, he suggests the idea that there
would be more of an opposition between the methods of the theory of
functions of a complex variable and algebra. On the other hand, Hilbert
has repeatedly expressed his firm belief in the possibility of finding a conjoint method of elaboration of algebra and analysis: eine methodisch einheitliche Gestaltung von Algebra und Analysis (1935, 57). We propose to show in
the pages that follow how modern mathematics is engaged in the process
of this unification of algebra and analysis, and that it is so by the penetration of increasingly sophisticated structural and finitist methods of the
algebra in the domain of analysis and the continuous. In sum, the conflict
between the methods of algebra and analysis are resolved in favor of
algebra. Weyl’s distinction between two kinds of mathematics thus seems
only to correspond to the historical conditions of the development of
mathematics, and leaves intact the unity of mathematics and the unity of
intelligence.
In the four chapters of this essay we try to show how, in the modern
theories of analysis, aspects which seem to characterize modern algebra
can be found. The algebraic idea of dependence of magnitude with respect
to the domain to which it belongs has led to the study of the influence of
the ‘dimensional’ structure of a set on the mode of individual decomposition of its elements, and to consider the importance of dimensional decompositions in the theory of functions (Chapter 1). The priority of the topology
of certain domains over the numbers attached to these domains is reflected
in the role of non-Euclidean metrics in the theory of analytic functions
(Chapter 2). We will then study (Chapter 3) the non-commutative algebra
which plays a part in the theory of equivalence of differential equations.
Finally we show how the consideration of finite and discontinuous algebraic structures can be used to determine the existence of functions of a
continuous variable (Chapter 4).
49
CHAPTER 1
The Structure of a Domain of Magnitudes and
the Decomposition of Its Elements:
Dimensional Considerations in Analysis
The decomposition of a mathematical entity can be envisaged from two
very different points of view: sometimes the decomposition brings to light
the particular properties of an entity within the set to which it belongs, and
it is characterized by the specificity of its very structure; sometimes on the
contrary the decomposition of the entities of the same set is done according
to a plan common to them all and thus reflects not only their particular
properties, but also their belonging to the same set whose global structure
is reflected in that of its elements.
The type of decomposition of the first kind is the arithmetic decomposition of a whole number into a product of equal or unequal prime factors.
This decomposition is possible in elementary arithmetic in only one way
and every number is thus uniquely characterized by the factors that are
proper to it.
Take as a type of decomposition of the second kind the algebraicgeometric decomposition of a vector X in an n-dimensional vector space R.
Let a system of n linearly independent vectors e 1 f e n be in this space, that
is, such that the equation:
a1 e1 f an en = 0
is only solvable if all the coefficients X cancel each other out. The space
being n-dimensional, these vectors form a basic system or fundamental
50
T H E S T R U C T U R E O F A D OM A IN OF M AGNIT UDE S
system for the set of vectors X of the space. This means that any vector X
can be represented by a decomposition of the following type:
X = x1 e1 + f xn en
These decompositions differ clearly from the decompositions of the first
kind. The coefficients x1 . . . xn characterize uniquely, it is true, the vector
X in the adopted basic system but the fact that any vector can be decomposed into n components reflects less an individual property of these vectors than their membership in an n-dimensional space. These are the global
dimensional properties of this space that thereby impose a uniform mode
of decomposition to each of its elements, and this influence of the structure
of the set on the nature of the elements is manifested in an even more
striking way when the space R can be represented as a direct sum of subspaces independent of one another:
R = R1 5 R2 f 5 Rn
Each Ri then contains all the vectors x i e i of the same index, so that any
vector X is thus obliged, by its affiliation with R, to have a component in
each of the n subspaces of the decomposition of R.
The decompositions of a second kind present yet another essential
aspect. The n base vectors of such a decomposition, e 1 f e n are, in the same
way as X, vectors of space R, whereas the factors of a decomposition of the
first kind are still, as we shall see, of a simpler nature than the entities that
they characterize. The vectors of space R therefore present these multiple
forms of solidarity, and a decomposition into components reflects not only
the dimensional structure of the global space, but also the choice of the
basic system adopted to support and organise the relations that unite all
the elements of the space to a fundamental set of n.
It is possible to find in algebra the two types of decomposition that have
just been distinguished. The theorem that has long been called the fundamental theorem of algebra is a theorem of ‘proper’ decomposition. It states
that any polynomial in x of degree m is equal to the product of a constant
by m equal or unequal factors of the form (x - a) :
f (x) = C (x - ai) (x - a2) . (x - am) .
To this decomposition, which brings to light the unique roots of a polynomial, we would oppose all ‘imposed’ decompositions in the modern
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
theories of algebra. Consider for example a hypercomplex system. Such a
system O is defined by a set b1 . . . bn of basic elements and a commutative
field P whose elements act as multipliers on the basic elements. This gives
the following decomposition for O
(I)
O = b 1 P + ff b n P ;
(P denotes the set of numbers of field P), and for each a of O
(II)
a = b 1 m1 + ff b n mn = m1 b 1 + ff mn b n) (mi fP) .
The law of individual decomposition (II) inscribed thus in each element a
of O the structural properties (I) of the system to which these elements
belong.
The two kinds of algebraic decomposition are in addition not totally
foreign to one another. The ‘proper’ decomposition of a polynomial into a
product of factors is closely connected to the global nature of the field in
which this decomposition is attempted. The polynomial x2 – 2 has no root
in the field k of rational numbers and is only decomposable into a field
which contains at least the field k ( 2 ) .1 The fundamental theorem of algebra relative to the complete decomposition into m factors of any polynomial of degree m is basically only the translation of a global property of
the field of complex numbers: that of being algebraically closed. Such a
dependency is even found with respect to the base field (Abhängigkeit vom
Grundkörper) in arithmetic decompositions. The number 21 is decomposable into a unique product of prime factors only in the field of rational
numbers, in which 21 = 3 x 7. On the contrary, it can be decomposed in a
second way in the field k - 5 , in which there is, in addition to 21 = 3 x 7,
the following decomposition: 21 = (1 + 2 - 5 ) (1 - 2 - 5 ) . The proper
decompositions are in a certain way connected to the basic domain as
‘imposed’ decompositions.
One essential difference nevertheless remains. The proper decompositions still study mathematical entities in isolation, independently of all
those that have the benefit of analogous properties, while the second kind
of decompositions impose on all elements of a set the same structure that
results from the dimensional structure of the set to which they belong, and
thus establishing a close dependency between the totality of entities studied
and a certain number of them taken as the basic system. The dependency
with respect to the base field that sometimes presents the decompositions
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T H E S T R U C T U R E O F A D OM A IN OF M AGNIT UDE S
of the first kind is thus very different from the interdependence of the
elements in the global system of contemporary algebra, and that brings to
light the decompositions of the second kind.
Before researching the respective importance of these two modes of
decomposition of a mathematical entity in analysis, we would like to show
how the linear interdependence of elements of a set is in close relation to
the theory of systems of linear algebraic equations.
Consider first of all the homogeneous system (without second member)
(III)
a 11 x 1 + ff a 1n x n = 0
fffffffff
fffffffff
a m1 x 1 + ff a mn x n = 0
Let r be the rank of the system, that is, the degree of the largest non-zero
determinant formed with the coefficients of system (III). A solution of the
system is any vector X with coordinates x1 . . . xn that satisfies the equations
(III), and it can be demonstrated that for r 1 n , there is a maximum
number of linearly independent n – r solutions forming a fundamental
system of solutions. Any solution X of the system (III) is a linear combination of fundamental solutions,
X = c 1 X 1 + ff c n - r X n - r
and the system (III) thus defines a vector space to n – r dimensions. We will
not discuss here the case of a non-homogenous system, that is, whose second members are not zero as in (III); but simply indicate that in the study
of these systems, the dimensional idea of the composition of a general
solution from any particular solution, and of a fundamental system of solutions from the corresponding homogeneous system is sometimes found.
This junction which thus appears between the theory of decompositions
of the second kind and the resolution of linear algebraic equations is in
our view very important, because it is through the consideration of such
systems of equations that the substitution of the synthetic point of view of
dimensional decompositions for the individual point of view of proper
decompositions occurs very often in analysis.
The domain of decompositions of the second kind is primarily formed
by the approximate representation of functions belonging to different
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
functional spaces. It is known that the trigonometric functions 1, cos x, cos
2x, . . . ; sin x, sin 2x . . . , form in the space of continuous and differentiable
functions f(x), for 1 - r # x # r with f (- r) = f (r) a system of basic functions, such that for any function f(x) of the space, there is the Fourier series
expansion:
n=3
f (x) = 1 a 0 + / (a n cos nx + b n sin nx) :
2
n=1
the a n and b n satisfying certain determined conditions.
We have also recalled elsewhere the analogous properties of the space of
functions of summable squares. This space is constituted by the set of functions f(s) defined on a measurable space E, such that
#
E
f (s) dn
exists, and that
#
E
f (s) 2 dn
has a finite value.2 Such a space is isomorphic to the Hilbert space (Cf.
Lautman 1938b, ch. 5); it is thus a vector space to an infinity of dimensions
and there exists in this space an infinity of basic systems {1 (s), {2 (s) f each
formed by an infinity of orthonormal functions3 such that every function
of the space is decomposable in series:
f (s) = x 1 {1 (s) + x 2 {2 (s) + ff
This result has played a large role in the history of the problems studied
here, because it is by the extension of these algebraic methods of decomposition of the second kind to unknown functions of integral equations
or partial differential equations that Hilbert conceived a possibility for the
conjoint elaboration of algebra and analysis. The unification of the mathematical disciplines must thus be carried out by the penetration of the
dimensional and finitist methods of algebraic origin into the domain of
analysis. The decompositions of the second kind being connected in algebra to the study of systems of a finite number of equations, of a finite
number of variables, Hilbert developed the theory of systems of an infinity
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T H E S T R U C T U R E O F A D OM A IN OF M AGNIT UDE S
of linear equations to an infinity of variables to find the theorems of linear
combination analogous to those of the finite, and applicable to the theory
of integral equations. Here is a passage in which Hilbert expresses with
admirable clarity the hope of unifying algebra and analysis by a similar
generalization of algebraic methods:
The theory of forms to an infinity of variables ‘is’ a new domain, and to
some extent intermediate between algebra and analysis, which by its
methods is based on algebra, but which belongs to analysis by the transcendent nature of its results. (Hilbert 1909, 62; 1935, 59)
We will now briefly show how Hilbert carries out the reduction of a problem of analysis to an algebraic problem.
Let the linear integral equation be:
f (s) = { (s) - m # k (s, t) { (t) dt
1
(IV)
0
in which (s) is the function sought after and in which f(s) and k(s, t) are
given. Hilbert first constructed a system, of arbitrary orthonormal continuous functions ~ p (s) (0 # s # 1) , that satisfy in addition, relative to an arbitrary continuous function u(s), the following conditions of completion:
#
0
1
u (s) 2 ds = ; # u (s) ~1 (s) dsE +; # u (s) ~2 (s) dsE + f +
2
1
2
1
0
0
Such a system forms a basic system in the space of functions u(s). Hilbert
then introduces the following expressions:
#
1
#
1
xp =
0
kq =
0
k pq =
{ (s) ~ p (s) ds, f p =
k (s, t) ~ p dt
# #
1
0
0
1
#
0
1
f (s) ~ p (s) ds
,
,
k (s, t) ~ p (s) ~ p (t) dsdt
which are such that / x 2p, / f 2p, / k q (s) 2, / k pq (s) 2 converge.
In these conditions, from the integral equation (IV), the system following from an infinity of linear algebraic equations, an infinity of unknowns
is deduced
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
(V)
(1 - mk 11) x 1 - mk 12 x 2 ff. = f1
- mk 21 x 1 - (1 - mk 22) x 2 f. = f1
ffffffffffff
ffffffffffff
The resolution of the integral equation (IV), a problem of analysis, is therefore reduced to the resolution of the system of algebraic equations (V).4
Let x1, x2 . . . be a set of values forming a solution to this algebraic system;
a continuous solution of the integral equation (IV) can be obtained by
forming the uniformly convergent series:
{ (s) = f (s) + m / k q (s) x q
q
and it is easy to show that this solution { (s) admits a decomposition into
Fourier series,
3
{ (s) =
(VI)
/x
p
~ p (s)
p=1
the basic system being the system of adopted ~p (s) and the coefficients xp
obeying the relations:
xp =
#
0
1
{ (s) ~ p (s) ds
The expression (VI) correctly represents a development of the function ‘of
the second kind’; { (s) is sought after, and it is this algebraic-looking result
that Hilbert had in mind in the constitution of his theory.
We will now try to determine the role of decompositions of the first kind
in the theory of analytic functions and see how in this domain too, the
dimensional considerations issuing from algebra could give new meaning
to the results of proper decomposition theorems of arithmetic inspiration.
The proper decomposition of a polynomial is the one that emphasizes its
roots. The analytic function that most resembles a polynomial is the integral function, which only has singularities at the infinite, and it is possible
to single out the zeros of such a function by a decomposition into products
analogous to the decomposition of a polynomial into factors of the first
degree. Let a 0, a 1, f , be an infinite sequence of points different to 0. The
product:
56
T H E S T R U C T U R E O F A D OM A IN OF M AGNIT UDE S
% = % '` 1 - az je
n=3
n=0
k
z + 1 z 2+f 1 z n
` j
` j
an 2 an
kn an
n
1
under certain conditions represents an integral function which admits the
points a 0, a 1, f a n f as zeros, and any integral function that also admits
these points as zeros is represented by the expression:
G (z) = e H(z) . %
in which H(z) is an arbitrary integral function.
Now envisage the meromorphic functions, that is, the functions that
admit, in the finite, a finite number of poles a i f a n , in the neighborhood
of which their expansion into series has the form:
f (z) =
A1
A2
+
+ f A m + G (z); (a i = a 1 fa n)
(z - a i)
(z - a i) m (z - a i) m - 1
,
G(z) being an integral function.
The set of fractional terms of this development constitutes the principal
part
gi c
1
m
z - ai
of f(z) corresponding to the pole a i .
This having been posed, it is possible to give in advance the poles and
principal parts of a meromorphic function and thus reconstruct this function from partial fractions (the different principal parts), each admitting no
more than one pole:
(VII)
f (z) = g 1 c
1
1
1
m + g2 c
m + f gp c
m + G (z)
z - a1
z - a2
z - an
The decomposition of an integral function into products, and the decomposition of a meromorphic function into partial fractions are essentially
decompositions of the first kind, because they reveal the very existence of
an infinity of functions admitting given points as zeros or poles, but they
reveal nothing about the global structure of this infinity. They are attached
exclusively in effect to the individual properties of the functions studied
and no at all to the organization of the class that these functions could
constitute. We will now see the dimensional considerations reappear at the
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
heart of these problems, and the algebraic solidarity between elements of
the same set be established once again.
The Riemann-Roch theorem proposes to study the structure of the set of
functions that admit as poles on a Riemann surface n given points a 1 f a n ,
which are uniform everywhere on the surface (Cf. Osgood 1907, part 2).
By virtue of the theorem of decomposition into partial fractions, the
functions studied can be put in the following form:
(VIII)
F (z) = A 1 Ya (z) + A 2 Ya (z) f + A n Ya (z) + C
1
2
n
in which Ya (z) are Abelian integrals of the second kind which each only
admit one pole, the point a i . This decomposition is absolutely equivalent to
(VII). The functions F(z) are presented in the form of a sum of functions,
from a certain point of view simpler than the function F(z) since they each
have only one pole. In any case these functions are not themselves parts of
the set of functions F(z). The decomposition obtained is therefore a decomposition of the first kind, and it is the determination of the coefficients
Ai that will introduce into this problem of analysis a dimensional point of
view of algebraic inspiration.
It is known that (Cf. Lautman, 1938b, ch. 3) a multiply connected
Riemann surface can be rendered simply connected by a ‘canonical cutting’.
Here is briefly what is meant by this: the Riemann surface can be brought
about by continuous deformation under the form of a disk with 2 sides of
pierced with p holes; p being the genus of surface. We can then trace on
this surface a maximal system of 2p closed curves that are not reducible to
a point by continuous deformation and such that none of these curves
divide the surface into two separate regions. By cutting the surface according to these 2p curves, a simply connected surface is obtained. When the
variable passes from one side of one of these cuts to the other, the abelian
integrals are subject to a jump which is manifested by the existence of a
discontinuity in the values of the function. The abelian integrals of the
second kind can be demonstrated to only have discontinuities different to
0 for the p half of the 2p retrosections traced on the surface.
Let Q 1 (a 1), Q 2 (a 1) f Q p (a 1) be the p discontinuities of the integral Ya (x) of
the formula (VIII); likewise, let Q 1 (a 1), Q 1 (a 2) f Q 1 (a n) be the discontinuities
of the functions Ya (z) f Ya (z) . So that the function F(z) is uniform on the p
cuts where each of these components is subject to a discontinuity, it is
necessary that there are the following p relations:
i
1
2
58
n
T H E S T R U C T U R E O F A D OM A IN OF M AGNIT UDE S
A 1 Q 1 (a 1) + A 2 Q 1 (a 2) + ff A n Q 1 (a n) = 0
A 1 Q 2 (a 1) + A 2 Q 2 (a 2) + ff A n Q 2 (a n) = 0
fffffffffffffffff
fffffffffffffffff
A 1 Q p (a 1) + A 2 Q p (a 2) + ff A n Q p (a n) = 0
There is therefore a system of algebraic equations to n columns and p rows.
To simplify, let p be the rank of this system. We know that n – p coefficients
A 1 f A n - p can be chosen arbitrarily and the p others are expressed in functions of Q i and the n – p arbitrary coefficients. The result of that which
precedes is that the most general function that admits the points a i f a n as
poles depends on n – p similar functions, that is, we have
(IX)
F (z) = A 1 F1 (z) + ff A n - p Fn - p (z) + C
This decomposition is eminently a decomposition of the type which we
have called the second kind. The functions F1 (z) f Fn - p (z) belong to the set
of functions F(z) sought after. They form in that space an (n – p)-dimensional basic system and all the others let themselves be obtained by the
linear combination of these basic functions.5
The comparison of decomposition (IX) and decomposition (VIII) is
extremely suggestive. It shows us how the same mathematical entities, in
this case the functions F(z), can be decomposed in two distinct ways: the first
decomposition characterizes their individual properties, the second the links
that they support between them. These are indeed two modes of thought of
different inspiration, and whose reunion is the work of Riemann.
The examples that have been given thus permit us to understand that
if there are different ways of thinking in mathematics, it is very unlikely
that these differences in method correspond to differences of domain. The
duality of the types of decomposition which have been stressed throughout this chapter is a certain fact that asserts itself for any observer, but this
duality of methods does not end up constituting two different mathematics, that which would be a promotion of the arithmetic of whole numbers,
and that which would be an extension of algebra. The same entities are
able to be studied in both ways and it is this encounter of methods that
gives rise to the profound unity of mathematics.6
59
CHAPTER 2
The Domain and Numbers: Non-Euclidean Metrics
in the Theory of Analytic Functions
We have shown, in the first chapter of our principal thesis (Lautman
1938b), the differences between the synthetic conception and the infinitesimal conception of geometry. The Riemannian definition of a space by
the formula that gives the distance between two infinitely near points is a
purely local conception, the entire space is constructed by the step by step
juxtaposition of infinitesimal neighborhoods, whose assemblage is not,
without new conventions, amenable to collective characterization. On the
contrary, the definition of the properties of a space by those of the group of
transformations whose properties in question are invariants, is a global
definition which envisages the possibilities of group action on the totality
of points of the space. It is also often possible to define an invariant metric
in ‘global’ spaces with respect to the group that operates on the space, but
this metric can be very different from the Riemannian metric.1 These in
effect only appeal to notions defined in the neighborhood of each point of
origin considered, while the formula for an invariant metric with respect to
a group, contains in general terms referring to properties of the whole
space. What is seen to appear here is a new aspect of the opposition of the
two mathematics. In the local conception, the expression of numbers
which form the metric of the surface are given prior to the constitution of
the surface since the surface results precisely from the bringing together of
neighborhoods defined by the value of the distance between two infinitely
near points. In the synthetic conception, space preexists the metric, since
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T H E D O M A I N A N D N U MBERS: NON-EUCLIDEAN
the geometry of the entire space influences the determination of the metric. Numbers are then posterior to the domain and it is particularly significant in this regard that Weyl alludes to projective geometry in the text that
these pages are intended to comment on.
Projective geometry has in effect provided the first example of a geometry in which a number is attached to any pair of points that is defined
independently of any reference to Euclidean distance and that satisfies
nonetheless certain formal conditions which allow it to be considered
abstractly as the ‘distance’ between two points on this space. Chief among
these conditions is the additivity of distance: let d (a, b) be the number d
attached to the pair of points (a, b) ; it is necessary that
d (a, b) + d (b, c) $ d (a, c)
The formal axioms that define the notion of distance can be satisfied by an
infinity of different systems of numbers, the object of abstract geometry is
to choose one that ensures the invariance of the metric with respect to the
overall transformations that are envisaged on the surface. In projective
geometry, for example, the distance between two points is defined by the
anharmonic ratio attached to the ‘play’ of the four following points: the
two given points and the intersections of the projective line passing through
these two points with the conic which is globally transformed in itself by a
sub-group of the group of projective transformations. It is well known how
Klein operated the union of notions of projective metric, due to Cayley and
non-Euclidean geometries: Riemann’s geometry corresponds to the case in
which the ‘absolute’, that is, the projectively invariant conic, is imaginary;
Lobatchewsky’s geometry corresponds to the case of a real conic; and
Euclid’s geometry to the case in which the conic degenerates into two
imaginary points. From the point of view that interests us here, the nonEuclidean geometries and projective geometry are conflated with one
another: what is seen in effect is the appropriation of a system of numbers
by a space that results from the prior consideration of the structural properties of the space, which thus receives a system of numbers on the space
made specifically for it.
What is proposed for the remainder of this chapter is the rediscovery of
the part played by non-Euclidean metrics in the theory of analytic functions, and the fact that it is of great philosophical importance. Weyl wanted
in effect to arrange all of analysis in the mathematics that issues from the
notion of whole number, in order to oppose it to the modern mathematics
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
of domains. It is certain that even the applications of analysis to geometry,
such as it is conceived in France by the school of Monge, and in Germany
by that of Gauss, are only geometric interpretations of analytical results
obtained independently of any reference to geometry.2 This concern to
eliminate geometric intuitions from analysis is moreover in the spirit of
Cartesianism, for which synthetic geometry is imaginative work and
remains well short of the infinite possibilities of calculation carried out
uniquely by the understanding.
This difference of planes between a purely intellectual analysis and a
synthetic geometry that’s more sensible has in our day almost completely
vanished, because topology and group theory have allowed the clarification of synthetic methods in geometry, which are just as intellectual as
methods of analysis. A reversal has thus been made in the relations of
analysis and geometry, and analysis has once again, not been ‘restricted to
the consideration of figures’ but more or less subordinated to a prior topological study of the domain of definition of the functions envisaged. This
new aspect of the theory of analytic functions is related to the development of the theory of the Riemann surface of an analytic function. We will
only envisage here the consequences relative to the metric.
When Poincaré used a non-Euclidean metric in a problem of analysis,
the first it seems, he thought he was witnessing the accidental and almost
inexplicable encounter between two orders of thought totally foreign to
one another. Non-Euclidean geometry had seemed up until then to be ‘a
simple mind game that was only of interest to the philosopher, without
being of any use to the mathematician’,3 and it found itself to be essential
in the uniformization theory of algebraic functions. It is necessary to insist
for a moment on the conditions that brought about the connection of the
old and the new mathematics.
We have already had occasion to define the problem of uniformization
(Cf. Lautman 1938b, ch. 3). Given an analytic function g = f (z) , this function in general on the plane of the complex variable z, possesses points of
ramification in the neighborhood of which it can take several distinct values. To uniformize the function g = f (z) is to find a complex variable t such
that g = { (t) and z = } (t) are two uniform functions on the complex
plane t. The resolution of this problem requires not only the construction
of the Riemann surface F corresponding to the function g = f (z) , but the
construction of the universal covering surface F of the surface F. It is possible in effect to establish a conformal correspondence between this simple
convex surface F and the totality or a portion of the plane of a complex
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T H E D O M A I N A N D N U MBERS: NON-EUCLIDEAN
variable t. From the existence of this conformal correspondence, the existence on the plane t of the sought after uniform functions can then be easily
deduced. There exist in this respect three types of universal covering surfaces: those whose conformal representation can be made on the open
complex plane; those whose conformal representation can only be made
on the plane completed by the point at infinity and are topologically equivalent to the complex sphere; and finally those whose representation can be
made on the unit circle. Let us now place ourselves in the latter case, by
envisaging for greater simplicity only a correspondence between interior
points of the surface and points interior to the circle. The surface F admits
a discontinuous group of internal transformations, that is, a point Po of this
surface can be made to correspond to as many points P1, P2 . . . as there are
on the Riemann surface of the given function, and as many classes of
closed paths, irreducible to one another by continuous deformation, and
issuing from the point po corresponding to Po. To the group of internal
transformations of the surface F there corresponds the discontinuous
group of linear transformations at the interior of the unit circle, and it
turns out that the invariant metric by this group coincides with the metric
of Lobatchewsky’s geometry at the interior of the circle z 1 1 . By agreeing arbitrarily to call the non-Euclidean distance between two points on
the surface F the non-Euclidean distance between corresponding points at
the interior of the unit circle, Poincaré thus obtained on the surface F a
‘hyperbolic’ metric connected to the topological fact that F was representable on the unit circle.
Let ds be the Euclidean length of an arc of the conformal representation
on the unit circle, r the distance from this arc to the center of the circle, and
dv the hyperbolic (or non-Euclidean) length of the arc, then
dv =
ds
1 - r2
In the case in which the surface F is of the elliptical type and therefore representable on the complex sphere, the spherical distance between two points
can be called the distance between corresponding points on the sphere,
dv =
ds
1 + r2
and this distance is invariant with respect to the group of rotations of the
sphere.
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
The Euclidean length of an arc is evidently unique, since the metric is
not defined with respect to special properties of the surface, but the nonEuclidean lengths being numbers attached by convention to an arc, it is
perfectly possible to confer several different non-Euclidean metrics to an
arc. Consider for example the simply connected surfaces representable on
the unit circle. Several different metrics can be conferred to them at the
same time; in particular a hyperbolic metric
dv =
ds
1 - r2
invariant under the group of transformations internal to the circle, and a
spherical metric
dv =
ds
,
1 + r2
that receives the unit circle when envisaged as a portion of the sphere. The
choice of metric is then controlled by the nature of the invariants that are
sought to be obtained on the surface.
We will now see, following the book of Nevanlinna (1936), how such
considerations have currently made a name for themselves in a central
problem of the theory of analytic functions, that of Picard’s exceptional
values. Consider a meromorphic function at the interior of any circle of
radius r, r 1 R 1 3 , with its center the point at the origin o of the complex
plane. It is a matter of studying the ‘affinity’ of the function w(z) for the
value a of this function. Nevanlinna introduced two expressions for this, of
which one, N (r, a) is related to the number of z values interior to the circle
r of center o for which w(z) effectively takes the value a ; and the other,
m (r, a) , measures as it were the average intensity with which the values of
w(z) gather around a on the circumference z = r . There exists then a
remarkable swing between the frequency with which the function w(z)
reaches any value a and the average of these deviations with respect to a .
Nevanlinna has in fact proved the following theorem:
m (r, a) + N (r, a) = T (r) + a bounded quantity;
in which T(r) is a constant relative to the circle z = r .
If in the circle z = r , w(z) only rarely takes the value a , the function of
frequency N (r, a) has a very low value which may be zero in the case in
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T H E D O M A I N A N D N U MBERS: NON-EUCLIDEAN
which a is an exceptional Picard value. On the other hand, the function
w(z) deviates very little in average from the value a on the circle z = r ,
the function m (r, a) is all the greater, and the sum m (r, a) + N (r, a) thus
reaches the constant value T(r). This theorem is of great important since it
shows that in the study of exceptional values, it does not suffice to seek the
values that a function never reaches, which is what was done previously,
but it is necessary to also consider how the function approaches them.
Consider for example the exponential function fz which allows as exceptional values a = 0 and a = 3 ; then, by calling O(1) a quantity which
remains bounded (of the order of 1):
N (r, a) = 0, m (r, a) = r for a = 0 and a = 3
r
N (r, a) = r + O (1), m (r, a) = O (1)
r
for an ordinary value of a , so that in all cases:
m (r, a) + N (r, a) = r potentially O(1),
r
r
r is the characteristic function T(r) attached to the function e z . For a fixed
value z = r , T(r) is an invariant and, through the notion of a nonEuclidean metric on a Riemann surface, it has been possible for Ahlfors
(1929) and Shimizu (1929) to give a geometric interpretation of this invariant.4 Consider the complex plane of the variable z and the complex
plane of variable w whose points represent the values of the meromorphic
function w(z) for z 1 R # 3 . Let F be the area on the plane w corresponding to the circle z 1 r of the plane z. Let us introduce a spherical metric
on the plane w. This metric associates a ‘spherical’ value to the area F, that
of the stereographic projection of F on the unit sphere. Let A(r) be the
spherical value (divided by r ) of this area F, which on the plane w corresponds to the circle z = r of the plane of the complex variable z. A(r) is
thus an invariant attached to the circle of radius r, and there is between
Nevanlinna’s invariant T(r) and the invariant A(r) of the geometric theory,
the relations:
dT (r)
= A (r) or again
d log r
#
0
r
A (t)
dt
t
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
The invariant of Nevanlinna’s analytic theory therefore gets its whole
meaning by the introduction on the envisaged Riemann surface of a global
metric specially appropriate for the topology of this surface: a spherical
metric.
A geometric interpretation of certain other theorems of the theory of
meromorphic functions can also be arrived at by introducing this time
a Lobatchewsky–Poincaré hyperbolic metric (Cf. Nevanlinna 1936, 242
[1970, 248]), and all the theorems of the theory can be retrieved by envisaging, as did Ahlfors, any metric satisfying the general axioms mentioned
at the beginning of this chapter. In all cases, what results from these examples, through the definition of the metric on a Riemann surface, is that the
theory of analytic functions is very much acquainted with the ideas of the
new mathematics, which coordinates in each domain a suitable system of
numbers, selected with regard to the structure of this domain, and thereby
establishes to a certain extent the primacy of geometric synthesis over
‘numerical’ analysis.
66
CHAPTER 3
The Algebra of Non-Commutative Magnitudes:
Pfaffian Forms and the Theory of Differential
Equations
One aspect of the new mathematics that we propose to examine here is
relative to the non-commutativity of multiplication in certain of the most
important modern algebraic theories. Let a and b be two magnitudes of a
theory. This theory is non-commutative if it has ab ! ba . This is a property
that profoundly distinguishes these theories from ordinary arithmetic and
algebra and whose richness is considerable, since it contains the key to
uncertainty relations in quantum mechanics.
Schrödinger’s mechanics associates in effect an abstract expression called
an ‘operator’ to any physical magnitude, and two magnitudes A and B are
simultaneously measurable only if the operators corresponding to them
are permutable, that is, if the multiplication of these operators is commutative. Now, if the operator Qk is made to correspond1 to a coordinate qk, in
which the operator Qk signifies multiplication by qk, and the operator
Pk =- h 2
2ri 2q k
is made to correspond to the quantity of movement according to the coordinate qk, then it is found that these operators are not permutable:
- h 2 .q k =- h q k 2 - h
2ri 2q k
2ri 2q k 2ri
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
or again: Q k Pk - Pk Q k = h
2ri
What follows from these equations is that a coordinate and the corresponding quantity of momentum are not simultaneously measurable and the
indeterminacy that affects the system of the two measures is itself measured by the value of a difference of the type ab - ba .
The operator theory of quantum mechanics, as Von Neumann has
shown (1935), is only a special case of the general theory of rings2 of
operators in Hilbert space, and the problems related to the commutativity
or non-commutativity of two elements in the study of the structure of
these rings play the same role as in the study of rings of algebraic numbers.
This structure is in effect characterized by the possibilities of dimensional
decomposition of the ring directly into components, analogous to the
global decompositions that was described in Chapter 1, and the point of
view of the commutativity of multiplication plays a part by the fact that the
global decomposition of the ring, in certain cases, is interdependent with a
global decomposition of the center of the ring. The latter is formed by the
set of elements a that ‘commute’ with all the others, and the connection
between the decomposition of the ring and that of its center is as follows:
if the ring O is decomposed directly into sub-rings (having certain specific
properties)3
O = o 1 + ff o n
the center Z of the ring is decomposed directly into the centers of different
sub-rings of the decomposition of O
Z = Z 1 + ff Z n
It was shown in Chapter 1 how the theories of modern algebra can be
characterized by the importance played by the theorem of dimensional
decomposition, and the close relation that unites the existence of these
decompositions to the distinction between commutative and noncommutative multiplication can now be seen.
In the rest of this chapter we intend to show how, thanks to the work
of Cartan and his predecessors, the mode of thought that is essential to
modern algebra, which is the calculus of non-commutative magnitudes,
has penetrated contemporary analysis. We will first briefly define the
calculus of Pfaffian forms and then envisage their application to analysis.
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T H E A L G E B R A O F N O N - C O M M UTAT IV E M AGNIT UDE S
Given two series of differentials4
(I)
dx 1, dx 2 ff dx n
(II)
dx 1, dx 2 ff dx n
let us say, following Grassmann’s notation of exterior products,
6dx i, dx [email protected] = dx i dx j - dx i dx j
Now let two Pfaffian forms be:
~ (d) = a 1 dx 1 + a 2 dx 2 + f a n dx n
s (d) = b 1 dx 1 + b 2 dx 2 + f b n dx n
the exterior product of these two forms, and by 6~, [email protected] the bilinear expression with respect to the two series of variables (I) and (II) is noted:
6~, [email protected] = ~ (d) s (d) - ~ (d) s (d) = / a i b i 6dx i, dx [email protected]
Such symbolic multiplication is essentially non-commutative and we
have:
6~, [email protected] =-6s, [email protected]
Cartan also established a ‘symbolic derivation’ of Pfaffian forms:
Let ~ (d) = a 1 dx 1 + f a n dx n be a form to n variables; by defining the symbolic derivative or exterior derivative of ~ (d) by d~ , we have:
n
(III)
d~ = d~ (d) - d~ (d) = / da j dx j - da j d j
i
2a j
m6dx j, dx [email protected]
= / c 2a k 2x j
2x k
(fk)
= / a jk 6dx j, dx [email protected]
The exterior products / a j b j 6dx i, dx [email protected] and the exterior derivatives
/ a jk 6dx i, dx [email protected] are differential forms of degree 2. Differential forms of any
degree can even be defined by successive multiplications or derivations,
and the set of these differential forms, univocally satisfying a law of
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
addition and a law of multiplication, constitutes a ring comparable to
operator rings and the rings of algebraic numbers which were spoken of
above.
We will now see the immediate connection that Cartan established
between the algebra of Pfaffian forms, as just defined, and the analytic
theory of differential equations.5
Consider the second order differential equation
d2 y
dy
= F c x, y, m
dx
dx 2
Let us pose
~1 = dy - yl dx
~2 = dyl - F (x, y, yl ) dx
Two Pfaffian forms are obtained to 3 variables dy, dy´ and dx; and the differential equation is reduced to the following Pfaffian system
~1 = 0
~2 = 0
This example allows us to understand how Cartan was able, in general, to
reduce the study of systems of differential equations and partial differential
equations, to the study of corresponding algebraic-geometric Pfaffian systems. This is a huge domain in which the methods of algebra are again seen
to penetrate the domain of analysis. We will try to give an idea of this by
first envisaging equivalence problems.
Consider two systems of Pfaffian forms ~1 f ~n , s1 f sn , the first constructed with the variables x 1 f x n and their differentials, the second with
the variables x 1 f x n and their differentials. The systems are said to be
equivalent if they are transformable into one another by analytical transformations making the variables pass from x i to x i = {i (x 1 f x n) , the {i
being analytic functions. This problem is doubly a problem of analysis: first
of all by the character of the functions {i whose existence is sought to be
demonstrated, and secondly because in the applications, the Pfaffian forms
envisaged always correspond to the systems of differential equations or
partial differential equations. Cartan’s methods then allow this problem of
analysis to be substituted with a problem of algebra: for the two Pfaffian
systems ~1 f ~n , s1 f sn to be equivalent as defined by analysis, it is
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T H E A L G E B R A O F N O N - C O M M UTAT IV E M AGNIT UDE S
necessary that in two corresponding points (x, x ) both systems of forms,
considered as algebraic forms in dxi and dx i , are transformable into one
another by a linear substitution, that is, that we have:
(IV)
si = a i ~i + f a in ~n
The existence of {i functions is connected to the conditions of solvability
of the algebraic system (IV). It is even possible to substitute for the forms
~i and si the forms ~i* = si* such that the problem (IV) is reduced to the
search for the conditions of compatibility of equations
(V)
si* = ~i*
In conformity with the general theory of Pfaffian systems, it is necessary to
add to equations (V) the equations between the exterior derivatives:
dsi* = d~i*
or again by virtue of (III)
(VI)
/c
i
jk
6~*j , ~k*@ = / c ijk 6s*j , sk*@
We see that the coefficients c ijk must be invariant. Let us now place ourselves in the most important particular case, that in which these invariants
are constants. It will be demonstrated that the constancy of these algebraic
coefficients is a necessary and sufficient condition for the existence of the
sought after analytic transformations. There is still more: the transformations {i form a finite continuous group to n parameters and the c ijk are the
algebraic structural constants of this group as defined by Lie and Cartan.
The problems relative to the equivalent of differential equations or of differential forms, problems of analysis, are thus resolved by the 3rd fundamental theorem of the algebraic theory of Lie groups: given a system of
constants c ijk satisfying certain determined algebraic relations, there is a
finite continuous group of transformations admitting the c ijk as structural
constants and leaving the Pfaffian forms, whose exterior derivatives have
these same constants as coefficients, invariant.
In a simple example given by Cartan, it will be shown how the existence
of continuous transformations can result from the algebraic conditions of
compatibility of two Pfaffian systems: that is to find the conditions of applicability of two surfaces defined by ds2. The ds2 of each surface is defined as
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
a quadratic differential form of coordinates x1, x2 for the first, x 1, x 2 for the
second, the surfaces are applicable to one another with conservation of the
metric, if there are specific biunivocal and bicontinuous correspondences
between xi and x i . It is possible to find the conditions of existence of these
continuous correspondences by considering the Pfaffian forms attached to
the ds2 surfaces.
The calculus of the exterior derivative of one of these forms gives in
effect:
d~*3 =- K 6~*1, ~*[email protected]
The quantity K playing the role of structural coefficients of formula (VI)
requires that the surfaces are applicable, that this quantity, formed from
the coefficients of the ds2 of each surface, is the same for both surfaces. This
is the total curvature as defined by differential geometry. This theory thus
gives, in a purely algebraic way, a classic result of the applications of analysis to geometry.
The role of the non-commutative calculus of Pfaffian forms in analysis is
therefore considerable. It allows, in many cases, a substitution of algebra
for analysis to be operated as follows: the theory of differential equations
reduces to that of Pfaffian forms; which merges with the theory of continuous Lie groups, and the latter has become, thanks to Cartan, an algebraic
theory. The fertility of these algebraic methods is even manifested in the
problems that have always seemed to be the center of analysis: those that
relate to the integration of differential equations or partial differential
equations. The infinitesimal transformations which constitute a Lie group
can in effect, in some cases,6 be defined as solutions to a system of partial
differential equations giving the transformed variables based on primitive
variables. It is however not directly by the integration of these equations to
partial derivatives that Cartan ensures the existence of the functions sought
after, but by the prior algebraic study of structural coefficients of Pfaffian
forms attached to the proposed equations. We encounter here considerations that will be further developed in Chapter 4: to a certain extent, the
existence of continuous functions depends on the structural properties
of a discontinuous system of algebraic magnitudes. The calculus of noncommutative magnitudes in analysis thus carries out a reconciliation of
the continuous and the discontinuous in which the distinction between
the two mathematics is erased.
72
CHAPTER 4
The Continuous and the Discontinuous:
Analysis and the Theory of Numbers
We’ve already seen in the Introduction to this paper how the distinction
between Weyl’s two mathematics could be interpreted in terms of an
opposition between the mathematics of the finite and that of the infinite,
or more exactly the countable and the continuous. These are the two
essentially distinct mathematics, since operations can be defined in the
domain of the continuous that have no meaning in the discontinuous, like
analytic continuity in the theory of functions. To establish a theory of relations of the continuous or the discontinuous, of arithmetic and analysis, is
then a fundamental problem for mathematical philosophy.
Two different mathematical disciplines can be interrogated in this regard:
mathematical logic and number theory, and both cases are found to be
faced with a similarly complicated situation.
One of the most important facts in the reconstruction of the foundations
of mathematics undertaken by the school of set theorists, Dedekind, Cantor, Frege and Russell, was certainly to support the definition of whole and
finite numbers of ordinary arithmetic over the consideration of infinite sets
or an infinity of sets. It is thus that the simple addition of whole numbers
in Russell is based on the application of the transfinite axiom of choice.
While paying tribute to the work of his predecessors, Hilbert judged the
route that they were committed to as impractical and showed how the
axiomatization of analysis presupposed the axiomatization of arithmetic,
without the latter supporting the former. It cannot yet be asserted that
Hilbert’s position has eliminated all possibility of envisaging the primacy of
the infinite over the finite, and this for reasons arising first of all from
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
mathematical logic itself. Cavaillès has moreover extensively studied the
meaning of Gödel’s results which demonstrated that it was impossible to
prove the consistency of arithmetic using only means borrowed from arithmetic. This logical discovery takes on a singular importance when it is
approached from the historical development of effective arithmetic. A very
large number of arithmetic results could only have been achieved by
appealing to very powerful analytic means. An analytic theory of numbers
has thus developed since Dirichlet, and whatever hope some arithmeticians have of one day eliminating transcendent proofs from number theory, the acquired fact remains of the close connection between arithmetic
and analysis.
The problem of the relations between the continuous and discontinuous
is therefore presented under a new aspect. It is no longer a matter of
whether or not the existence of an analytic number theory is compatible
with the logical priority of the axioms of the discontinuous with respect to
the axioms of the continuous, but rather to study, within analytic number
theory itself, the mechanism of the connections that are asserted between
the continuous and the discontinuous.
Certain modern mathematicians distinguish in this respect two kinds of
problems: there are those, such as Fueter, in which the methods of the
theory of functions are of use in the solution of purely arithmetic problems, and in which the theory of ideals (that is, higher arithmetic and
algebra) is essential for the construction of certain analytic functions
(Fueter 1932, 83). There is would be in short a distinction between problems in which analysis is of use to arithmetic and in which arithmetic is of
use to analysis. That the two kinds of problems exist is certain, but they
are, as we shall see, reciprocal to one another to the extent that it is impossible to study them separately. This seems to be the opinion of other
authors, such as Hecke, who, a student of certain functions of considerable
importance in arithmetic, considers that advances in analytic number theory now require the deliberate departure from arithmetic to construct
these functions (Hecke 1927, 213). It is impossible to proceed here with a
study of the whole of analytic number theory, which is among the most
difficult theories of all mathematics. We would simply like to show, with a
few precise examples, how it can be argued that if certain analytic functions are of use to arithmetic, then the definition of these functions is
already based on the arithmetic structure of these fields whose study is
contributed to by these functions. The two theories distinguished by
Fueter, analytic number theory and what might be called the theory of the
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T H E C O N T I N U O U S A ND T HE DISC ONT INUOUS
arithmetic origins of certain analytic functions, are probably only one
because the latter absorbs the former. In this domain, as with most of those
discussed above, the unity of algebra and analysis is effected by the role of
the structural and finitist characteristics of the algebra in the genesis of the
continuous.
Let k be the field of rational numbers. Riemann defined on this field a
function
3
g (s) =
/ n1
s
n=1
in which n is any rational number > 0 of k, and in which s is a complex
variable whose real part is always > 1. This function also admits a representation in the form of a product:
g (s) = %
1
1 - 1s
p
in which p is any prime of k. We can guess by this the close link that unites
the function g (s) to the distribution of primes p in the field k. Thus Riemann
was able, by means of this function, to construct a function F(x) which
gives the number of primes less than an arbitrary positive number.
Dirichlet and Dedekind generalized the function g (s) by defining, for an
any field K, a function gK (s) also able to be represented as a sum and as a
product of an infinity of terms. These terms are the ideal norms of the field
K. Here’s what should be understood by this: what’s called the ideal of a
field of algebraic numbers is any set of numbers of the field such that:
a) If a is part of the set, then so is ma , whatever the integer m .
b) If b is another element of this set, a + b is also a part.
c) There is an integer n ! 0 such that for all a of this set, na is an
integer.
Let A be an ideal of an algebraic field K. The norm N (A) of this ideal is a
fixed number, attached to A in the field K in which it is considered.1
That being said, for the function gK (s) of any field we have:
gK (s) = /
1
1
=%
1
6N (A)@s
16N (P) [email protected]
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
the sum being extended to all ideals numbers of the field K different from
0, the product has all the prime ideals of K. It is generally difficult to use
these formulas to express the function gK (s) of any field. An expression
like
%
1
1-
1
N (P) s
can in effect be formed from the function
gK (s) = %
1
1 - 1s
P
relative to the field of rationals only if the laws of decomposition in K of
rational primes are known. On the other hand, if we can know the function gK (s) of a field K, or at least some of its properties, other than by the
knowledge of the way that rational primes decompose into prime ideals of
K, this function can then be used to study the laws relative to the prime
ideals of K. For example, we can prove by means of the function gK (s) and
certain others that are connected to it, that there exist an infinity of prime
ideals in any class of ideals2 of the field K.3
This result is characteristic of analytic number theory. An analytic function is seen in effect to be of use in the determination of results relative to
arithmetic notions of the discontinuous: numbers and prime ideals. This
priority of analysis with respect to arithmetic is, however, only apparent,
and we’ll see how the arithmetic utilization on the field K of the function
gK (s) is possible only because the determination of this function already
implies the knowledge of certain structural and discontinuous properties of
the base field. For example, the theorem mentioned above and relative to
the distribution of prime ideals according to the classes of ideals of K, is
based, as has been said, on a prior knowledge of certain properties of
gK (s) , which are those that result from the formula:
lim
(s - 1) gK (s) = h\
s=1
in which h is the number of classes of the field K and \ is another invariant
also attached to the field K. What in our view is essential, in the previous
equality, is that the determination of the function gK (s) is thus based
on the prior arithmetic decomposition of the ideals of K in h ideal classes.
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T H E C O N T I N U O U S A ND T HE DISC ONT INUOUS
This is no longer, strictly speaking, analytic number theory, but rather a
fact of the arithmetic theory of origins of certain analytic functions. In sum
the following schema is obtained: the decomposition into classes of field
K permits the conclusion that the function gK (s) exists, the knowledge
of which can, by a rebound effect on the base field, be of use in a more
profound study of this field.
The previous example again shows only a global relation between the
number of classes of a field and the analytic functions gK (s) . It is possible,
in certain cases, to associate the genesis of certain analytic functions more
closely to the discontinuous domain that results from the decomposition
of a set into classes of equivalent elements. It may then be that these
functions are later likely to be of use in exploring the more hidden arithmetic properties of this field on which they depend, but their very existence
as analytic functions connected to a domain of discontinuity is of considerable mathematical and philosophical interest. Considering that Weyl
described the primacy of a domain with respect to algebraic entities defined
on this domain as a characteristic fact of the new mathematics, we see a
dependency with respect to a basic domain appear by analogy with the
Abhängigkeit vom Grundkörper (Dependency on the base-field) of algebra,
even within the theory of functions, in which the priority of the idea of
geometric domain with respect to that of number or function is equally
asserted in analysis.
Let us further study in this regard the case of automorphic functions.
Consider in the plane of the complex variable z the substitutions defined by
relations:
zl =
az + b
cz + d
a, b, c, d being whole numbers such that ad - bc = 1
These substitutions are demonstrated to form a discontinuous group G,
what’s called the nodular group, and any function f(z) such that:
f (z) = f e
az + b
o, ad - bc = 1
cz + d
is called an automorphic function with respect to this group.
To better see the connection between the discontinuity of the group and
the existence of a continuous automorphic function, we will define what
is called the domain of discontinuity of the group.
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
The group operates a division into classes of points in the complex plane,
and two points belong to the same class, when a substitution S of the group
exists for which we have z´ = Sz. The transferable set of points by a substitution of the group obviously forms a discontinuous set. It is difficult to
demonstrate such a set, but the discontinuity of the group can nevertheless
be materialized as follows: to do this let us envisage a domain such that
every point in the complex plane is equivalent to one and one only point
of this domain. A similar domain which contains only points that are nonequivalent attests to the discontinuity of the modular group since any
transformation of the group concerned with a point of the domain of discontinuity transforms this point into a point that does not belong to this
domain.
The figure below thus represents a domain of discontinuity or fundamental domain of the modular group. It is located outside of the unit circle
and is limited by the straight lines (Figure 1):
z = ! 1 + ic
2
y
i
x
+1
Figure 1
This domain of discontinuity permits the immediate definition of an automorphic function connected to this modular group. There exists in effect
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T H E C O N T I N U O U S A ND T HE DISC ONT INUOUS
a function w = J(z) (the modular function) that assures the conformal representation of the domain of discontinuity of the modular group on the
totality of the complex plane w with two cuts - 3 " 0 and 1 " + 3 , and
this modular function is an automorphic function of the group G. We have
in effect J(z) = J(Sz), S being a substitution of G.
The modular function J(z) whose determination is thus based on the
discontinuity of a basic domain, is, by what we’ve called a kind of rebound
effect, amenable to very rich applications in number theory. In effect, let
K be an imaginary quadratic field (obtained by adjoining a number
m , m 1 0 , to the field of rationals). Each class of ideals of this field is
defined by a number of the form
i = c + m [a, c, m are integers and rational, in addition
a
m 1 0 and c 2 - m / 0 ].
For two classes of ideals to be equal, it is necessary and sufficient that the
numbers i and il which correspond to them are equivalent to each other
with respect to the modular group G. There are therefore as many numbers
i1 f ih of the field k´ in the domain of discontinuity of G as there are classes
of ideals in the field. Let h be the number of classes. The function J(z), for
the h values i1 f ih , takes h different ‘singular’ values J (i1) f J (ih) , and
the algebraic interest of these values is considerable. They belong in effect
to an extension field k´ of major importance: the field of classes (Cf. Lautman
1938b, ch. 3).
We therefore obtain a schema of relations of the continuous and the
discontinuous analogous to the one described above. The continuous function J(z) is, without doubt, of use in determining a discontinuous set of
numbers of a new algebraic field,4 but the very existence of this function is
already based on a fundamental discontinuity: that of the domain of discontinuity of the modular group. In this pure example of analytic number
theory, we can therefore observe how the essential moment of the theory
resides in the genesis of a continuous function from the discontinuous
structure that gives rise to it.
79
Conclusion
The introduction to this essay showed how the distinction between Weyl’s
two mathematics tended to oppose the synthetic methods of modern algebra to the analysis of the infinite. Then, over the course of four chapters, it
was shown that this distinction should not be conceived in the sense of an
essential opposition between irreducible disciplines, since it is possible, in
modern theories of analysis, to retrieve the points of view that characterize
algebra. The theory of equations with infinitely many variables, the arithmetic theory of algebraic functions, the theory of non-Euclidean metrics
that are invariant by a discontinuous group, the calculus of Pfaffian forms
and continuous group theory and analytic number theory, these are many
of the intermediate theories between algebra and analysis whose methods
are algebraic and whose results apply to analysis. It is therefore legitimate
to oppose the mathematics of the twentieth century to that of the nineteenth century, but they are opposed like the physics of the continuous
and the discontinuous. Until about 1905, the coexistence of two distinct
areas of facts could have been believed in. That which presented itself as
continuous, light waves; and that, on the contrary, in which a discontinuous structure appeared, material atoms. Since Einstein introduced discontinuity into the study of light, and de Broglie the continuity of waves into
the study of matter, it is impossible to maintain the old idea of domains of
physical facts that are separate from one another. The physics of the continuous represents a mode of treatment by differential equations of physical facts. The physics of discontinuity represents a mode of treatment of the
same facts by other methods: group theory, calculation of matrices, quantum
statistics, etc. There thus exists a certain analogy between contemporary
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physics and contemporary mathematics, in that they offer each other the
spectacle of facts amenable to being studied at the same time by the calculus of the continuous and by the calculus of the discontinuous. But considering that this duality of methods relative to the same facts is the source of
the main problems of contemporary physics, it is contrary to the testimony
of the profound unity of the mathematical sciences.
The continuous and discontinuous in quantum physics, are, in effect,
complementary points of view in Bohr’s sense, that is, our knowledge of
continuous aspects of matter increases to the extent that those aspects of
the discrete decrease, and conversely. They are, writes de Broglie,
like the faces of an object that cannot be contemplated at the same time,
and that nonetheless requires each in turn to be envisaged in order to
describe the object completely. These two aspects, which Bohr called
‘complementary aspects,’ understanding by this that these aspects, on
the one hand, contradict each other and, on the other hand, complement each other. (De Broglie 1937b, 242)
The relations of the continuous and the discontinuous, and of the finite
and the infinite, are very different in mathematics. We do not merely consider special problems, like those of analytic number theory, which was
discussed in Chapter 4, but fully intend to envisage here the general problem of the relation between analysis and algebra. There exist, in this
respect, two classical positions. For one, the continuous emanates from the
discontinuous like the infinite from the finite, by a kind of progressive
enrichment of the finite and the discontinuous. What then happens is
that, in the conditions where the passage to the limit is legitimate, new
properties, which are connected to the continuous and the infinite, and
which have no equivalent in the finite, are suddenly discovered. Consider
for example a polynomial in z (z being a complex variable) that contains a
finite number of terms. A convergent series with an infinity of terms, in
certain cases, represents an analytic function, and, as noted by Montel
(1927), the transition from a finite to an infinite number of terms introduces the new fact of the existence of a singular point without which the
analytic function is reduced to a constant. Analysis thus presented goes
beyond algebra and its results have long seemed to analysts incapable of
being retrieved by simple methods.
The priority of the continuous and the infinite can also be asserted, and
the finite and the discontinuous seen either as a limit of infinity, or as an
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
approximation of infinity. This attitude is perhaps more philosophical than
mathematical: the Cartesian infinite is first with respect to the finite, like
continuity in Bergson is first with respect to the discontinuous. It has in
addition been shown how the infinite is retrieved especially in those disciplines of mathematics that are most in contact with philosophical thinking:
Cantor’s set theory and the mathematical logic of Frege and Russell. It is
no less true that an image of such relations can be retrieved in the authentically mathematical theorems of approximation. We were reminded, for
example, in Chapter 1 of our principal thesis (Laumtan 1938b), how a
continuous function can be approached by polynomials; the continuous
function in this theory is therefore correctly conceived as given anterior to
the infinite discontinuous polynomials that it approaches.
We think it possible to observe in the movement of the mathematics of
the twentieth century a third way of conceiving the relations between
analysis and algebra, the continuous and the discontinuous, the infinite
and the finite. By seeking to give rise to the infinite by the dilation of the
finite, or the finite by the constriction of the infinite, the finite is always
considered to be in extension as a part of the infinite, and the impossibility
of exhausting the infinite can only incessantly be come up against. A very
different attitude is held by many contemporary mathematicians,1 who
see in the finite and the infinite not the two extreme terms of a passage to
be carried out, but two distinct kinds of entities, each endowed with a specific structure, and disposed to support relations of imitation or expression
between them. We understand by relations of imitation the cases where
the internal structure of the infinite mimics that of the finite, and by relations of expression, the cases where the structure of a discontinuous finite
domain envelops the existence of another continuous or infinite domain,
which thus finds itself expressing the existence of this finite domain to
which it has adapted. Some of the examples have shed light on relations of
imitation; others on relations of expression. When Hilbert transports the
dimensional methods of decomposition of algebraic origin into analysis, he
imposes a structure on the functional space which mimics that of a space
that has a finite number of dimensions. When Poincaré envisages a discontinuous group of transformations and the continuous automorphic function attached to this group, he brings together two kinds of entities that are
entirely foreign in nature, but the existence of the continuous function
expresses no less the properties of the domain of discontinuity used to
define it. By thus focusing each time, not on the quantity of the elements,
but on the existence or framework of the entities being compared, structural
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analogies and reciprocal adaptations are thus discovered between the finite
and the infinite, with the result that the unity of mathematics is essentially
that of the logical schemas that govern the organization of their edifices.
We thus rediscover considerations that agree with those that are developed
our principal thesis (1938b). Our Essay on the notions of structure and existence
in mathematics (1938b) tends in effect to show that it is possible to retrieve
in mathematical theories, logical Ideas incarnated in the very movement of
these theories. The structural analogies and adaptations of existence
between analysis and algebra, that we have tried to describe here, have no
other aim than to contribute to shedding light on the existence of logical
schemas at the heart of mathematics, which are only knownable from
within mathematics itself, and to secure both its intellectual unity and
spiritual interest.
83
BOOK II
Essay on the Notions of Structure and
Existence in Mathematics
Introduction:
On the Nature of the Real in Mathematics
This book arises from the sentiment that in the development of mathematics, a reality is asserted that mathematical philosophy has as a function
to recognize and describe. The spectacle of most modern theories of
mathematical philosophy, in this regard, is extremely discouraging. In
most cases, mathematical analysis reveals only very little and that very
poorly, like the search for identity or the tautological character of propositions.1 It is true that in Meyerson the application of rational identity to a
variety of mathematics presupposes a reality that resists identification.
There seems therefore to be an indication here that the nature of this real
is different from the too simplistic schema that is used to try to describe it.
On the contrary, the development of the notion of tautology in Russell’s
school completely eliminated the idea of a reality specific to mathematics.
For Wittgenstein and Carnap, mathematics is no more than a language
that is indifferent to the content that it expresses. Only empirical propositions refer to an objective reality, and mathematics is only a system of
formal transformations allowing the data of physics to connect to each
other. If one tries to understand the reasons for this progressive disappearance of mathematical reality, one may be led to conclude that it results
from the use of the deductive method. By trying to construct all mathematical notions from a small number of notions and primitives logical
propositions, we lose sight of the qualitative and integral character of the
constituted theories. Now, what mathematics leaves for the philosopher to
hope for is a truth which would appear in the harmony of its edifices, and
in this domain as in all others, the search for the primitive notions must
yield place to a synthetic study of the whole. It seems to us in this regard
very strange that after having lead the most complete investigations on
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theories related to number and space, Poincaré had claimed to see in mathematics only a game of symbols devoid of meaning (Cf. Poirier 1932). He
seems to have approached them with the intention of asking of them an
enrichment of the indications that suggest the external perception or inner
sense of the real. The real is foremost to him that of immediate experience,
and abstract theories give us no hold over it. Poirier almost reproaches
these theories for their excessive perfection. The ease with which they
correspond to one another gives the aspect of each of them an arbitrary
character, possible among many others. Nowhere is there impressed upon
the mind the sentiment of a necessity resulting from the nature of things,
and one never finds only formal procedures, which do not respond to a
‘natural and intuitive classification’ of their objects.
We believe it is possible to arrive at less negatives conclusions, and contemporary mathematical philosophy has moreover committed, on two
different routes, to a positive study of mathematical reality. This reality can
in effect be characterized by the way in which it can be grasped and organized, which it can equally be intrinsically, in terms of its own structure. We
will first try to briefly summarize the basic ideas of the two methods.
There is no philosopher today more than Brunschvicg who has developed the idea that the objectivity of mathematics is the work of intelligence, in its effort to overcome the resistance that is opposed to it by the
matter on which it works. This matter is neither simple nor uniform, it has
its folds, its edges, its irregularities, and our conceptions are never more
than a provisional arrangement that allows the mind to go further forward.
Mathematics is constituted like physics: the facts to be explained were
throughout history the paradoxes that the progress of reflection rendered
intelligible by a constant renewal of the meaning of essential notions.
Irrational numbers, the infinitely small, continuous functions without
derivatives, the transcendence of e and of r , the transfinite had all been
accepted by an incomprehensible necessity of fact before there was a
deductive theory of them. They had the fate of these physical constants like
c or h which were essential in an incomprehensible way in the most different domains, up until the genius of Maxwell, Planck and Einstein knew
to see in the constancy of their value the connection of electricity and
light, of light and energy. We thus understand the defiance of Brunschvicg
vis-a-vis any attempts that would deduce the whole of mathematics from
a small number of initial principles. Brunschvicg, in Les étapes de la philosophie mathematique, rose up against the reduction of mathematics to logic,
just as much as against the idea that there might be general mathematical
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principles like Poncelet’s principle of continuity or Hankel’s principle of the
permanence of formal laws. Any effort of a priori deduction tends for him
to reverse the natural order of the mind in mathematical discovery.
Brunschvicg’s mathematical philosophy should not, however, be interpreted as a pure psychology of creative invention:
Between the vicissitudes of invention that are solely of interest to an
individual consciousness, and the forms of discourse that concern above
all the pedagogical tradition, mathematical philosophy delimits the
terrain where the collective acquisition of knowledge is produced, it will
recognize the pathway traced by creative intelligence. (Brunschvicg
1912, 459)
Between the psychology of the mathematician and logical deduction, there
must be room for an intrinsic characterization of the real. It must partake
both of the movement of the intelligence and of logical rigor, without
being mistaken for either one, and this will be our task, to attempt this
synthesis.
The structural point of view to which it is thus also our duty to refer is that
of Hilbert’s metamathematics. We know the difference that separates the
Hilbertian conception of mathematics from that of Russell and Whitehead’s
Principia Mathematica (1910). Hilbert has replaced the method of genetic
definitions with that of axiomatic definitions, and far from claiming to
reconstruct the whole of mathematics from logic, introduced on the contrary, by passing from logic to arithmetic and from arithmetic to analysis,
new variables and new axioms which extend each time the domain of
consequences. Here is, for example, according to Bernays (see Bernays
1935, 196–216), who in the complete works of Hilbert published a study of
all his work on the foundations of mathematics, all that is necessary to
be given to formalize arithmetic: the propositional calculus, the axioms of
equality, the arithmetic axioms of the ‘successor’ function ( a + 1 ), the
recurrence equations for addition and multiplication, and finally some
form of the axiom of choice. To formalize analysis, it is necessary to be
able to apply the axiom of choice, not only to numeric variables, but to a
higher category of variables, those in which the variables are functions of
numbers. Mathematics thus presents itself as successive syntheses in which
each step is irreducible to the previous step. Moreover, and this is crucial,
a theory thus formalized is itself incapable of providing the proof of its
internal coherence. It must be overlaid with a metamathematics that takes
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the formalized mathematics as an object and studies it from the dual point
of view of consistency and completion (see Chapter 4). The duality of planes
that Hilbert thus established between the formalized mathematics and the
metamathematical study of this formalism has as a consequence that the
notions of consistency and completion govern a formalism from the
interior of which they are not figured as notions defined in this formalism.
To express this dominant role of metamathematical notions with respect to
formalized mathematics, Hilbert writes:
The axioms and provable theorems (i.e. the formulas that arise in this
alternating game [namely formal deduction and the adjunction of new
axioms]) are images of the thoughts that make up the usual procedure
of traditional mathematics; but they are not themselves the truths in the
absolute sense. Rather, the absolute truths are the insights (Einsichten)
that my proof theory furnishes into the provability and the consistency
of these formal systems. (Hilbert 1923; 1936, 180 [1996, 1138])
The mathematical theory thus receives its value from the metamathematical properties that its structure incarnates.
The structural conception and the dynamic conception of mathematics
seem at first to be opposed: one tends in effect to consider a mathematical
theory as a completed whole, independent of time; the other, on the contrary, does not separate it from the temporal stages of its elaboration. For
the former, the theories are like entities qualitatively distinct from one
another, whereas the latter sees in each an infinite power of expansion
beyond its limits and connection with the others, by which the unity of the
intellect is asserted. In the pages that follow, we would however like to try
to develop a conception of mathematical reality which combines the fixity
of logical notions and the movement with which the theories live. In
Hilbert’s metamathematics, we propose to examine mathematical theories
from the point of view of the logical notions of consistency and completion, but this is only an ideal toward which the research is oriented, and it
is known at what point this ideal currently appears difficult to attain (Cf.
Cavaillès 1938). Metamathematics can thus envisage the idea of certain
perfect structures, possibly realizable by effective mathematical theories,
and this independently of the fact of knowing whether theories making
use of the properties in question exist, but then only the statement of a
logical problem is possessed without any mathematical means to resolve it.
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This distinction between the position of a logical problem and its mathematical solution has sometimes seemed not very fertile, because what
matters is not knowing that a theory could be non-contradictory, but
rather being able to effectively decide whether or not it is. It seems to us
nevertheless possible to envisage other logical notions, equally likely to be
potentially linked to one another within a mathematical theory, and which
are such that, contrary to previous cases, the mathematical solutions to
the problems they pose can entail an infinity of degrees. Partial results,
comparisons stopped midway, attempts that still resemble groupings, are
organized under the unity of the same theme, and in their movement
allow a connection to be seen which takes shape between certain abstract
ideas, that we propose to call dialectical. Mathematics, and above all modern mathematics, algebra, group theory and topology,2 have thus appeared
to us to tell, in addition to the constructions in which the mathematician
is interested, another more hidden story made for the philosopher. A dialectical action is always at play in the background and it is towards its
clarification that the following six chapters tend.
The first three deal more specifically with the notions of mathematical
structure. In Chapter 1 (‘The Local and the Global’), we study the almost
organic solidarity that pushes the parts to organize themselves into a whole
and the whole to be reflected in them. Then, in Chapter 2 (‘Intrinsic
Properties and Induced Properties’), we examine if it is possible to reduce
the relations that a mathematical entity maintains with the ambient milieu
to properties of inherence characteristic of that entity. In Chapter 3 (‘The
Ascent towards the Absolute), we show how the structure of an imperfect
entity can sometimes preform the existence of a perfect entity in which all
imperfection has disappeared. Then come three chapters on the notion of
existence. In Chapter 4 (‘Essence and Existence’), we try to develop a new
theory of the relations of essence and existence which shows the structure
of an entity interpreted in terms of the existence of entities other than
the entity whose structure is being studied. Chapter 5 (‘Mixes’) describes
certain mixed intermediaries between different kinds of entities, whose
consideration is often necessary to effect the passage from one kind of
entity to another kind of entity. The last chapter (‘On the Exceptional
Character of Existence’) describes finally the processes by which an entity
can be distinguished from an infinity of others.
We claim to show that the ideas which are inscribed at the head of each
of the chapters and which seem to dominate the movement of certain
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mathematical theories, though conceivable independently of mathematics,
are nevertheless not amenable to direct study. They exist only with respect
to a matter that they penetrate with intelligence, but it can be said, on
the other hand, that it is they who confer on mathematics its eminent
philosophical value. The method that we follow is essentially a method
of descriptive analysis, mathematical theories constitute for us a given
within which we try to identify the ideal reality with which this matter is
involved.
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SECTION 1
The Schemas of Structure
CHAPTER 1
The Local and the Global
One of the characteristic traits of the development of mathematics since
the mid-nineteenth century is that the most diverse mathematical research
has been able to be pursued from a dual point of view, the local point of
view and the global point of view. The local study is directed towards the
element, most often infinitesimal, of reality, which it seeks to determine in
its specificity. Then, following its course step by step, gradually establishing
strong enough connections between its different parts thereby recognized,
so that an idea of the whole emerges from their juxtaposition. The global
study seeks instead to characterize a totality independently of the elements
that compose it. It immediately tackles the structure of the whole, thus
assigning elements their place before even knowing their nature. It tends
mainly to define mathematical entities by their functional properties, arguing that the role they play confers on them a much more assured unity
than that resulting from the assemblage of parts.
The duality of the local point of view and the global point of view was
first presented to mathematicians as an opposition between two modes of
study, irreducible to one another. It seemed necessary to choose between
these two incompatible conceptions, and in fact, the division that was thus
set up in mathematics still remains today in many domains. We would like
to briefly show what is in the theory of analytic functions, in geometry and
in the theory of differential equations.
The conception of the analytic function according to the ideas of Cauchy
and Riemann is a global, or at least regional, conception. It is based in effect
on the consideration of a whole domain of the plane of the complex variable z = x + iy. A complex expression g = u + iv, for Cauchy, represents an
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analytic function over the whole of this domain, if, at each point of the
domain, the existence of a unique derivative of g in relation to the complex variable z can be defined. We know that for such a derivative to exist,
it is necessary that the functions u and v are continuous functions of x
and y possessing continuous first order partial derivatives and satisfying the
differential equations (of Riemann):
2u = 2v ; 2u = 2v
.
2x
2y 2y
2x
The analytic function thus defined by the unicity of the derivative at each
point is not yet a defined notion of the global point of view, but it leads to
the theory of the integral, which is a global notion of the highest degree.
The value of an analytic function at an interior point z of a domain limited
by a closed curve C is determined by the value of the function on the
boundary curve:
f (z 0) = 1
2ri
#
C
f (z)
dz
z - z0 ,
if the domain in question is simply connected, that is, only has as its boundary the sole closed curve C. The structural properties of simple connection
relative to the topology of the basic domain are thus seen to play a part in
this definition. We will discuss in a later chapter the importance that these
topological considerations have taken with Riemann and how they have
allowed linking the existence of analytic functions to the existence of basic
domains, defined in their totality by their topological properties. The conditions relative to the existence of the derivative at each point no longer
plays the primary role, and the function is no longer so much defined by
its properties at each point of the domain as because it is appropriate to the
entire domain.
The global conception of Riemann is opposed to the local conception of
Weierstrass. An analytic function is essentially defined for Weierstrass in
the neighborhood of a complex point z0, by a power series with numerical
coefficients, which converges in a ‘circle of convergence’ around the point
z0. The method of ‘analytic continuity’ in turn helps construct step by step
a whole domain in which the function is called ‘analytic’ and this is done
in the following manner: take as a new center a point inside the first circle,
both a new series and a new circle of convergence is thus obtained that can
extend beyond the first. The new series extends the first if their values
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coincide in the common part of the two circles. The series can thus be
extended in all directions up to the points in the immediate neighborhood
in which the series obtained diverge. Thus we see that in this method the
domain is not circumscribed in advance, but rather results from the infinite
succession of local operations.
Weierstrass’s theory is developed in deliberate opposition to the integral
conception of Cauchy and Riemann. If some authors today, like Bieberbach,
give a presentation of the whole of the theory of analytic functions in which
the two points of view are intimately intertwined, others on the contrary,
like Goursat or Courant (Cf. Courant 1925), argue that it is necessary to
maintain a separation between the conceptions of Cauchy-Riemann and
those of Weierstrass. We shall have an opportunity later to see the importance that this separation of points of view also has for Hermann Weyl.1
As concerns geometry and group theory, we can do no better than to be
inspired by the famous articles published by Cartan on the question.2 At
the forefront of his general exposés on geometry, he always puts the
profound differences that, up until the theory of relativity, separated the
global conception of space as defined by Felix Klein in his famous ‘Erlanger
program’ of 1872 (Klein 1893) and Riemann’s infinitesimal conception
developed in the essay of 1854, On the hypotheses which lie at the bases of
geometry (Riemann 1868). A geometry by Klein’s definition is the study of
the properties of figures that are conserved when space taken as a whole is
subjected to certain transformations, forming what is called a transformation group. Euclidean geometry is thus the study of properties of figures
that are conserved when all the points of the space are subject to the same
displacement. It is found that these displacements conserve all the properties of the figures and in particular their metric properties. Affine geometry
studies the properties that are conserved by a linear transformation: the
parallelism of two straight lines, for example. Projective geometry studies
the invariant properties in relation to a homographic transformation like
the anharmonic ratio of four points on a straight line or the degree of an
algebraic curve. Whatever the properties with which the invariance in
relation to a group of transformations is researched, the essential characteristic of Klein spaces is their homogeneity. The group operates in the
same way on all points of space. Riemann spaces are on the contrary devoid
of any kind of homogeneity. Each one is characterized by the form of
the expression which defines the square of the distance between two infinitely near points. This expression is called a quadratic differential form
which generalizes the Euclidean formula of the distance between two
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points: ds2 = du12 + du22. The Riemannian ds2 to two dimensions is of the
following form ds2 = g11du12 + g12du1du2 + g21du2du1 + g22du22. In an ndimensional manifold we have the general formula:
n
ds 2 = / g ij du i du j
i, j
The gij are the absolutely arbitrary coefficients, which vary from point to
point. The result, as Cartan said, is that ‘two neighboring observers can
locate the points in a Riemann space that are in their immediate neighborhood, but they cannot, without new convention, be located with respect to
one another’ (Cartan 1924, 297). Each neighborhood is therefore like a
small bit of Euclidean space, but the connection from one neighborhood to
the next neighborhood is not defined and can be done in an infinity of
ways. The most general Riemann space is thus presented as an amorphous
collection of juxtaposed pieces that aren’t attached to one another. The
distinction that thus exists between a Klein geometry and a Riemann
geometry is found between the special theory of relativity and the general
theory of relativity. Special relativity is of the Kleinian type, it studies, in
the Minkowski four-dimensional universe, the invariants of the Lorentz
group. General relativity is a Riemannian geometry in which the gij depend
at each point of the distribution on the matter at that point. The space of
the general theory of relativity however does not present this complete
absence of organization that characterizes the most general Riemann
spaces. A physics in which the laws of the universe would vary from point
to point is in effect inconceivable. Einstein’s Riemannian space has what
Cartan calls a Euclidean connection, that is, it is possible to locate step by
step the different positions of an observer from each other. We discuss in
the next chapter the philosophical problems that are related to this Euclidean connection of Riemann spaces. If the purely local point of view is
exceeded, no knowledge of the universe as a whole is even obtained. The
gap between the local point of view and the global point of view still
remains, and, for Cartan, it is from this disparity that the principal difficulties of the unified field theory arise, as presented by Einstein in 1929. The
metric of the universe gives rise to a system of partial differential equations
for which Einstein sought solutions without singularity existing in all of
space. This would require the knowledge of the topological properties
of space–time taken in its totality, like knowing for example whether it is
open or closed. ‘This shows,’ Cartan said, ‘that the search for local laws of
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physics cannot be dissociated from the cosmogonical problem. It cannot in
addition be said that the one precedes the other. They are inextricably
mingled with one another’ (Cartan 1931, 18). Global integration is not an
extension of local integration. The solution of the local problem requires
prior knowledge of the structure of the universe.
Departing from an opposition of points of view which seemed proper to
geometry, the same conflict is found in problems of considerable philosophical importance, because the interpretation of the determinism of
physics depends on their solution. They are problems that are related to
the conditions of the existence of solutions of differential equations or partial differential equations.
The analysts of the nineteenth century were able in most cases to establish the theorems of existence that allowed the assertion of the existence
and possibly the uniqueness of the solution to a differential equation or a
partial differential equation, defined in the whole domain in which a certain inequality holds, and this by relying solely on the knowledge of local
data, in a point of origin for example.
It is thus that a second order differential equation3 of the form:
d2 y
dy
= f c x, y m
dx
dx 2
admits in general one and only one solution corresponding to the given
initial conditions, namely that for x = a , y takes a given numerical value b
and
dy
dx
takes the value b´. The solution to such a problem is thus determined by
the local conditions, according to Cauchy. Kovalevsky (1875) established
a theorem for second order partial differential equations analogous to
Cauchy’s theorem for differential equations: If the equation of partial
derivatives
i, k = 0, 1, 2 f n
2
F c x, x 1, f x n, u, 2u , 2 u m = 0
2x i 2x i 2x k
x0 = x
2
can be resolved with respect to 2 u2 so that we have
2x
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2
22 u
= f c x, x 1, f x n, u, 2u , 2 u m, i, k = 1 f n
2x 2
2x i 2x i 2x k
function f being holomorphic4 with respect to x, x1, . . . xn, u, and to all the
other derivatives, this equation admits one and only one solution, holomorphic in x, x1, . . . xn, and satisfying, for x = 0, the conditions
u = g (x 1 f x n), 2u = h (x 1 f x n)
2x
functions g and h being holomorphic in x1 . . . xn. If the set of points x = 0
are then envisaged as determining a plane or a surface of n-dimensional
space (n being the number of independent variables in the equation), the
Kovalevsky theorem can be interpreted5 in terms of classical determinism.
Knowing the value of a function and one of its derivatives at all points of a
surface S allows the assertion of the existence and the regularity of this
function in a certain neighborhood of the surface S.
While the analysis thus establishes the local theorems of existence,
the direct study of certain physical phenomena led to the consideration
of very different problems These problems are all the same type as the
famous Dirichlet problem, in which one is led to prove the existence of a
function at the interior of a volume V satisfying a certain partial differential
equation,
2
2
2
(the Laplace equation: 2 u2 + 2 u2 + 2 u2 = 0 )
2x
2y
2z
and, at the boundary of the domain, taking the values given in advance.
Similar functions are encountered in the study of electric or calorific equilibrium, when it is a matter of, for example, determining the temperature
which will eventually be established within a domain, on the boundary of
which were distributed a continuous succession of temperatures invariable
over the course of time. The physical fact that an equilibrium temperature
ends up effectively being established gave mathematicians the certainty of
the existence of the sought after function even before they had a rigorous
proof of it.
If one reflects upon the nature of the conditions at the limit of the
Dirichlet problem, one realizes, remarks Hadamard, that, between them
and the initial data of the Cauchy-Kovalevsky theorems, there is a profound contradiction. Since knowledge of the value of the function at each
point of the surface S completely determines this function in the case of the
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Dirichlet problem, ‘it appears obvious that one has no right to give the
length of S the value of u and that of one of its derivatives’ as one does in
the general statement of the theorem. The apparent contradiction from the
mathematical point of view is avoidable, on the one hand, because the
initial data of the general theorem of existence is subjected to the rigorous
conditions of analyticity, while the data ‘at the limits’ is of a much more
general nature, and, on the other hand, because the solution of which
Kovalevsky’s theorem asserts the existence is only defined for a more or
less immediate neighborhood of the surface S, while in the case of a problem ‘à la Dirichlet’ the desired solution must be defined and regular in the
whole volume V whose surface S is the boundary. It is no less true that the
theory of partial differential equations governs the clearly different physical processes. Some, like the propagation of light, are in free evolution, and
the deterministic schema can be perfectly applied to them. Others, on the
contrary, are circumscribed by the data at the limits. Not only must the
initial data be known in advance, but also the extreme limits between
which the phenomena studied can oscillate. If the function sought after is
not a constant, it is in effect shown to be on the boundary of the field, and
that, for the values given in advance, in the case of a Dirichlet problem, it
attains its maximum and minimum. It seems then that for a physical phenomenon whose evolution is ‘directed’ and that precise limits enclose all
parts, it is necessary to call upon philosophical interpretations, unexpected
in mathematics, in which the physical system is comparable to an organic
unity. We will find in addition considerations of this kind in Chapter 4, and
see again how the search for maximum and minimum could sometimes
suggest the idea of a finality inherent in certain mathematical theories and
in certain physical phenomena.
The study of these questions, which in addition emerge more from the
philosophy of physics, will not be discussed here, we will confine ourselves
to the purely mathematical aspect of the duality of the local study and the
global study.
The observation of this duality naturally suggests to mathematicians the
search for a synthesis. Given that the elements composed by some process
of progressive development cannot give rise to any entity amenable to
global characteristics, it is necessary, to be sure to reach a result, that the
topological structure of the whole is reflected in the properties of its parts.
This can give rise to two kinds of problems: either we start from the set
whose structure is known and we search for the conditions that must be
satisfied for the elements to be elements of this set; or we provide elements
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having certain properties and try to read in these local properties the structure of the set in which these elements are able to be arranged.6 In both
cases we try to establish a connection between the structure of the whole
and the properties of the parts by which the organizing influence of the
whole to which they belong is manifested in the parts.
Considerations are thus met in mathematics that may at first sight seem
foreign to mathematics and brought there as reflections of certain specific
conceptions in biology or sociology. It is evident that the mathematical
entity as we understand it is not without analogy to a living being. We
however believe that the idea of the organizing action of a structure on the
elements of a set is entirely intelligible in mathematics, even if transported
to other domains it loses its rational clarity. The obstructions that the philosopher sometimes meets with in regards to arrangements that are too
harmonious comes not so much from the subordination of the parts to the
idea of a whole which organizes them, than from the manner in which this
organization of the whole is carried out sometimes as a naive anthropomorphism and sometimes as a mysterious obscurity. Biology like sociology
often lack in effect the logical tools necessary to constitute a theory of
the solidarity of the whole and its parts: we shall see on the contrary that
mathematics can render to philosophy the eminent service of offering the
example of interior harmonies whose mechanism satisfies the most rigorous logical requirements.
Three theories will briefly be reviewed that will provide us as many
models in which this implication of the whole in the part is realized: differential geometry in its relation to topology; group theory; and the theory
of the approximate representation of functions. These three examples
seem to us to be particularly suggestive because they allow the same conclusion to be arrived at regarding the conditions that must be fulfilled by
the structure of a mathematical entity so that within this entity reigns like
an organic solidarity.
1. DIFFERENTIAL GEOMETRY AND TOPOLOGY
The study of the relations between topology, an eminently synthetic
study of geometrical objects,7 and differential geometry has given rise to
a large amount of methodically pursued research under the leadership of
W. Blaschke in Hamburg. What is proposed here is a brief analysis of a
paper by Hopf (1932) which contains the essential ideas that dominate the
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question. Posing the general problem of knowing what the connections
are that can exist between the topological properties and the differential
properties of a surface, Hopf is led to distinguish two reciprocal problems
whose interest has been pointed out above: a problem of metrization and
a problem of extension. Their statement is of such logical interest that we
cannot but reproduce the very text of the author:
The problem of metrization is as follows: given a topological surface F, it
is a matter of determining on this surface a differential metric (a ds2). . . .
What are the conditions to be satisfied by the metric of the surface?
What metric properties are prescribed in advance by the topology of F?
By what limitations is the arbitrariness with which I can fix the gik to the
place where I begin to fix the metric of the surface restrained?
The inverse problem of extension is as follows:
Given a small piece of a surface F, I can examine this piece with all the
precision possible, but, on the other hand, it is not possible to study the
surface as a whole. What conclusions can I draw from the knowledge
that I have of the small piece of the surface, as regards the total surface
and in particular its topological structure? (Hopf 1932, 209)
These problems can only be addressed if the meaning of the expression ‘the
total surface’ is specified. A surface is only total if it cannot be ‘extended’ in
turn into another surface, and for this to be so Hopf and Rinow (1931)
state four equivalent conditions each of which is sufficient to make the
surface an independent ‘whole’.
Only one of these conditions will be focused on because the necessity of
an analogous condition will be found in all the examples in this chapter. It
is necessary that the surface is complete as defined by the metric. Here’s
what should be understood by complete surface as defined by the metric.
A fundamental sequence on a surface is called an infinite sequence of
points a 1, a 2, f a n f , so that, from a certain rank, the distance between two
points is infinitely small. The sequence is said to be convergent to a limit
A if from a certain point the distance of points of the sequence to this
point A also becomes infinitely small. If they happen to be real numbers,
every fundamental sequence would be convergent, by virtue of Cauchy’s
theorem, that is, it tends towards a limit which would also be part of the
set of real numbers. When it comes to points on a surface, it is no longer
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always the case and the surface is said to be rightly complete as defined by
the metric, when any fundamental sequence converges towards a limit
also located on the surface. When a set of points, a surface for example, is
not complete, it can be completed by adding to it the points which it lacks,
namely the limits of its fundamental sequences. Now what is essential in
the results of Hopf and Rinow is that the topological properties of a surface
are only reflected in the properties of the parts if the surface is not likely to
be completed. It is only on this condition of completion that the results that
we will now present are valid.
Let it be the case that a surface is simply connected, that is, such that any
closed curve by continuous deformation can be reduced on this surface to
a point. This is a topological property of the surface. A surface is said to
have a constant curvature if a certain quantity is defined at each point
using the coefficients gik of ds2 of the surface and that the so-called curvature is the same for all values of gik. The curvature being a purely local
notion, the constancy of the curvature is also a local property, defined for
each element of the surface. That being so, we have the following theorem:
For any curve K there exists, to a near isometry, a sole simply connected
surface of constant curvature K, namely the surface of a sphere, the
Euclidean plane or hyperbolic plane according to whether we have K 2 0 ,
K = 0 or K 1 0 . If the requirement of simple connection is abandoned, the
surfaces can be arranged in three topological classes, the classes C+, C0 , C–.
The class C+ contains two types: the sphere and the projective plane. The
class C0 contains five types: the plane, the cylinder, the torus, the nonorientable cylinder, and the non-orientable closed surface of genus zero.
The class C– contains all surfaces with the exception of the four closed surfaces contained in C+ and C0 (sphere, projective plane, torus and nonorientable closed surfaces just defined). We then have the following
theorem: only surfaces of class C+ can have a constant positive curvature,
the surfaces of class C0 have a constant zero curvature, and those in class
C– have a constant negative curvature.8 The metric properties are therefore
closely related to the topological class of the total surface. It is thus that, for
example, a surface of constant positive curvature is necessarily closed
(problem of extension) or that on a closed surface of constant curvature,
the sign of the curvature is the same as that of a global topological invariant
of this surface, the Euler characteristic (problem of metrization) (Hopf 1932,
213). The importance of such results can be seen for the cosmogonical
problems which were discussed earlier. The ds2 of the space is not determined
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by the form of the space, but the choice of ds2 is subject to very restrictive
conditions that fix the topology of the global space.
2. THE THEORY OF CLOSED GROUPS
The results due to Weyl and Cartan will now be presented where the
metric, in certain cases, can be entirely determined by a global property of
the group associated with the space in question. The importance of these
results comes from them sometimes allowing the reconciliation of Kleinian
type spaces and Riemannian type spaces, despite all the differences in
structure that have been recognized in them above.
The Weyl theorem used by Cartan is as follows: a closed linear group
with real coefficients leaves invariant a quadratic form defined as positive.9
We immediately see the immense interest of this theorem: the group is
defined by global characteristics since it is closed; the quadratic form can be
used on the other hand as a local metric, of ds2, in a space associated in a
convenient way with the group in relation to which this quadratic form is
invariant. In the problem studied by Hopf, the global properties of the surface could only be registered in the nature of the infinitesimal elements of
the surface if this surface was complete. We end up with an condition of
completion analogous to the theorem of Weyl. A continuous group can in
effect be conceived as giving rise to a topological manifold if a point of the
manifold is associated with each transformation of the group. It is not necessary for this manifold to be defined for it to possess a metric, it is sufficient that it is constituted by a set of ‘neighborhoods’ satisfying certain
conditions set forth by Cartan (Cartan 1930, 3). For the manifold of a
group, a condition of closing or closure can then be defined that plays, in
the case of spaces defined by a system of neighborhoods, the role of completion, which, in the case of metric spaces, is played by the property of
being complete: a point A being the accumulation point for an infinite set
of distinct points of a manifold, if every neighborhood of A contains at least
one point of the distinct set of A, the manifold is said to be closed if every
infinite set of points on this manifold admits an accumulation point. This
property of completion as defined by topology is distinct from completion
as defined by the metric, but is at least sufficient to confer on the structure
of a group a character of closure whose presence is indispensible for this
structure to reflect itself locally, in the metric of the space with which the
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group is associated. Weyl’s theorem cannot be applied directly to Klein
groups that operate on a given homogeneous space E, since these groups
are in general not linear. Also Cartan proceeds as follows: he adjoins to
group G the transformations that operate on the space E, the group C of
automorphies internal to group G, this adjoint group is found to be linear,
as are all its subgroups. Cartan then considers a certain subgroup c of C ,10
and proves the following theorem: If linear group c is closed, there exists
in the homogeneous space E a Riemannian metric invariant under G
(Cartan 1930, 30, 43). There is here an essential connection between the
topology of the Klein group and the local metric of a Riemann space from
which Cartan has identified enough consequences to be able to write in
the terms of an article that we have been cited above:
Departing from the antagonism between Kleins geometries and Riemann
geometry, we arrive after a long detour at this finding that it is in the
Riemannian form that Kleins geometries best show their fundamental
properties. (Cartan 1927, 222)
3. APPROXIMATE REPRESENTATION OF FUNCTIONS
The domains in which we are now going to study a solidarity between the
structure of the set and the individual nature of the elements are sets of
functions. We will first consider the set K (f) of continuous functions, the
value of which is a complex number and which is defined on a closed set
of points f . It is possible to envisage this set of closed functions as forming
a space C (f) , satisfying the axioms of vector spaces, and which is called the
space of continuous functions on f . The global properties of this space are
essentially the completion properties comparable to those already defined
above. The individual properties of functions, which are elements of this
space, concern a mode of decomposition that is applied to each of them,
and we see the close link that unites the properties of the set to the properties of the elements.
Completion as defined by the metric is defined for a function space in an
analogous way to the completion of a space of points. It is sufficient to
define for any two functions f and g a number f - g , which is said to be
the distance of these two functions. The fundamental sequences and the
converging sequences (as defined by uniform convergence) can then be
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defined on the space of functions.11 The space is complete if every fundamental sequence is convergent, and the space C (f) of continuous functions is precisely such a complete space.
To study the individual properties of decomposition of continuous functions we begin with the simple case in which the space of points f is a
closed interval [a, b] of Euclidean space. Weierstrass proved that if f(x) is
continuous in [a, b] , then this function can be approximated as closely as
desired by a polynomial in x
n
P (x) = / c k x k
0
We thus have for every f 2 0
f (x) - P (x) 1 f
This approximation of any continuous function by a polynomial is immediately interpretable in terms of decomposition for the function in question. It proves in effect that one has the right to deduce from the
inequality12
f - P 1 f the equality f (x) =
k=3
/cx
k
k
k=0
The function is thus decomposed into a uniformly convergent series of
infinite terms. If we now try to clarify the meaning of this result we will see
that the global point of view of completion and the point of view of individual decomposition are united in it. The equality
3
f (x) = / c k x k
0
concerns the particular mode of decomposition of the function f(x), but if
the series
3
/c x
k
k
0
is considered, an infinity of polynomials P0 . . . Pn . . . of a finite number of
terms
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k=n
/c x
k
k
k=0
and of increasing degree can be distinguished. The uniform convergence of
this sequence of polynomials towards a limit f(x) results this time, no longer
in the individual properties of this limit, but in the global property of closure of the space of continuous functions. In other words, the fact that a
polynomial P(x) is found in the infinitesimal neighborhood of f(x) rightly
concerns the function f(x) considered in isolation, but this fact is immediately related to the totality of analogous cases. In the space of continuous
functions, the set of polynomials in x is everywhere dense, that is, in the
neighborhood of every continuous function a polynomial is found, and the
proof of this theorem appeals to the closure of the space.
There exist, in analysis, other examples in which such a connection
between the global structure of a set of functions and the mode of individual decomposition of these functions is shown. In this way, for example, any continuous and differentiable function in the interval - r 1 x 1 r
is representable as a convergent series of trigonometric polynomials (or
Fourier series):
3
f (x) = 1 a 0 + / (a n cos nx + b n sin nx)
2
n=1
the an and the bn being connected to the expression of f(x) in the interval
considered by the formulas
#
r
-r
f (x) cos nxdx ; b n = 1
r
#
r
-r
f (x) sin nxdx
an = 1
r
This theorem can be presented as a consequence of Weierstrass’s theorem:
the set of continuous and differentiable functions in the interval
- r 1 x 1 r also forms a complete space, and this global property of
completion is translated by a new mode of individual decomposition of
functions that belong to this set.
The preceding studies show that it is impossible to consider a mathematical ‘whole’ as resulting from the juxtaposition of elements defined
independently of any overall consideration relative to the structure of the
whole in which these elements are integrated. There thus exists a descent
from the whole towards the part, as a ascent from the part to the whole,
and this dual movement, illuminated by the idea of completion, allows the
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observation of the first aspect of the internal organization of mathematical
entities. If one claims to admit that the study of such structural connections
is an essential task for mathematical philosophy, one cannot fail to notice
the differences that separate mathematical philosophy thus conceived from
the entire current of logicist thought that developed after Russell had
discovered the paradoxes of set theory. The logicians have since always
claimed to prohibit non-predicative definitions, that is, those in which the
properties of an element are supportive of the set to which that element
belongs. Mathematicians have never been willing to admit the legitimacy
of this interdiction, rightly showing the necessity, to define certain elements
of a set, to sometimes call upon the global properties of this set. The whole
chapter that has just been read tends towards showing the fecundity of
that point of view. We thus hope to make evident this idea that the true
logic is not a priori in relation to mathematics but that for logic to exist a
mathematics is necessary.
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CHAPTER 2
Intrinsic Properties and Induced Properties
When the properties of a geometric object are studied, one is led to
distinguish between properties which result in the consideration of this
entity’s intrinsic nature, and those that confer on it its relations with the
environment that surrounds it. The intrinsic properties of an entity are
independent of the position of this entity in space; they are even independent of the existence of other entities. They belong literally to the entity
under consideration. The properties of relation on the contrary can only be
attributed to a mathematical entity if one is referred to something other
than it. It is sometimes a reference system common to several entities,
sometimes an ambient space whose properties can be defined independently of any content, sometimes again a certain number of other entities
that support, with the former, relations of neighborhood, of impact, of
orientation, etc. The properties of relation in sum translate the solidarity
between one entity and the universe within which it is embedded.
It is known that one of the essential differences between the mathematical philosophy of Leibniz and that of Kant resides in the opposition of
their conceptions relative to the extrinsic properties of geometric entities.
We would like to briefly summarize here a debate that, we will see later,
retrieves the relevance of this opposition today in differential geometry
and in topology.
The Leibnizian conception of the monad, just as much as concerns
geometry as concerns the existence of created things, is based on the reduction of relations that the monad sustains with all other monads to internal
properties, enveloped in the essence of the individual monad. There are
two important moments in this reduction: that in which Leibniz conceives
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the universal sympathy of all substances; and that in which he inscribes
this sympathy in the law of the internal becoming of each monad considered as isolated from all others. In the letter to Arnauld of September 1687,
he explains that one thing can express another as ‘a projection in perspective expresses a geomtric figure’, and further on he writes: ‘every substance sympathizes with all the others and receives a proportional change
corresponding to the slightest change which occurs in the whole world’
(Leibniz 1969, 339). The notion of monad only takes shape, however, in
A New System of the Nature and Communication of Substances, and of the Union
of the Soul and Body published in 1695:
[E]very substance represents the whole universe exactly and in its own
way, from a certain point of view, and makes the perceptions or expressions of external things occur in the soul at a given time, in virtue of its
own laws, as if in a world apart, and as if there existed only God and
itself . . . There will be a perfect agreement among all these substances.
(Leibniz 1989, 202)
All Kant commentators have shown the importance, for the formation of
the Kantian theory of space, of two texts of the pre-critical period, one
from 1768 Concerning the ultimate ground of the differentiation of directions in
space, the other from 1770, the dissertation Concerning the form and principles
of the sensible and intelligible world, which sketch the ideas of the transcendental aesthetic. The respective positions that bodies occupy in space with
respect to one another are not described uniquely in terms of mutual relations between these bodies. They are necessarily referred to a system of
privileged and universal references that establish in space the fundamental
distinctions of left and right, top and bottom, front and rear of the human
body. The distinction of the left hand and right hand is the most important.
It imposes on space like lines of cleavage to which all bodies in space are
subjected. If we look in effect at the way in which the hair ‘grows in a spiral
on the head’, the spirals of snail shells uncoil, or ‘hops wind around their
poles’ (Kant 1768, 368), we observe everywhere in nature a privileged
movement of left to right, which no artifice can erase. One of the more
famous consequences of this opposition of the left and the right, is the
incongruity of symmetric figures. The left hand and right hand possess,
when examined in themselves, exactly the same parts, they are arranged
in each hand in the same way, both hands are therefore identical and similar (gleich und ähnlich) but if they are by all means symmetric, they are not
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superimposable. This incongruity of symmetrical figures that relates to the
structure of our own body is found in pure geometry, and, in support of his
thesis, Kant invokes the case of spherical triangles that can be perfectly
identical and similar, without however overlapping. There are here sensible facts, of which no rational analysis of internal properties of bodies can
account for and which result from the difference of place that these bodies
occupy in sensible space. The dependence of bodies with respect to ambient space is therefore narrowly connected for Kant to the fact that reason
can only characterize the intrinsic properties of geometric bodies in an
abstract way, and that those arising from their position in space can only
be grasped by sensible intuition which refers to the orientation of space as
a whole: ‘Which things in a given space lie in one direction and which
things incline in the opposite direction cannot be described discursively
nor reduced to characteristic marks of the understanding by any astuteness
of the mind . . . ’ and later: ‘It is, therefore, clear that in these cases the
difference, namely, the incongruity, can only be apprehended by a certain
pure intuition’ (Kant 2003, 396). We shall return later to these possible
intrinsic limitations of geometry.
It is hardly necessary to recall how the constitution by Gauss and
Riemann of a differential geometry that studies the intrinsic properties of a
manifold, independently of any space in which this manifold is embedded,
eliminates any reference to a universal container or to a center of privileged coordinates. Gauss’s General Investigations of Curved Surfaces (1827)
envisages a real surface of two dimensions and defines a metric on that
surface from the point of view of an observer bound to the surface and
who consequently could not consider it from a position of space exterior to
it. Riemann’s point of view, which was presented in the previous chapter,
generalizes, in the case of a ds2 of n variables, Gauss’s ‘superficial’ point of
view. The notions of distance, curvature and geodesic have an intrinsic
meaning, since they are defined step by step, without exiting the manifold.
The distinction between space and the manifold disappears; it subsists
only as the space of this manifold. It is known how the theory of relativity
reinforces this identification of the container and the contents: matter is no
longer considered to be located in space, the properties of space at each
point being determined by the density of matter at this point. Geometry
and physics are constituted interdependently, so that it is impossible to
separate space, the Riemannian manifold and matter.
Such an intrinsic characterization of geometric objects is not brought about
without causing some difficulties of interpretation among philosophers.
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The idea that this Riemann manifold, which is the Einstein space, would be
closed often evokes the image of a closed surface that the intuition could
not help but locate in an infinite 3-dimensional space, and yet outside of
this surface, by an incomprehensible paradox, there could be no matter,
nor even space. The paradox disappears when one realizes that a manifold
on which a ds2 of more than two dimensions is defined is in no way amenable to an intuitive comparison with a surface. The notions of intrinsic
differential geometry are purely intellectual, they characterize a mode of
mathematical exploration of a manifold by following a path on this manifold, in opposition to the extrinsic method which considers this manifold
as embedded in a Euclidean space to a sufficient number of dimensions.
It is in effect always possible to realize a ds2 in a Euclidean space, but it is
a Euclidean space of:
n (n + 1)
2
dimensions whose geometry is as abstract as that of the manifold it contains. What is of interest here is the existence of two points of view that are
as clearly distinct from one another as the intrinsic point of view and the
point of view of insertion. This new duality leads in effect to the first of
the problems that we intend to study in this chapter: Is it possible to reduce
the induced properties on a Riemann surface by the ambient Euclidean
space to the purely intrinsic properties of this manifold?
1. PARALLELISM ON A RIEMANN MANIFOLD1
The question posed is authentically mathematical and the technical aspects
of it can be seen in Cartan’s expositions that have been cited above. It is of
no less considerable philosophical interest that Cartan himself stresses in
sober terms, which confer on his remark all the precision of a scientist’s
assertion. The point of view of induced properties, he tells us, is ‘philosophically inferior’ to the intrinsic point of view. The lesser result, in the sense of
the reduction of the extrinsic to the intrinsic, tends in effect to inscribe in
the structure of an entity the relations it maintains with the ambient space
and thus to restore the vision of the Leibnizian monad. Here again, it seems
that mathematics offers a privileged domain to the movement of a thought
which attempts to reconcile two opposed logical notions.
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We saw the beginning of the first chapter how, in a Riemann space
defined by a certain ds2, two neighboring observers can, by means of a trirectangular trihedron, ‘locate the points that are in their immediate neighborhood, but cannot, without new convention, locate with respect to one
another their trihedrons of reference’ (Cartan 1924, 297). We also indicated the necessity, where the theory of relativity is found, to endow the
Riemann spaces that it considers with a certain homogeneity so that the
laws of physics can be independent of any attachment to particular points
in space. This Euclidean connection of Riemann spaces is defined by
Levi-Civita (1917) with his conception of parallelism on any manifold. The
connection of neighborhoods of different points is no longer indeterminate, the pieces of space are oriented closer and closer to one another, so
that it is always possible to define the parallelism of two vectors issuing
from two infinitely near points. Cartan explains how, in Levi-Civita’s theory, this parallelism is ‘induced’ on the manifold by the Euclidean space to
n (n + 1)
2
dimensions in which it is embedded. To represent intuitively the way that
properties can be induced on a manifold of ambient space, Cartan envisages first of all a curve in Euclidean space. This curve differs from a straight
line only for an observer exterior to it. The kinematics of a mobile point on
this curve is identical to the kinematics of a mobile point on a straight line.
This correspondence between axes of the curve and segments of the
straight line comes from the possibility of unrolling the curve on a straight
line which would initially be a tangent at a point on the curve. But the
operations of rolling or unrolling a curve on a straight line are only possible
by a series of successive projections of the curve on the straight right,
effected in the space that contains the two. Let us now define the parallelism of two vectors tangent to a Riemann surface Vn and issuing from two
infinitely near points A and A´. Cartan presents the point of view of LeviCivita by considering the tangent to the surface at A (Cartan 1924, 297).
This plan contains the tangent vector Vn issuing from A. We can project
orthogonally on this plane the vector issuing from the plane infinitely near
A´, and the two vectors from A and A´ are called parallel if the projection of
the latter on the tangent plane at A is said to be parallel, in the ordinary
meaning of the word, to the vector issuing from A. Knowledge of the ds2 of
the surface is sufficient to determine the coordinates of A´ with respect to
A, but if local systems of references are attached to A and A´, the latter axes
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are subject to a rotation with respect to the former and it is possible to
determine the angles of this rotation so that the conditions of parallelism
of two infinitely near vectors is always satisfied. As in the case of the curve
embedded in Euclidean space, the exterior tangent planes, the projections
and the rotations implied by the parallelism of Levi-Civita only make sense
with respect to the space in which the manifold is embedded. Cartan shows
later in his article the differences that separate this view from that of Weyl,
who could give the parallelism a purely intrinsic definition. In the case of
two dimensions, it presents itself as follows: two directions issuing from
two adjacent points A and A´ are parallel if they form the same angle with
the geodesic (line of minimum length) passing through A and A´ (Cartan
1924, 298). This definition does not call upon any operation in the exterior
space and it works out to be the same result as that of Levi-Civita.
This result is all the more remarkable because, when both the intrinsic
point of view and the extrinsic point of view are possible, it is not at all
necessary that they work out to confer a same connection to a same
surface. Cartan in effect generalizes the point of view that is derived in the
work of Levi-Civita by seeking to attribute to a manifold embedded in
a space other than the Euclidean space (the affine space, projective or
conformal) a connection which gives the simplest account of the relations
of this manifold with ambient space (Cartan 1924, 317). Let, for example,
a surface be embedded in conformal space (in which the notion of plane is
replaced by that of sphere). It is possible to consider a ‘conformal connection’ induced on this surface by the ambient conformal space as follows:
instead of tangent planes in two neighboring points, envisage the spheres
of curvature passing through two points and connect the two spheres by a
kind of orthogonal projection of one on the other. This extrinsic conformal
connection differs essentially from the purely intrinsic conformal connection since the latter identifies every plane and every surface with a sphere.
The reduction of the extrinsic to the intrinsic thus conflicts with the facts
that show the limits faced by the elimination of any reference to a universal container, and confer an even greater interest on surprising cases in
which this elimination succeeds.
2. STRUCTURAL PROPERTIES AND SITUATIONAL PROPERTIES
IN ALGEBRAIC TOPOLOGY
The duality of the extrinsic point of view and the intrinsic point of view
that has been observed with respect to certain problems of differential
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geometry is rediscovered in algebraic topology and occupies such an important place there that the authors of the most recent treatises on topology,
Seifert and Threlfall (1934), like Alexandroff and Hopf (1935), put it at the
center of their general considerations relative to topology.
The geometric properties studied by topology are those which are conserved by biunivocal and bicontinuous transformations. Two figures are
said to be homeomorphic in this regard if such a correspondence can be
established between the points of one and those of the other. Two types of
homeomorphism must then be distinguished: those that are realizable by a
deformation of the two figures that makes them coincide in space; and
those that exist between the points of two figures that no deformation in
space can get them to coincide. Consider any two closed curves; they are
homeomorphic in the sense that it is possible to establish a biunivocal and
bicontinuous correspondence between the points of one and the points of
the other. If they are situated in the same plane, this correspondence
can always be realized by bringing one of the two curves, in a series of
intermediate positions, to coincide with the other. If they are not situated
in the same plane, it may be impossible to get them to coincide without
tearing. Consider, for example, the circle and the node in the form of a
clover (Figure 1).
These are two homeomorphic curves since a punctual correspondence
can be established from one to the other and yet it is impossible to get them
to coincide by a continuous deformation in space. There thus exist for the
figures: internal properties (Eigenschaften Innere in Seifert and Threlfall
1934) or structural properties (Eigenschaften Gestaltliche in Alexandroff and
Hopf 1935) like being a closed curve, and which are independent of any
Figure 1. (Seifert and Threlfall 1934, Figure 2)
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reference to the ambient space; and insertion or situational properties
(Einbettungseigenschaften, Lageeigenschaften) that are conferred on a figure by
the relations it maintains with all of the other figures in the space. The
results that we will present all relate to this eminently Leibnizian problem:
Is it possible to determine the properties of situation by the knowledge of
the structural properties? The very term Analysis situs, due to Leibniz,
expresses well this hope of determining what concerns the ‘situation’ by an
analysis of the internal properties of the figure.
There is, at least in 3-dimensional space, a classic case of situational
property completely reducible to a property of intrinsic structure. Consider
what’s called the Mobius ring. It is the figure obtained by welding the two
ends of a band twisted once on itself. If a line is drawn on this surface by
stopping only when it rejoins the starting point, and then if the ring is split
to unfold it in its full length, a band is obtained whose two sides are traced
on by the line drawn. If the ring were to be colored step by step, it would
be seen, by unfolding the ring again in space, that both sides of the obtained
band have been colored (Figure 2).
The Mobius ring therefore only has a single side, and that is an essentially extrinsic property since, to be realized, it is necessary to split the ring
and untwist it, which implies a rotation around an axis exterior to the surface of the ring. It is nevertheless possible to characterize this ‘unilaterality’
by a purely intrinsic property. Consider in effect an arrow perpendicular to
the line traced on the ring and move this arrow along the line, so that it is
Figure 2. (Seifert and Threlfall 1934, Figure 12)
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always situated on the surface. By presuming the surface transparent
enough we realize that the arrow arrives at a point to cover its departure
position with an inverse orientation. The surface is said to be non-orientable
and this property could be observed by an observer bound to the surface,
which would neither split the ring, nor untwist it. In an n-dimensional
orientable space, the 3-dimensional Euclidean space for example, it can be
proved that for an (n – 1)-dimensional manifold, a two dimensional ring
for example, there is equivalence between the fact of being two-sided and
the fact of being orientable, or between the fact of being unilateral and that
of being non-orientable. This junction of the situational point of view and
the intrinsic point of view is, in this case again, all the more interesting
because it does not necessarily take place in all cases. Orientability or nonorientability being intrinsic properties cannot be removed at a surface by a
modification of its relations with the ambient space, but it is not as before
due to being bilateral or unilateral, properties that depend on this ambient
space. It is thus that an orientable surface is bilateral in Euclidean space
but could cease to be so in another 3-dimensional space.
This example hints at the philosophical interest of algebraic topology (or
even combinatorial topology): the geometric properties of relation to a
very large extent let themselves be expressed in intrinsic algebraic properties and, to the extent that the intellectualization of the relations of a
figure and of ambient figures is successful, the Kantian distinction between
an aesthetic and an analytic is seen to vanish. The most triumphant success
of topology in this regard is the theory of the duality that we will now
present.
3. DUALITY THEOREMS
If a polyhedron is envisaged in Euclidean space Rn, the situational properties that we are led to attribute to it concern in the first place the repercussions that involve, on the global structure of the space, the presence in
it of the geometric entity considered. The specific structure of the polyhedron is an intrinsic property of the polyhedron, independent of all the
possible situations in which it can be found, but its introduction2 within a
space changes the internal structure of this space, by submitting the elements to new connections and thereby establishing the relations between
the space and itself that characterize its mode of insertion in the space.
Duality theorems in topology allow the determination of this action of the
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polyhedron on the space solely from the structural knowledge of the polyhedron. The polyhedron thus enjoys certain properties of the Leibnizian
monad, and we will try to show this by drawing upon only the indispensable definitions.
The object of study of topology is not the polyhedron but the simplicial
complex. Here’s what is meant by that: a point is a 0-dimensional simplex;
a line segment AB is a 1-dimensional simplex determined by the 2 points A
and B; a triangle is a 2-dimensional simplex; more generally n + 1 vertices
determine an n-dimensional simplex. This simplex possesses faces of 0, 1,
2 . . . n – 1 dimensions obtained by considering successively its vertices, its
edges, its sides. Now consider a figure formed by a set of simplices satisfying the following conditions: a) any point of the figure belongs to at least
one simplex; b) any point belongs only to a finite number of simplices; c)
given two simplices, either they are without common points, or they have
a common face; d) the neighborhoods of a point in the different simplices
to which it belongs are ‘reunited’ in a single neighborhood. This figure is
what is called a simplicial complex, whose dimension is that of the highest
simplex that appears in the ‘simplicial decomposition’ of the complex.3
Naturally, there is no distinction between a complex and the figures
that are topologically equivalent to it.4 Thus, for example, the surface of a
sphere is a 2-dimensional complex whose simplicial decomposition is
obtain by considering the tetrahedron inscribed ABCD. The tetrahedron is
a 3-dimensional simplex: the 4 triangles that limit it form a 3-dimensional
complex topologically equivalent to the surface of the sphere where 4 2dimensional simplices occur: the 4 triangles ABC, ABD, BCD, ACD, six 1dimensional simplices (6 edges) and four 0-dimensional simplices (vertices).
It can even be demonstrated that most of the important figures of topology
are complexes. Any structural study of a complex is based on knowledge
of certain numbers, called Betti numbers, attached to this complex, that
are invariant under topological transformation. Despite the abstract character of this theory it is absolutely necessary to clarify the nature of the
geometrical objects that these numbers measure on a complex. On an ndimensional complex, certain combinations of 0-dimensional, 1-dimensional,
to n-dimensional simplices are defined, forming, relative to each dimension, what are called cycles.5 Then, what is understood as the independence of several cycles of the same dimension is defined and, for each
dimension from 0 to n, the Betti numbers measure the maximum number
of independent cycles of this dimension. It is possible to give an intuitive
representation of the meaning of Betti numbers of dimension 0 and
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dimension 1. The Betti number of dimension 0 measures what’s called the
components of the complex, that is, the number of isolated parts that constitute it.
Thus, for example, the Euclidean space from which the points situated
on a circular annulus are removed has its Betti number of dimension 0
equal to 2, because a point on the shaded exterior part and a point of the
interior parts cannot be connected by a continuous path. They therefore
belong to two distinct components (Figure 3).
The Betti number of dimension 1 measures the maximum number of
independent closed curves that are not reducible to a point by continuous
deformation. Consider for example the Euclidean plane pierced by the
hole z. The edges of the holes are curves irreducible to a point, the Betti
number of dimension 1 is equal to 2.
We can now turn to the presentation of duality theorems. The first
duality theorem, due to Poincaré, refers exclusively to the intrinsic structure of a complex. Alexander’s duality theorem is a direct consequence of
Poincaré’s theorem, and immediately attains the end to which we intended
to arrive. It reduces the structural study of the space that receives the complex to a structural study of a complex (Figure 4).
It is Alexander who operates the reduction of situational properties to
intrinsic properties, and it is necessary for us to show how this is latent in
Poincaré’s ‘internal’ theorem.
Poincaré’s theorem proves that for an n-dimensional closed multiplicity
(complex satisfying certain conditions) the Betti numbers of dimension k
Figure 3.
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I N T R I N S I C P R O P E RT I E S A N D INDUC E D P R OP E RT IE S
Figure 4. (Lefschetz 1930, Figure 13)
are equal to the Betti numbers of dimension n – k. This internal symmetry
between the Betti numbers of a same complex Q comes from the Betti
numbers of dimension k being Betti numbers of dimension n – k for a different complex to the first, its dual complex. The dual complex Q* of a
complex Q results in another cell decomposition of the same points as
those figured in the first cell decomposition of Q (Figure 4). Each (n – k)dimensional cell of Q* is in intersection with a k-dimensional cell of complex Q. This notion of the dual complex is essential to assure the passage
from the internal case to the external case. Considering that we were party
to a conception in which the Betti numbers of a multiplicity were the very
characteristics of this multiplicity, here’s the place where the studied object
divides in two and that the Betti numbers of the new complex can be
determined from those of the former. This duality within the same entity
therefore already has the sense of a relation between two discernable entities albeit still indissolubly connected to one another. A simple change of
perspective will dissociate and transform an internal symmetry into a true
correspondence of two distinct entities (Figure 5).
Consider in effect a k-dimensional complex Q embedded in an n-dimensional Euclidean space R n. Let R n – Q be the complementary space of the
complex Q, that is, the space whose points belonging to Q are removed.
Considering that there was earlier a duality between the Betti numbers of
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Q
C
D
C
D
Figure 5. (Alexandroff and Hopf 1935, Figure 35)
the same complex, Alexander’s theorem proves a new duality between the
Betti numbers of Q and those of its complementary space Rn – Q.6 Let pr(Q)
be the Betti number of the dimension r of the complex Q, we have pr(Q) =
pn–2–1(Rn – Q). The Betti number of dimension r of Q is equal to the Betti
number of dimension n – r – 1 of Rn – Q, except in the case in which r = n – l,
for which we have pn–1Q = p0(Rn – Q) – 1. It is necessary to insist on the
meaning of this result. In n-dimensional Euclidean space, all the Betti
numbers are zero except that of dimension 0 which is always equal to 1 in
the spaces composed of a single part. It is only because a complex Q is
introduced in Rn that a more complicated structure than the primitive space
results for the space Rn – Q. Thus in the case of Figure 5, it is only because
the space Q, pierced with two holes C and D, is removed from the space Rn
that, for Rn – Q, there results the existence of cycles C´ and D´, of dimension
1, enlaced with C and D, and not reducible to a point by continuous deformation. The Betti numbers of Rn – Q, which express the structure of this
space, therefore express in the same way the nature of the action that the
complex Q exercises over Rn. Alexander’s theorem can therefore predict in
advance the result of this action of Q on Rn by the knowledge of the specific
structure of Q. How Alexandroff and Hopf can write the following can now
be understood:
Alexander’s duality theorem belongs without doubt to the most important discoveries of topology in recent years. Everything we know about
the situational properties of polyhedra and closed sets in multidimensional space is derived from him. The situational properties of a set F in
a space R are primarily the structural properties of the complementary
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space R – F, and the duality theorem teaches us to determine these properties, in the case of a polyhedron embedded in Euclidean space, provided that they are allowed to be expressed by Betti numbers and torsion
groups7. (Alexandroff and Hopf 1935, 449)
Here is the simplest case in which these considerations are applied: Jordan’s
theorem teaches that a closed curve in the plane divides the plane into two
separate regions of which it is the common boundary. The theorem manifestly characterizes the action of the curve on the space. Brouwer and
Lebesgue have shown, long before Alexander had proclaimed his general
theorem, how it was possible to link this action of an n-dimensional closed
multiplicity on an (n + 1)-dimensional ambient space to the intrinsic properties of the multiplicity. With Alexander’s formula, Jordan’s result is
immediate: a closed curve C is a 1-dimensional complex, its Betti number
of dimension 1, p1, is equal to 1. In the plane in which n = 2, by calling
p0(R2 – C) the number of isolated components of the space whose points
situated on the curve C are removed, we therefore have:
p0(R2 – C) = p1(i) + 1 = 2
The action of the curve on the space is therefore determinable from the
knowledge of the structural invariants of the curve considered intrinsically. What this result had from the moment of its discovery that is extraordinary is highlighted by Pontrjagin in the introduction to a very important
article on duality theorems. Pontrjagin writes in effect:
In his celebrated memoir of 1895 on Analysis situs published in the Journal de l’Ecole Polytechnique, Poincaré proved the duality theorem that now
bears his name and which established the identity of the rth and the
n – r th Betti number of an n-dimensional orientable multiplicity. At about
the same time, Jordan stated for the first time the theorem related to
closed curves. No one suspected then that two totally different theorems
belonged to the same circle of ideas. (Pontrjagin l931, 165)
Nobody could indeed have suspected, before the development of algebraic
topology, that the properties of internal structure discovered by Poincaré
would someday explain the extrinsic situational properties expressed by
Jordan’s theorem.
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4. THE LIMITATIONS OF REDUCTION
Duality theorems allow the determination of the structural properties of
the complementary space of a polyhedron embedded in Euclidean space by
the knowledge of the structure of the polyhedron in question ‘provided
that they are allowed to be expressed by Betti numbers and torsion groups’
(Alexandroff and Hopf 1935, 449). Now the facts have come to show that
this restriction was essential and that the reduction of situational properties to structural properties could never be completed. It seems as though
topology could only have developed by siding with Leibniz, but it always
encounters facts that side with Kant and require the search for new
methods. The discovery in question is that of Louis Antoine (Antoine 1921,
221). Studying the case of two Jordan curves F and f each situated in spaces
F1 and f1, he examines to what extent the homeomorphism that always
exists between two Jordan curves can be extended into their neighborhood. Three cases are possible: 1) that in which the homeomorphism of
the curves extends to any space; 2) that in which the homeomorphism
extends to a neighborhood that exceeds the curves without covering any
space; 3) that in which the homeomorphism of the curves can be extended
for any region exterior to the curves. In the case of plane curves, they are
always of the first case and the structure of the curve completely determines the structure of the space that contains it. On the other hand, for
curves in 3-dimensional Euclidean space, three cases can be presented and
Antoine effectively constructs a Jordan arc on a torus whose correspondence with a line segment does not extend to any neighborhood. The two
curves F and f being homeomorphic, their structural invariants, by virtue
of Alexander’s theorem, determine structural invariants identical to their
respective complementary spaces F1 – F and f1 – f, but the identity of these
invariants is not sufficient for the spaces F1 – F and f1 – f to be homeomorphic.
They are not determined univocally by their Betti numbers, therefore by
the curves that can be inserted into them, and their structural differences
are irreducible. If it can be observed that the two 3-dimensional Euclidean
spaces F1 and f1 are identical and that it is the introduction of homeomorphic curves F and f in each of them that makes them profoundly dissimilar,
this accounts for all that the invariant structure of curves lets escape from
the relations between the curve and the space. The situational properties,
reducible to structural properties in the case of two dimensions, cease to be
so in the case of three dimensions. At this level of reality, the distinction of
an aesthetic and an analytic subsists.
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CHAPTER 3
The Ascent towards the Absolute
In Cartesian metaphysics there is an essential dialectical reasoning: the
passage from the idea of imperfection to the idea of perfection and to God
who is the cause of the presence in us of the idea of the perfect. We will
focus mainly on two stages in this passage, as found in the fourth part of
the Discourse on Method. The first is where Descartes asserts the logical anteriority of the idea of the perfect with respect to the idea of the imperfect.
The imperfect being can only be understood by reference to the perfect
being whose existence is thus enveloped in its very own:
Next, reflecting upon the fact that I was doubting and that consequently
my being was not wholly perfect (for I saw clearly that it is a greater
perfection to know than to doubt), I decided to inquire into the source
of my ability to think of something more perfect than I was; and I recognized very clearly that this had to come from some nature that was in
fact more perfect. (Descartes 1985, 127)
Gilson compares this text to a passage from the interview with Burman
where the statement of this rule is found: ‘And every defect and negation
presupposes that of which it falls short and which it negates’.1 Not only
does imperfection presuppose perfection but, and this is the other point
that we insist on, imperfection being only a privation, it is possible, solely
by consideration of the imperfect, to determine the attributes of the perfect
being. This is what the text of the Discourse shows:
For, according to the arguments I have just advanced, in order to know
the nature of God, as far as my own nature was capable of knowing it,
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I had only to consider, for each thing of which I found in myself some
idea, whether or not it was a perfection to possess it; and I was sure that
none of those which indicated any imperfection was in God, but that all
the others were. (Descartes 1985, 128)
The distance that separates perfection from imperfection is thus inscribed
in the very nature of the imperfect being. The mind rises to the absolute
in a movement whose steps are controlled by the end that is perceived
from the starting point. The structure of the imperfect being then takes
on its true meaning: its complication or its obscurity are only deviations
with respect to the transparent simplicity of the final vision, the ascent
towards perfection seems to go through in reverse the stages of an anterior
degradation.
It is possible to retrieve in some theories of modern algebra similar relations between perfection and imperfection, we will study in this chapter
the necessity of this reference to an absolute which lets itself be seen in the
imperfect nature of certain mathematical entities, and this ascent toward it
in a series of steps each of which effaces some impurity, up to the last
where every defect is rectified. This is a very different mode of thought to
those arising in ordinary arithmetic. Arithmetic is in effect the domain of
recurrence to infinity, whereas what is characteristic of the movement of
the theories that will be considered is the existence of an end conceived
in advance as a term of the ascent. In class field theory, that goal is the
absolute class field, in the theory of the uniformization of algebraic or analytic functions on a Riemann surface, it is the universal covering surface.
The logical framework of each of these two theories is derived from Galois’
theory of algebraic equations. Although several authors2 have also already
presented the philosophical importance of this, we think it should be shown
anew how it contains the mathematical tools necessary for the passage to
the absolute.
1. GALOIS’S THEORY3
Let there be a number field k, that is, a set of arbitrary numbers satisfying
the axioms of addition, multiplication and division, and let f(x) be a polynomial in x of degree n and with coefficients in k. This polynomial is irreducible in k if k contains none of its roots a1 f an . Thus, for example, the
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polynomial x2 – 2 is irreducible in the field of rational numbers since its two
roots + 2 and - 2 are not contained in it.
Let K be the field obtained by adding to k a root a1 of the polynomial.
This extension K of k is written k (a1) . If the n fields conjugated k (a1) f k (an)
coincide, the unique field K thus defined contains all the roots of the polynomial f(x) and is said to be Galoisian over k. The degree of the extension
K over k is equal to the degree n of the irreducible polynomial f(x). The
Galois group G of polynomial f(x) is formed by the internal transformations
(or automorphies) of the field K which leaves the elements of k (all contained in K) fixed and permutes between them the roots ai of the proposed
polynomial. As n conjugated roots a1 f an can correspond to a root ai , the
group contains n substitutions. The order of this group is thus equal to the
degree of the extension field K.
These definitions will allow us to understand what could be called the
‘imperfection’4 of the base field with respect to a given polynomial. This
imperfection resides in that it requires an extension of degree n to pass
from field k to field K which contains all the roots of the polynomial in
question and is measured by order of the Galois group attached to the
equation. We are going to see how, by ‘ascending’ from k to K, the intermediate field k´ can be considered such that k 1 kl 1 K and the imperfection of which decreases as K is approached. In effect, let k´ be such an
extension field, such that K is no more than degree m with respect to
kl (m 1 n) . Galois’s theorem univocally associates a subgroup g´ of G to this
intermediate field, defined as follows: g´ leaves all the elements of K which
are in k´ invariant and permutes only those that remain to be integrated. In
addition, the order of this subgroup is equal to the degree m of the extension that has yet to pass from k´ to K, and thus measures what imperfection
remains in k´. The ‘ascent’ from k to k´ is therefore accompanied by a
‘descent’ in the order of the groups attached to these fields. And, to the
final field K, in which all imperfection has disappeared, since it contains all
the roots of f(x), corresponds the smallest subgroup of G, the unit 1 group
which only contains the identical transformation. The interest of the logical schema of Galois’s theory is considerable. A certain number of other
theorems are encountered in algebra that, for a basic domain imperfect
from a certain point of view, assert the existence of an extension in which
this imperfection has disappeared. Thus, for example, there exists for all
number fields, an extension X which is ‘algebraically closed’, that is, such
that any polynomial in x with coefficients in X is completely decomposable
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into first-degree factors with coefficients in X , but the completion theorem
of Galois’s theory is infinitely richer. It associates with each stage of the
ascent of k to K a number measuring the remaining difference between the
stage in question and the final stage. The forecast of the end is even more
precise when the equation f(x) = 0 is solvable by radicals, that is, when the
field K that contains all the roots of the proposed equation can be constructed by successive adjunctions of magnitudes n a , a belonging each
time to the field already obtained. In this case a increasing series of fields
can be ordered from k to K having special properties k 1 kl 1 k m f 1 K to
which correspond term for term a decreasing series of groups G 2 gl f 2 I
such that this series cannot be extended by the adjunction of any inserted
element. The initial data implies then not only the existence of the end and
the difference that separates it from the base field, but also the exact
number of stages to be executed to arrive at it. The two essential moments
of the passage to the absolute are indeed found in the Cartesian Meditations: first, the vision of the perfect being whose existence is implicated
by that of the imperfect being; and secondly, the consciousness that the
reasoning to be effectuated to attain the absolute is, to some extent, given
with the imperfect being proposed, whose structure is thus called a complete model in which its defects are effaced.
It is by placing ourselves at this Cartesian or Galoisian point of view that,
class field theory will now be examined.
2. CLASS FIELD THEORY5
Class field theory issuing entirely from the genius of Hilbert is an extremely
abstract theory of algebra that calls upon a large number of notions that are
difficult to grasp, but it stands out as one of the clearest examples of the
new mathematics in which successive edifices tend toward an end that
their movement anticipates. It presents once again the base of an ascent
from fields to extension fields up to a maximum entity: the absolute class
field that has, with respect to an initial base field, the greatest simplicity of
which an extension is possible. Its philosophical richness is moreover inexhaustible in this ascent towards the absolute, because this structural solidarity between the elements of a whole and the whole to which they
belong, which was described in Chapter 1, can be found in it. The two
problems are also so closely connected that it is impossible to present the
contribution of one or the other separately.
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The ‘elements’ of a ‘whole’ are here the ‘ideals’6 of an algebraic number
field. They remain ‘ideals’ in the successive extensions of this field, but
their internal structure varies with the extension that each time is considered and expressed as a global characteristic of the field in which they are
embedded. If any of these fields can then be conceived as the ultimate term
of an ascent, it is because in it finally all the ideals of the base field find
the most uniform and simplest internal structure. More precisely, given a
prime ideal of a base field k, this ideal does not remain prime in an extension K of k. Knowing the degree of extension of K with respect to k, can the
mode of decomposition of prime ideals of k in K be predicted? Conversely,
knowing the laws of decomposition of ideals in an extension field, can
the nature of the envisaged extension be characterized? These are two
reciprocal problems of solidarity between a whole and its parts, problems
comparable to problems of metrization and extension envisaged above
(Chapter 1), but there is a new element added, namely the ascent towards
a maximal field which establishes a connection between the problem of
Galois’s theory.
Hilbert stated in 1898, without proof, the principal theorems of class
field theory in the case of particular extension fields (non-ramified abelian
extension fields). These results have been proven and extended to much
more general categories of extension fields by Furtwängler in 1907 and
Takagi in 1920. We confine ourselves here to simple cases envisaged by
Hilbert and would like to briefly highlight the connection established
between the set of extension fields, which contain the field k, and the set
of groups of ideals that are contained in this field.
The definition of groups of ideals of a field k can be made as follows: let
A be the group of all ideals of the field k and let S be the group of principal
ideals of the field, which is evidently a subgroup of the group A. All subgroups H of A, which contains the group S of principal ideals, can be envisaged as groups of ideals. These sub-groups are thus arranged between a
maximal group A and a minimal group S, and they each determine a division into classes of ideals of A, two ideals being in the same class with respect
to H when their quotient is contained in the considered subgroup H.
The notion of ‘groups of ideals’ being thus clarified, Hilbert called ‘class
fields’, for the group of ideals H of a field k, a certain algebraic extension
field K such that only the prime ideals of k belonging to H decompose in K
following a certain simple law that he indicates.7 The other ideals are
decomposed according to a more complicated law. Furtwängler (1907) and
Takagi (1920) then prove the two reciprocal theorems that allow for a
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biunivocal correspondence to be established between groups of ideals in k
and the extensions over k: for any group of ideals H situated in k, there
exists an extension K of k that is a class field for H. And conversely: any
field K which is relatively abelian over k is a class field for a certain group
of ideals H situated in k. The degree of the extension K over k is equal to
the order of the group A/H of classes of ideals determined by the division
into classes that the subgroup H operates in the group A of all the ideals of
the field k. In addition, this biunivocal correspondence of fields and groups,
like the Galois theory, establishes a connection between the ascent in the
extension field and descent in the groups. It shows in effect that the relation K l 2 K m entails the inverse relation H l 1 H m for the groups of ideals H´
and H´´ corresponding to two distinct class fields K´ and K´´.
These results therefore establish first of all a close solidarity between the
laws of decomposition in K of prime ideals of k and the overall features of
the extension K, but they also give us a richer result. They let us foresee the
existence of an absolute class field that contains all the class fields for all
groups of ideals H of k, and that is not contained in any. Since there is in
effect a minimal group in the series of subgroups, the group S of principal
ideals, and that to a descending hierarchy between groups corresponds an
ascending hierarchy between corresponding fields, there is a maximal class
field, one corresponding to the group of principal ideals. It turns out then
that the maximal class field is such that all the ideals of k will be subject to
the same modification of structure: they all become principal ideals. To the
fact of being the last, the maximal class field combines the fact of being the
simplest: the ascent comes to an end at a stage where certain of the most
important differences between the ideals of the base field have disappeared,
and the existence of this stage was implicated from the moment the function was established between the groups of ideals and class fields. From the
theory in its entirety emerges the same Cartesian movement as Galois’s
theory.
3. THE UNIVERSAL COVERING SURFACE
The theory that will now be presented has a far greater philosophical
importance than the previous theories, because the ascent towards the
absolute has as a consequence, not only to confer on a mathematical entity
the greatest simplicity of internal structure possible, but to make it able to
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give rise to entities other than itself. The entire second part of this essay
will be dedicated in effect to the study of the procession of mathematical
entities one with respect to the other, and, as we shall see, this movement
is only possible if the structure of the entity from which the other entities
proceed was brought to a certain prior state of perfection. In so far as we
will have shown how the existence of a universal covering manifold is
implied by the structure of any manifold whatsoever, the ascent towards
this maximal surface is still a problem concerning the completion of an
internal structure, and the elimination of entanglements that it could primitively present. But, insofar as the universal covering manifold immediately
gives rise to certain functions whose existence was impossible on all surfaces covered by this universal manifold, the passage to the absolute confers on the surface a power of production that it did not possess at earlier
stages; the structure of the perfect entity is radiant in surprising richness.
To study the covering, we can taken as an object of study either: the
n-dimensional complex of combinatorial topology, as defined in Chapter 2
in the section devoted to duality theorems; or, the n-dimensional manifold
of ‘set theoretical’ topology, defined by the axioms of neighborhood that
Threlfall presents as follows: 1) to any point P of a set is associated a neighborhood of points of the set such that any subset containing this neighborhood is equally a neighborhood of P; 2) to each neighborhood V is associated
a biunivocal correspondence between the points of V and the points of a
hypersphere in n-dimensional Euclidean space; 3) the manifold is in one
piece (2 points can always be reunited by a continuous path). We will confine ourselves in the following to manifolds of two dimensions, that is, to
surfaces.8 What we will consider as an inherent imperfection in the structure of a given surface is the fact that this surface cannot be simply connected. A surface is said to be simply connected if any closed curve on this
surface is reduced to a point by continuous deformation. The Euclidean
plane is simply connected, but the surface of the torus is not. By arranging
in effect all closed curves reducible to one another by a continuous deformation in the same class, at least two classes of closed curves on the surface
of the torus that are not reducible to one point are discovered: those surrounding the central ‘hole’ and those that go around the surface like the
circle AOB (Figure 1). The set of classes of closed curves issuing from any
point O of a surface forms a group: the fundamental group of the surface,
defined by Poincaré, and whose structure thus measures as it were the
multiplicity of the internal connection of the surface.
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B
0
A
Figure 1. (Seifert and Threlfall 1934, Figure 4)
We’ll now see how it is possible to define on a multiply connected surface F a series of manifolds, each of which ‘covers’ the surface F. These
manifolds lie between F and a universal covering manifold that covers all
the preceding without being covered by any, and on which the multiplicity
of connection to the basic surface is completely erased.
A surface F is said to cover a given surface F if:9 1) at any point of F there
corresponds at least one point of F which covers it; 2) for each point
P 1, P 2, f P n that covers in F a point P of F there exist neighborhoods
V (P 1), V (P 2), V (P n) that can be represented topologically on the neighborhood V(P) of P in F; 3) if Q is a point in the neighborhood V(P) of a point
P on F, and Q a point covering Q in F , then Q belongs to one of the neighborhoods V (P 1), V (P 2), V (P n) that cover V(P) in F.
There exist in general several covering surfaces of the same surface. It is
possible for them to become covered by each other and for each to possess
like the basic surface a fundamental group that expresses its degree of connection. The essential theorem of the theory then establishes an isomorphism between the fundamental groups of different covering surfaces of a
given surface F and the subgroups of the fundamental group of F. The set
of these covering surfaces is thus in biunivocal correspondence with the
set of subgroups of the fundamental group. By considering then the
smallest subgroup of the fundamental group, that is, the unit group, which
corresponds to the class of paths reducible to one point by continuous
deformation, the dual result expected is obtained. There exists a maximal
or universal covering surface that covers all the other covering surfaces,
and this surface possesses a character of absolute simplicity since any closed
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path on it, its fundamental group being the unit group, is reducible to a point.
This universal surface, the only one of all the surfaces that cover the initial
surface, is therefore simply connected. It can be characterize as follows: to
a given point P on the initial surface F, there correspond as many conjugate
points Pl, P2 . . . on the covering surface F as there exist on F of closed
contour issuing from P, not reducible to one another. In sum, each point of
the multiply connected surface is only unique in appearance, it possesses a
hidden complexity, which is reflected by the degree of connection of the
surface and by dissociating all the confounded points to scatter them separately on the covering surface, the ideal simplicity of perfect surfaces is
restored to the surface. We will see in the next paragraph how by restoring
to the surface its simplicity, its fertility has simultaneously been restored.
4. THE UNIFORMIZATION OF ALGEBRAIC FUNCTIONS
ON A RIEMANN SURFACE
The problem of the uniformization of analytic functions (we’ll only be
occupied with analytic algebraic functions)10 is in itself a problem whose
logical signification is comparable to that of the preceding problems: it is
still a matter of eliminating the imperfections of certain mathematical entities by the passage from what they are primitively to an ideal of absolute
simplicity, whose existence is implicated in the entanglements of their
structure. An algebraic function g = f (x) is said to be multiform around an
algebraic point of ramification of order m if, when the variable turns around
this point, the function can take different m values successively before
returning to the first of these values. It is therefore said that m branches of
the function end up in this point. A uniform function on the contrary can
only take one value at one point. The problem of uniformization consists
then in finding a complex parameter t such that g and z are left to be
expressed as uniform functions g = { (t) and z = } (t) of the new variable
t. The passage from the old to the new variable therefore has the consequence of making the points of ramification of the algebraic function disappear, and, by presenting the theory of Riemann surfaces of algebraic
functions, it will be shown how the possibility of solving the problem of
uniformization is connected to the fact that these Riemann surfaces contain within them the elements of the construction of a universal covering
surface.
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If the complex plane z is itself taken as a domain of existence of the function g = f (x) , we have said that the function could then be ramified into
several branches around certain points: thus the elliptic function
g = A0 z4 + A1 z3 + A2 z2 + A3 z + A4
is divided around each of the 4 supposed distinct roots of the polynomial
under the radical, in 2 distinct branches. Instead of admitting that the distinct branches of g end up in a same point of the complex plane, Riemann
imagines a surface composed of different sheets of the complex plane,
welded in crosses along the cuts joining two by two the points of ramification. In the case of the elliptic function just considered, the Riemann surface is composed of two sheets welded in a cross along the 2 cuts uniting
two by two the 4 points of the ramification. This gives, in a general way,
the Riemann surface of the algebraic function in question. This surface,
from the point of view of its structure, is infinitely more complicated than
the primitive complex plane. The cuts which it is subject to, the cross weld
of the edges of each cut, make it not simply connected. It is possible in
effect to trace closed curves, called retrosections, on this surface, none of
which divides the surface into two regions, so that to pass from one to the
other it is necessary to meet the curve, and which are irreducible to a point
by continuous deformation. These topological complications of Riemann
surfaces are compensated for by the fact that the comportment of the
algebraic function on this surface is much nearer to uniformization than
on the complex plane. If a global uniformizing function still doesn’t exist,
there is already at each point on this surface a ‘local uniformization’.
Let there be for example the function g = m z . The Riemann surface of
this function possesses, at the complex point z = 0, m superposed sheets
connected so as to establish a circular permutation of m values of the function at the origin. If we let t m = z, we have g = t , there is punctual biunivocal correspondence between the m sheets of the surface welded together at
the point z = 0 and m separated portions of the simple complex plane of the
variable t, in the neighborhood of the point t = 0. The variable t thus uniformizes the function g , near the origin. The Riemann surface can even be
defined as it was by Weyl (1913), without proceeding to this complicated
superposition of sheets and by only appealing to the existence, in the
neighborhood of each point p0, of a local uniformization t(p) such that if
f(p) is a regular or branched analytic function in the neighborhood of p0,
f(p) can be represented in the form of a power series in t(p):
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f (p) = a 0 + a 1 t (p) + a 2 [t (p)] 2 + f
The Riemann surface thus constituted (Weyl 1913, 36 [1964, 36]) by a
juxtaposition of neighborhoods in which local uniformizations are defined,
seems to Weyl comparable to those n-dimensional multiplicities of Riemannian differential geometry, defined by the value of their ds2 in the
infinitesimal neighborhood of each point. This comparison is only possible
thanks to the new definition of the Riemann surface proposed by Weyl,
which itself recognizes in addition (Weyl 1913, 36 [1964, 36]) that nothing
in the writings of Riemann suggests that Riemann had established the
connection between the spaces that he introduced in geometry and the
surfaces that he introduced in analysis. An essential difference can even
justifiably be seen between Riemann spaces and Riemann surfaces. The
spaces defined by their ds2 are explored in a purely local way, and we have
indicated which richness of connections their elements remained amenable to. The principal characteristic of the Riemann surface of an algebraic
function, on the contrary, is to possess that global topological structure
which confers on it the cuts and the retrosections mentioned above, and
whose consideration is essential to the problem of uniformization. It is
possible in addition to intuitively represent this structure of the Riemann
surface of an algebraic function. It can be shown in effect that a similar
surface, by continuous deformation, can always take the form of the surface of both sides of a disk with holes (Figure 2).
If the disc has p holes, there are 2p possible independent retrosections,
that is, 2p closed curves, irreducible to one another by continuous deformation, and none of which divide the surface into two distinct regions.
R
Q
Q'
P
R'
Figure 2.
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Consider, for example, the case in which p = 2. At each of the two holes
there corresponds a pair of retrosections Q and Q´ for the first, R and R´ for
the second: it is easy to see that the surface (when the retrosections have
not been cut) is not simply connected and possesses at each point a fundamental group in the sense defined by Poincaré. Let us join in effect any
point P to the point common to two conjugated retrosections. Already four
classes of closed paths are obtained issuing from a same point and irreducible to one another. These classes generate the fundamental group of the
surface. All that remains for us is to show that all the elements necessary
to prove the existence on this Riemann surface of a global uniformization
are thus possessed.
For this we will turn to the considerations announced at the beginning
of this section. There is a close connection between the fact of a surface
being simply connected and the fact of giving rise to a function that ensures
the conformal11 representation of this surface either on the totality of the
complex sphere, on this sphere from which a point is removed, or on the
unit circle of the complex plane. This is a theorem of considerable mathematical importance, as much by the difficulty of the means that Riemann,
Poincaré, Hilbert, Koebe and many others made use of to prove it, as by the
immensity of the new horizons that it allowed to be discovered. Let p0 be
any point on the simply connected surface, t0 the corresponding point of
the complex plane, the function that ensures the conformal representation
of the surface on the plane has the value t0 for the value p0 of its variable,
and there exists an inverse function which, taking the value p0 on the
surface for the value t0 of its variable thus ensures the reciprocal representation of the plane on the simply connected surface. The problem of uniformization is therefore solved for a simply connected surface. The theorem
of conformal representation in effect provides this function t(p) whose
inverse p(t) is a uniform function on the complex plane t. In addition, any
function f(p) on the simply connected surface is also a uniform function
of t. The new variable is therefore a global uniformization in the sense
defined above.
When the Riemann surface of the algebraic function in question is
not simply connected, the brilliant idea of Poincaré and Koebe was to consider the universal covering surface of this surface: since this surface is
itself simply connected, the theorem of conformal representation is itself
applicable, and it gives rise to a global uniformization of the function f(p)
attached to the primitive Riemannn surface, which solves the problem
completely.
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We therefore reach the conclusion of this study on the uniformization of
algebraic functions at a dual result. The universal covering surface is presented first with a character of perfection with respect to the primitive
Riemann surface, because the ascent towards it has made the ramifications
of the algebraic function under question disappear. But this is still only an
internal perfection that results from a simpler structure letting itself to be
seen in the design of the primitive structure. Another perfection is manifested that is no longer only of interior completion, but of creative power.
It is that privilege that certain domains have to give rise to a new world of
functions and integrals, which will occupy the second part of this essay.
The studies that we have pursued during these three chapters thus show
the variety of logical connections that are manifested within mathematics.
The solidarity of the whole and its parts, the reduction of relational properties to intrinsic properties, the passage from imperfection to the absolute,
here are so many attempts at structural organization that confer on mathematical entities a movement towards completion by which it can be said
that they exist. But this existence is not only manifested in what the structure of these entities imitate, the ideal structures to which they allow
themselves to be compared. It happens to be that the completion of an
entity is at the same time the genesis of other entities, and it is in the logical
relations between essence and existence that the schema of new creations
is inscribed.
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SECTION 2
The Schemas of Genesis
CHAPTER 4
Essence and Existence
1. THE PROBLEMS OF MATHEMATICAL LOGIC1
The problems of the passage from essence to existence, which will occupy
us henceforth until the end of this essay, belong to questions that have
been raised for a long time by the development of mathematical logic.
It doesn’t seem to us however that logic has the benefit in this regard of a
special privilege. It is in effect only one mathematical discipline among
others, and the geneses that are manifested there are comparable to those
observed elsewhere. The presentation that we will make of what could be
called the metaphysics of logic therefore has above all the value of an
introduction to a general theory of connections that unite the structural
considerations to assertions of existence.
Two periods can be distinguished in mathematical logic: one, the naive
period, ranging from the early work of Russell until 1929, the date of the
metamathematical work of Herbrand and Gödel which marks the beginning of what could be called the critical period. The first period is that
where formalism and intuitionism are opposed in discussions that extend
those raised by Cantor’s theory of sets. Proponents of the actual infinite
claim the right to identify, for a same mathematical entity, the essence of this
entity, as a result of its implicit definition by a system of non-contradictory
axioms, with the existence of this entity. We know, on the other hand, the
attitude adopted towards an entity whose construction would require an
infinite number of steps, or a theorem that is impossible to verify, by those
that Poincaré called the pragmatists: in his Last Essays, the famous mathematician says that ‘they see in it only unintelligible verbiage’ (Poincaré
1913, 66). Poincaré relies mostly on the assertion of existence contained in
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the Zermelo theorem: ‘there is a way of well ordering the continuous,’ to
show how this statement could be meaningful only if the manner in which
it is necessary to proceed for well ordering the continuous was really
known. Asserting the possibility of an unrealizable operation is to assert
something which is either meaningless, or false, or at least unproven.
In his essay on the Infinite, Hilbert (1926) justifies the introduction of
transfinite elements in mathematics in putting them side by side with the
ideal elements introduced by Kummer in algebra, which were mentioned
in Chapter 3. Ideal numbers are no more parts of the numbers of a field
than transfinite elements of axiomatized mathematics are determinable in
a finite number of steps. But, just as the consideration of ideal numbers
is essential to generalize the theorem of decomposition into prime factors,
the consideration of transfinite elements is necessary to generalize the
application of the excluded middle. Requiring their elimination would
imply abandoning the rule of contradiction in logic. Here’s how, in an
essay from 1923, the connection is presented between the transfinite
axiom of choice and the excluded middle for infinite sets. The axiom is
introduced as follows:
A (xA) " A (a)
The object xA is an object such that if the property A agrees with it, then it
certainly agrees with all the objects a . For this, it is necessary to conceive
the object xA as that in which there is the least chance that property A is
applied. Thus, for example, if A is the property of being bribable, xA is the
least bribable of men (Hilbert 1923; 1936, 183 [1996, 1141]). It is then
evident that if this ideal man (in every sense of the word) is also bribable,
then all men will be. Hilbert and Bernays proved that it is possible to
deduce the axiom of the excluded middle from the transfinite axiom
(a ) A (a) " (Ea) A (a)
(If all a do not possess the property A, there exists at least one a that does
not possess it.)
The object xA , distinguished in the set of entities that have property A,
is obviously not likely to be constructed in a finite number of steps, since it
is defined by comparison with the infinity of entities of the set. It is no less
true that the admission of the axiom, where it is formally defined, is equivalent to the use of the excluded middle. If therefore it is possible to prove
the consistency of a system of axioms containing the transfinite axiom, it
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is, in the same way, as legitimate and advantageous to talk of the existence
of the object xA as the existence of the point at infinity in geometry, of
imaginary numbers, or of ideal elements of a number field. It can therefore
be said that in 1926 the problems of mathematical logic arose again in the
same terms as the discussions at the beginning of the century, relating to
the existence of the transfinite. True to its Leibnizian origins, formalism
always considered that the passage from essence to existence should consist uniquely in the proof of the ‘compossibility’ of essences, of the consistency of the axioms that define them.
It is precisely in the research of the critical period with respect to the
consistency of arithmetic that we seem to see asserted a theory of the relations between essence and existence as different from the logicism of the
formalists as from the constructivism of the intuitionist. We will first recall
the principal features of this evolution internal to logic and then try to
extract a philosophy of mathematical geneses, whose scope goes far beyond
the domain of logic. The considerable work that Bernays wrote on Hilbert’s
logic is in this regard a source of inexhaustible richness.2
We have already indicated at the beginning of this essay how Hilbert
conceived the necessity of a metamathematics whose object would be to
prove the consistency and completeness of systems of axioms of formalized
mathematics. Metamathematical research was undertaken from two different points of view, and the connection between the two methods is in our
view an essential fact: one of these methods employs the extensive processes of set theory, the other more in line with the directives given by
Hilbert himself, constitutes what Hilbert called ‘proof theory’.3
From the point of view of proof theory, a mathematical theory contains,
as given with the initial system of axioms, the set of propositions that can
be obtained from these axioms by applying the rules of substitution and of
passage, just like the ‘conclusive schema’:
S
S
T
T
In these conditions, to prove the consistency of a system of axioms amounts
to proving the compatibility of the set to its consequences, that is to obtain
the certainty that it is not possible to encounter in this set the contradictory
proposition: p. p (p and not p). Similarly, the system is said to be complete
if any proposition of the theory is either provable, or refutable (by proof
of its negation), that is, if one always knows which proposition p or p is
provable. The properties of ‘consistency’ and ‘completion’ for a system of
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axioms, and of ‘provable’, ‘refutable’, ‘irrefutable’ for the propositions of a
theory, are called structural properties (Cf. Carnap 1934) because their
attribution to a system or to a proposition requires an internal study of the
set of consequences of the considered system.
We will now define the second method of metamathematical study of a
formalized theory, the ‘extensive’ method, which adjoins to the system of
axioms studied the consideration of individual domains that may be used
as values for the arguments of logical functions of a formula of the theory.
Certain metamathematical notions exist that can only be defined in the
extensive method. The principals are those of ‘universal validity’ (Allgemeingüligkeit) and ‘possibility of realization’ (Erfüllbarkeit) (Hilbert and Bernays
1939, 8, 128). A formula is said to be universally valid if, whatever the
manner in which predicates are substituted for the variables representing
logical functions and whatever the individual domains substituted for the
argument variables, a true proposition is always obtained. A formula is said
to be realizable if there is at least one individual domain and one mode of
similar substitutions capable of making the formula a true proposition.
Thus, for example, the formula:
(x) P (x, x) " (x) (Ey) P (x, y)
is universally valid. Whatever the relation P and the considered domain,
for all x in this domain such that the relation P takes place from this object
to that object, there is always at least one object y such that P takes place
from object x to object y, the latter could be none other than x itself. On the
other hand, consider the system of the following three axioms (Hilbert and
Bernays 1939, 14):
(x) R (x, x)
(x) (y) (z) R (x, y) & R (y, z) " R (x, z)
(x) (Ey) R (x, y)
Suppose that R means ‘to be smaller than’. The first axiom requires that
the relation ‘smaller’ does not take place from one individual to that individual itself, the second, that it is transitive, the third, that there always is,
for all x, at least one y such that x 1 y . It is impossible to verify this system
in a finite domain, since the third axiom cannot be verified, the last number
of a finite collection not being smaller than any number. The system is thus
achieved only in an infinite domain. The properties of universal validity
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and of possibility of realization are in duality: a proposition is universally
valid only if its negation is not realizable and, by supposing the excluded
middle to be admitted, a formula is realizable only if its negation is not
universally valid.
The structural conception and the extensive conception being thus
defined, we will now present the importance of the passage that modern
mathematical logic operates from one to the other.
Consider first of all the links that exist between the structural conception
of consistency and the extensive notion of ‘realization in a given domain’.
When the given domain only contains a finite number of individuals, by
trying all possible combinations, the question (entscheiden) of knowing
whether there is a choice of values of the variables allowing the proposition obtained by the conjunction of all the axioms of the proposed system
to be verified can be settled. It can be said that this choice realized the system of axioms, and it can easily be proved by relying on the principle of the
excluded middle in the finite, that if a conjunction of axioms is thus realizable, its negation is not provable. This latter property is none other than
the consistency of the system. There is therefore equivalence in the finite
between the non-contradictory structure of a system of axioms and the
existence of a domain containing a number of determined individuals and
such that the system is realizable in this domain (Hilbert and Bernays 1939,
17). This result, so simple that it seems almost obvious, contains the seed
of a new theory of the relations of essence and existence. These two notions
cease to be in effect relative to the same mathematical entities. It is no
longer a matter of knowing whether the definition entails existence, but of
inquiring whether the structure of a system of axioms can give rise to a
domain of individuals who support among themselves the relations defined
by the axioms. The envisaged essence is rightly that of the system of axioms, but the existence that the internal study of the system allows to be
asserted is that of the ‘interpretations’ of the system, the domains in which
they are realized.
When the system of axioms can only be realized in an infinite domain of
individuals, there is no longer necessarily equivalence between the consistency of the system and the existence of an interpretation of this system.
The principle of the excluded middle is not necessarily applicable in the
narrow form that it takes in the case of finite domains, and the conjunction
of the axioms of a system can be irrefutable, hence the non-contradictory
system, without there existing in fact an interpretation of the axioms of
the system. In sum, the possibility of realization is a stronger requirement
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than consistency. A dissociation is established between the extensive point
of view and the structural point of view. The problem of realization must
leave room, according to Bernays, for a purely internal search for irrefutability. This is what Bernays calls the negative conception of consistency.
Bernays has in this sense proven (Hilbert and Bernays 1939, 156) that for
the predicate calculus there was a strict equivalence between consistency
and irrefutability of the conjunction of axioms, without implying in the
least the existence of an interpretation of the system.
The facts show that it is nevertheless difficult to develop a purely structural theory of consistency. In this regard, the most authentic attempt at
a structural proof of the consistency of arithmetic is that by Gentzen
(1936). We do not propose to discuss it here in detail, referring for this to
Cavaillès’s thesis (1938b). We would like only to recall a key element of it.
Far from seeking to realize the axioms of arithmetic, as in the extensive
method, Gentzen remains faithful to the conception of Hilbert’s proof
theory and intends to follow each of the operations involved in a proof, to
prove that a contradictory proposition could not slip in at any moment
whatsoever. For this, Gentzen undertook, as Herbrand had moreover
already tried, to reduce each of the operations of a proof to increasingly
simple operations until the final formula of the proof had been put in
the form of an evidently true expression. For this, Gentzen coordinates an
ordinal number to each proof and proves an essential theorem that
amounts roughly to this: the reductions succeed because the set of these
ordinal numbers are ‘well ordered’ as defined by set theory. The proof is
thus not based on the existence of an interpretation of the system, it resorts
no less to the consideration of a set of numbers whose existence is equivalent to the completion of the proof of consistency. Whatever the intentions
of the author, the extensive point of view reappears associated with the
structural point of view. The structure of a theory is expressed anew by the
existence of entities other than the elements of the theory.
These considerations apply in an even more manifest way when one
moves from the study of consistency to that of the completion of a system
of axioms. In the purely structural sense of Hilbert, the completion of a
system is equivalent to the fact that any formula of the theory possesses,
as we have seen, either the property of provability, or the property of
refutability. In regard to the predicate calculus, Gödel proved a theorem of
completion that immediately had extensive repercussions (1930). He in
effect established that any formula of predicate calculus is either refutable,
or realizable in the domain of whole numbers. So, for the envisaged system
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of axioms, there is equivalence between the structural properties of irrefutability and the extensive property of realization. In fact, says Bernays, a
finitist theorem of completion can be extracted from Gödel’s proof, according to which there always exists, for an irrefutable formula, a formal realization within the framework of arithmetic (including the excluded middle).
Conversely, Herbrand’s theorem allows the deduction of the irrefutability
of a proposition from the existence of a domain of individuals who realized
this proposition. The domain envisaged by Herbrand to tell the truth does
not realize a logical formula like the numerical values realize an algebraic
value. Nor do they contain the possible values of arguments and of logical
functions in the general sense defined above. We shall see however in the
following chapter how it is possible nevertheless to consider them from a
certain point of view as interpretations of the system of axioms. In any
case, they are constructed in a rigorously intuitionistic way, they possess a
nature proper to domains, fundamentally heterogeneous to the nature of
the propositions of the system of axioms, and their interest results from
their existence being linked closely to the internal structure of the system
to which they are associated. Herbrand’s theorem in effect can be summarized as follows: the existence ‘in extension’ of an infinite domain in which
P is realizable is equivalent to the structural fact that P is not part of the set
of provable propositions of the theory.4 As in the study of consistency, we
encounter, at the threshold of the theory of completion, theorems that
identify the structural properties of a system of axioms with the existence
of a domain in which the propositions of the system are realized. The connection in that sense is complicated by the fact that the structural properties of P are expressed in extensive properties of P and conversely. It does
not appear to us any less like the first schema of all the mathematical
expansions that we will describe later. It is offered to us in effect like a pure
case of solidarity between a set of formal operations defined by a system
of axioms and the existence of a domain in which these operations are
realizable. It seems that a certain restriction still adheres to this logical
schema; the genesis only takes place in effect in one sense, the operations
of the domain. Now, if, between the domain and the operations definable
on it, a rigorous appropriation can be established, then one can seek just as
well to determine the operations from the domain as the domain from the
operations. We have just seen how in logic the operations rely spontaneously on a domain that they seem to summon by the very organization of
their structure. We will see that conversely, in a very large number of cases,
the domain appears already prepared to give rise to certain abstract operations.
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Our intention being to show that completion internal to an entity is
asserted in its creative power, this conception should perhaps logically
imply two reciprocal aspects: the essence of a self realizing form within a
matter that it would create; the essence of a matter giving rise to the forms
that its structure designs.
This symmetric presentation of things sometimes seems justified when it
is a matter of certain simple axiomatic theories like arithmetic or domain
theory: there is absolute reciprocity between the domain of whole numbers and the Peano axioms, between the domain of rational numbers and
the axioms that define the four arithmetic operations. One can look for
the mathematical entities that satisfy the conditions of axioms just as well
as for the axioms that implicitly define the domain in question. The expression of a closed domain with respect to certain operations shows well in
this case the tight equivalence between the ‘material’ point of view of the
domain and the ‘formal’ point of view of the operations.
In fact, the schema of geneses that we are going to describe within more
complicated theories abandons the too simplistic idea of concrete domains
and abstract operations that would possess in themselves for example a
nature of matter or a nature of form. This notion would tend, in effect, to
stabilize the mathematical entities in certain immutable roles and ignore
the fact that the abstract entities that arise from the structure of a more
concrete domain can, in their turn, serve as a basic domain for the genesis
of other entities. It is therefore only within a determined problem that
distinct functions can be assigned to different kinds of entities. The essential fact is the genesis of kinds of entity from one another, and it is in a
purely relative sense, to account for these mutual relations, that we always
use the term ‘domain’ for the given structure and that of ‘creation’ for the
final object of the described genesis. Thus, for example, in regards to mathematical logic, despite all the ‘material’ meaning attached to the expression
of ‘realization’, it seems to us more congruous with the logical schema that
we describe to say that these are the axioms that form the domain and that
the interpretations come from the domain, as the entities that it creates.
2. EXISTENCE THEOREMS IN THE THEORY OF
ALGEBRAIC FUNCTIONS
We will now turn to the study of purely mathematical examples, in which
one sees a structure preform the existence of abstract entities on the
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domain that this structure defines. Our first example is borrowed from
Riemann’s theory of algebraic functions. We saw in Chapter 3 how he had
been led to associate, to an algebraic function, a surface composed of a
certain number of sheets welded in crosses along certain cuts, and at each
point of which the function is locally uniformizable. By continuous deformation, this Riemann surface can be given the form of a two sided disk,
pierced with holes. The number p of these holes is a topological invariant
of the surface that determines the 2p retrosections of the surface, that is,
the maximum number of independent closed curves that can be traced on
this surface without dividing it into two separate regions. This number p
defines the genus of surface. That being posed, Riemann’s train of thought
is not so much to construct the Riemann surface of an algebraic function
that would be defined by its algebraic equation, but to choose an arbitrary
surface to later find out if an algebraic function exists of which this surface
is the Riemann surface. The Riemann method does not proceed directly, it
is true, to the determination of the algebraic functions of a surface, but to
the search for functions that are integrals of algebraic functions. These
integrals are called abelian integrals, and there are three kinds of them: the
integrals of the first kind that never become infinite; those of the second
kind that have poles; and those of the third kind that have logarithmic
singularities.5
Riemann’s essential theorem is the following: the number of linearly
independent integrals of the first kind is equal to the topological genus of
surface. There is a connection of extraordinary importance here between
the topological structure of the surface and the existence of abelian integrals that are everywhere finite on this surface. Also, mathematicians cannot help but use the most admiring terms to describe this dual role of the
genus. Here is what Weyl writes, for example:
One cannot refuse to recognize the importance of the number p in the
theory of functions, and yet this number is, as for its nature and its origins, a magnitude of Analysis situs. The main lines of the theory of functions . . . are all inspired by the divine lawgiver (göttliche Gesetzgeberin)
who hovers unseen above the functional reality: Analysis situs. (Weyl
1913, 134)6
We would like to go into the detail of this genesis of the integral from the
domain and show the moment in which, imperceptibly, the passage from
the structure of the surface to the existence of integrals that are everywhere
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finite is carried out. It is proven that if any two points a and b are taken on
the surface, and if the surface is cut from a to b, there exists a potential
function7 u a,b that admits the two points a and b as logarithmic singularities, and that, by crossing the cut that goes from a to b, is subject to a
determined discontinuity and is continuous, finite and uniform everywhere else on the surface. It is easy by departing from here to define a
potential function that has no logarithmic singularity (nor any pole): all
that is required is to apply the preceding theorem of existence to the case
of a succession of points z´, z˝, z n–1, z´, such that the last coincides with the
first, and which are situated on one of the 2p closed curves that does not
divide the surface into separate regions (a retrosection of the surface). The
following functions are thus obtained
u zlzm, ff u z
n-1
zl
which, by virtue of a theorem of addition, give rise to the potential
function
u (z) = u zlzm + ff + u z
n-1
zl
This potential function admits as a line of discontinuity the entire closed
cut z´z˝. It therefore no longer possesses the logarithmic singularities since
the singularities of the functions of which it is the sum cancel each other
out in pairs, and it is not uniformly zero since, when the variable z jumps
over this cut, the value of the function undergoes a determined increase.
In addition, the cut does not divide the surface into two regions, the function is finite, continuous and uniform over the whole surface. As there are
2p retrosections, the potential functions that are everywhere finite can
thus be defined, and by associating them in pairs, p linearly independent
abelian integrals of the first kind are thus obtained. It can then be demonstrated that others can be obtained, which completes the proof of the dual
role of the number p.8
The importance of the ‘canonical cutting’ of the surface is seen in this
theory: the brute surface leaves nothing to appear on it. On the other hand,
the structure it receives from its retrosections renders it apt to a creation.
The precise moment of the genesis resides in the act by which it confers on
the structure a dual interpretation: insofar as the retrosections render the
surface simply connected, they have a topological value, insofar as they
require the variables of certain functional expressions to jump, they already
distribute in advance these integrals on the surface. The passage from
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essence to existence thus becomes a connection between the structural
decomposition of an entity and the existence of other entities that this
decomposition gives rise to. We will show that this result is reflected in the
theories in which the problems of decomposition play an essential role:
algebraic theories.
3. EXISTENCE THEOREMS IN CLASS FIELD THEORY
An aspect of class field theory has already been discussed in Chapter 3. We
shall now see how the theorems of existence of this theory are closely
related to a decomposition into classes of ideals contained in the base field
k. The set of ideals of k forms a group with respect to multiplication. In this
group a division into classes is established as follows: envisage the subgroup of principal ideals (those which are generated by the multiples of a
number of the field) and two ideals are said to belong to the same class
when their quotient is a principal ideal. In the group of ideals of a field,
h distinct classes of equivalent groups are thus obtained. This number h
therefore gives the most important structural notion relative to the ideals
of the field k. In class field theory it plays an analogous role to the genus
p of the theory of Riemann surfaces, and like it, can be immediately interpreted in terms of being created.
In dealing with the universal covering surface, we have seen how
certain extension fields K of k, called class fields over k, could be put in
relation with the subgroups H of the group of all the ideals of k. Hilbert’s
quadratic class field theory studies the class fields that can be generated
from the base field when the square root of an element n contained in k is
adjoined to this base field. In sum, k contains n , but does not contain n
, and the conditions under which the extension k ^ n h can be a class field
for a group of ideals contained in k are sought. An essential result of the
theory, that we can only indicate without proof, establishes a very precise
connection between the number h of the classes of ideals in k and the existence in k of a number n such that k ^ n h is a class field: it is necessary, for
such a number to exist, that the number of classes h is even.9 Just as in the
theory of Riemann surfaces, in which the existence of abelian integrals
that are everywhere finite was linked to the number p, in class field theory
the properties of the number h, which measures the classes of an internal
decomposition of the field, have repercussions by the assertion of existence
relative to a number n which generates a quadratic class field k ^ n h.
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We compared a theory in which existence arises from the cutting of a
surface, to a theory in which existence arises from the division into classes
of ideals of a field, because the analogies of these theories seemed to shed
light on a same conception of the relations of essence and existence, in
which the existence of an entity emerges from the structural decomposition of a basic domain. We showed how this notion could be related to the
metamathematical distinction between structural properties and extensive
properties. Now these notions have only taken shape during the last
ten years, and since 1900, at the Paris Congress, Hilbert established the
comparison of the two mathematical theories which we have just spoken
about! Mathematicians have always admired the prophetic power of a genius who, in 1900, could state 23 problems to be solved, most of which were
later solved by him. We will quote a fragment of this Conference that concerns the problem that we are currently dealing with, and where it seems
that Hilbert outlines the philosophy of mathematical genesis that should
suggest to us the logic that he went on to develop twenty years later:
The analogy between the deficiency of a Riemann surface and that of the
class number of a field of numbers is also evident. Consider a Riemann
surface of deficiency p = 1 (to touch on the simplest case only) and on
the other hand a number field of class h = 2. To the proof of the existence
of an integral everywhere finite on the Riemann surface, corresponds
the proof of the existence of an integer a in the number field such that
the number a represents a quadratic field, relatively unbranched with
respect to the base field [that is, a class field]. (Hilbert 1900; 1935, 312
[2000, 423])
This text does not perhaps contain the terms of the philosophical commentary that we have given, it certainly contains the spirit. From the genus of
the surface or of the number of class fields to the existence of the integral or
to that of the number a , there is in effect a passage from one kind of entity
to another kind of entity, and the possibility of this passage results from the
discovery of a way to structure a domain that renders it apt to a creation.
4. THE THEORY OF THE REPRESENTATION OF GROUPS
The theory of the representation of groups will show us how the role of
domain and that of the created entities, do not depend on the intrinsic
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nature of the mathematical entities in question, but are conferred on them
by their respective functions in a process of genesis. Historically, the notion
of ‘representation’ has a more ‘concrete’, more ‘material’ sense than the
notion of group, but as representations arise from the structure of groups,
it will be more consistent to our conception to assign to the group the
nature of domain, and to see in the representations, the abstract entities
created on this domain that is called the space of the group. We will see, in
addition, that this change of perspective corresponds well with the current
evolution of the notion of representation.
The notion of group has its origin in the groups of transformations.
A group of transformations presupposes a space of points E on which are
defined the operations of the group. A set of transformations forms a group
if, given a transformation S that passes from point p to point p´ and a transformation T which passes from point p´ to point p˝, a transformation exits,
considered as the product ST of the first two, that passes from point p to
point p˝. It is necessary in addition that the group contains the identical
transformation, which makes that point itself correspond to each point,
and finally, that for any transformation, an inverse transformation exists.
This being posed, in the transformations, the sense of operations defined
over a space of points can be neglected, and the elements of an abstract set
within which a certain law of composition is defined are no longer seen.
What is then obtained is an abstract group whose elements no longer have
any intrinsic meaning and can be regarded as the points of an abstract
space: the space of the group. We can then inversely pass from this abstract
group to a group of transformations by restoring a nature of transformations to elements of the group. It is in this sense that a group of transformations realises an abstract group. In particular, when the realization of the
group makes a linear and homogeneous transformation of the space of
primitive points (see Chapter 1) correspond to each element of the group,
we have what is called a representation of the group. A representation is
therefore a correspondence between the elements s of an abstract group
and the transformations U operating on the points of the space E, and is
defined by equations of the following type:
xl1 = u 11 x 1 + ff u 1n x n
fffffffff
fffffffff
xln = u n1 x 1 + ff u nn x n
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Weyl’s train of thought is without any doubt to consider (1913, ch. 3) that,
from transformations to the abstract group, there is impoverishment, and
that, in the opposite direction, the descent of the abstract group to representations that operate on a space of points, nourish with matter the
slightly empty form of the abstract group. The representations of the group
thus play the role of interpretations of a system of axioms in logic with
respect to the abstract group. We will see that despite this concrete character that comes from their geometric origins, the representations can be
considered as abstract entities defined on the group space. We will then see
how their existence is linked to the structure of this space.
The representations that will be envisaged are those that are unitary and
irreducible. A representation is unitary if the transformations that constitute it leave invariant a Hermitian form x 1 x 1 + x n x n .10 It is in addition irreducible if it does not leave invariant any subspace of the space on which it
operates.
Let there then be s, a variable element of the group, and U the corresponding transformation for this element s in a unitary and irreducible
representation of the group. This transformation is characterized by an
array of coefficients
u 11 ff u 1n
ffff
ffff
u n1 ff u nn
and we designate any element of this table by uik(s) to show that the coefficients of a transformation depend on an element s of the group. They are
thus presented as functions of one variable that is none other than the
variable element of the group. These are therefore like functions defined
on the space of the group, and an essential theorem of the theory establishes that the coefficients of all unitary representations, irreducible and
non-equivalent11 among themselves, form on the space of the group a
complete system of basic functions. This means that all functions definable
on the space of the group can be represented as a linear combination of
coefficients of these representations. If a geometric sense is still attached to
the notion of representation, this sense is totally absent from the notion of
function defined on a group space, and under this new aspect of basic functions, the envisaged coefficients of representations are seen to have in
effect the sense of being defined on the abstract group space.
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E SSE NC E A ND E X IST E NC E
All that now remains for us is to indicate how, in the case of finite groups,
these abstract entities are not only defined on the group space but emanate
directly from its structure. The structure of a group is essentially the
number of its classes. Let a be a fixed element and s any element of a
group, the element asa –1 is said to be conjugated by s with respect to a . Two
conjugate elements are said to belong to the same class, and the group is
thus decomposed into classes of conjugate elements. This division into
classes can be interpreted in terms of genesis, in regard to the representations of the group. For finite groups, we have in effect the following theorem: the number of representations that are irreducible and non-equivalent
among themselves12 is equal to the number of classes of the group (Cf.
Weyl 1913, 130 [1964, 147]). The number of these classes characterise an
internal decomposition of the group. It also has a creative sense since it
determines the number of representations whose coefficients form a basic
system for the functions of the space of the group. Its role is therefore in all
respects comparable to that of the number p in the theory of Riemann
surfaces.
We have just studied the clearest case in which the structure of a domain
is immediately interpretable in terms of existence for certain functions
defined on that domain: these are cases in which this structure is characterized by a number attached to the decomposition of the basic domain.
The mathematical geneses are based upon the dual meaning of this number,
structural and creative. We therefore have here a means to distinguish, in
the relations that support these two different kinds of mathematical entities, what plays the role of concrete domain and what can be conceived
as abstract entities created on this domain. It is perfectly possible that, in a
schema of genesis, a same kind of entity plays the role of abstract with
respect to a concrete base and is, in another genesis on the contrary, the
concrete of a new abstract. Thus, for example, the group space can be conceived both as being abstract with respect to a space of fundamental points,
and as a concrete domain with respect to the representations of the group.
The essential element in the passage from essence to existence is not so
much the nature of the role assumed by each kind of entity present, than
the very existence of the passage between two kinds of the entity.
A presentation of things like this implies such a reversal as regards to the
habits of classical thought, that, in closing, we must again insist upon the
new meaning that the expressions concrete and abstract receive here. The
‘abstract group’ in the mathematical sense of the term, just as the systems
of axioms of formalized mathematics are often conceived as purely formal
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structures, ‘abstract’ as defined by the concept of classical philosophy, and
independently of their arithmetic or geometric realizations. It seemed to
us, as we have seen, that these structures, abstract in the classical sense,
were so profoundly engaged in the genesis of their realizations that it was
better to abandon any reference to an ontology in the use of the expressions concrete and abstract in order to designate respectively by these terms
only the basic structures and the entities whose existence is determined by
these structures.
This conception of the relations of the concrete and the abstract seems to
be particularly adapted to express not only the engagement of the concrete
in the genesis of the abstract, but also, the relations of imitation that can
exist between the structure of this abstract and that of the concrete base.
We shall give only one example: for certain systems of numbers (called
hypercomplex systems), as well as their representations, a mode of complete decomposition into irreducible elements can be defined. An essential
theorem of the theory then establishes a close relation between the complete decomposition of certain of these hypercomplex systems and that of
all their representations (Cf. van der Waerden 1931, 180). We thus see
how, in certain cases, the genesis of the abstract from a concrete base is
asserted up to the realization of an imitation of structure between these
kinds of entities that arise from one another.
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CHAPTER 5
‘Mixes’
What characterizes the geneses described in the previous chapter is that
created entities originate directly from the basic domain, and that from one
kind of entity to another there is thus immediate passage. The structure of
the domain plays a dual role in these creations, since it is interpretable both
as a decomposition of the domain and as a distribution of new entities on
this domain. It is not however a third kind interposed between the domain
and the created entities because it is entirely inherent to the domain that
receives it.
Certain mathematical geneses are however able to be described by
schemas of this type. They obey more complicated schemas in which the
passage from one kind to another requires the consideration of mixed
intermediaries between the domain and the entity sought after. The mediating role of these mixes is going to result from their structure imitating
that of the domain on which they are superimposed, so that their elements
are already the kind of entities that arise on this domain. Requiring the
adaptation of these radically heterogeneous realities to one another, mathematics recognises in its own development the logical necessity of a mediation, comparable to that of the schematism of the Transcendental Analytic,
intermediate between the categories and intuition. The text in which
Kant defines the schematism is in this respect of an importance that goes
far beyond the special problem of the philosophy of the understanding. It
contains as a general theory the mixes that we’ll see be applied almost
perfectly to the needs of mathematical philosophy. Here is the text:
Now it is clear that there must be a third thing, which must stand in
homogeneity with the category on the one hand and the appearance on
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the other, and makes possible the application of the former to the latter.
This mediating representation must be pure (without anything empirical) and yet intellectual on the one hand and sensible on the other. Such
a representation is the transcendental schema . . . Now a transcendental
time-determination is homogenous with the category (which constitutes
its unity) insofar as it is universal and rests on a rule a priori. But it is on
the other hand homogenous with the appearance insofar as time is contained in every empirical representation of the manifold. Hence an
application of the category to appearances becomes possible by means of
the transcendental time-determination which, as the schema of the concept of the understanding, mediates the subsumption of the latter under
the former. (Kant 1998, 272)
The essential moment of this definition is that in which the schema is
conceived from two different points of view as homogenous to the nature
of two essentially distinct realities and between which it serves as a necessary intermediary for any passage from one to the other. The mixes of the
mathematical theories assure the passage from one basic domain to the
existence of entities created on this domain by the effect of a similar internal duality.
It is possible to find a preliminary outline of these mixes in the theories
we have mentioned in the previous chapter. Consider first of all domain
theory in mathematical logic. The extensive property for a formula to be
realizable in a domain of k individuals implies, as we have seen at the
beginning of Chapter 4, that the domain of variation of the formulas variables contains at least k individuals, and that it is possible to find a choice
of values of the variables that make the formula, a true formula in the
ordinary sense. This research relies on the fact that in the case of finite
domains (in the ordinary sense) any logical formula can be put in the form
of a disjunction of terms, none of which contain any variables. If any of
these terms is true, and it is determinable by a finite number of steps, the
disjunction is an identity and the original formula is then realizaible. For
example, in a domain containing only the two variables 1 and 2, the
expression (x) P(x, x) is equivalent to P(l, l) 0 P(2, 2). The simple inspection of two arithmetic expressions P(l, l) and P(2, 2) allows one to see if
either of these expressions is true, and, in the same way, resolves the problem of knowing if the formula is realizable. We have already indicated
that when the variables can take an infinity of values, it becomes impossible to form the disjunction that results from all the possible substitutions.
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‘M IX E S’
The substitution that realizes the given proposition cannot be discovered
by a finite process, and the passage from the extensive point of view of the
realization to the structural point of view of irrefutability is due to this
impossibility, as we have shown. The extensive point of view however
reappears in Herbrand’s research, but under the form of metamathematical domains, intermediate between the signs of formulas and mathematical
domains of their effective values.
Herbrand’s intention is to reduce any arithmetical proposition, and this
irrespective of the extension of the domain of definition of the variables, to
a finite disjunction of terms, no longer containing variables and whose
examination thus permits, as in the case of finite mathematical domains,
the decision as to whether this disjunction is a logical identity. It was therefore necessary for him to find a way to define, for a variable that can take
an infinity of mathematical values, a finite number of metamathematical
values that thus symbolize the existence of this infinity so difficult to handle. Consider for example the variable y playing a part in the ‘particular’
form: (y), (there is a y such that . . . ). This y can be equal to any of the
terms of an infinite set. Herbrand (1930, 99 [1971, 150]) however only
envisaged as a domain of values of this variable a domain C1 composed of
a sole value a 1 . Now let the variable z play a part in the ‘general’ form: (z)
(for any value of z). Herbrand associates a certain index function fz to this
general variable that has the restricted variables preceding the general variable in question for arguments. The values of this index function are situated in a domain C2 which only contains the number of elements necessary
so that all elementary functions (descriptive functions given in the formula
and index functions) make the different values a 1 f a n of C1 correspond to
the different values a n + 1, a n + 2, a n + 3 f in the domain C2. If we have, for example, the following succession of signs of variables: (y) (z) (there is a y such
that for any value of z), and if a 1 is the value of (y), to z corresponds the
function fz(y) to which, as a value, is given by convention the letter a 2 of a
domain C2, a 2 represents the ‘value’ of z for the value a 1 of y. A finite series
of domains of ascending order is thus constructed by induction, each of
which contains the values of descriptive functions and index functions to
arguments taken in the domains of lower order whose value has not yet
been given. The elements of these domains are therefore in close correspondence with the signs of the variables of the formulas. They constitute
a system of new signs that are substituted for the first, rather than a set of
true values for the variables designated by these signs. On the one hand,
they do not possess any less a nature of domains independent of the
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formula they realize, and thus present a first aspect of mixes insofar as they
are intermediate between the formal signs and their effective mathematical value. From this first character, a second can be drawn, infinitely precious for the general conception of mixes in mathematics. Intermediate
between the signs and their values, these domains are, on the one hand,
homogeneous to the finite discontinuity of the signs, since to a sign of variable (y) there only corresponds a value a 1 , and, on the other hand, they
symbolize an infinity of mathematical values, since the letter a 1 represents
any current mathematical value of the variable y when it plays a part in the
particular form (y). A mediation therefore takes place via these domains
from the finite to the infinite, which, in the cases treated by Herbrand,
allows the domination of the infinite, and this is the role that we recognize
in the mixes that will now be considered. We will see in effect with respect
to the Hilbert space how, between the continuity of a basic domain and the
discontinuity of solutions to certain expressions defined on this domain, a
mix is interposed that takes the continuous by the origin and the topology
of its elements, the discontinuous by its structural properties, and allows
the connection of the one to the other.
1. HILBERT SPACE
We saw in Chapter 1 how certain differential equations or partial differential
equations could only be solved if the boundary conditions, initial and final,
are also given, which the functions that are solutions of the proposed equation must satisfy. Consider for example a vibrating membrane whose edges
are firmly embedded in a fixed wall. The equation that translates the oscillation of a point of the membrane is the classic equation Tu + mu = 0 . It is
evident that any solution u must be zero at the boundary of the membrane.
Consider also in quantum mechanics the Schrödinger equation which
expresses the amplitude a of a corpuscle of mass m oscillating in an exterior
domain of intensity V and possessing an energy E. This equation1 is written
Ta 8r 2m 6E - V (x, y, z)@ a = 0
h
2
Given that the amplitude a is the amplitude of a wave associated with a
corpuscle, it is natural to assume that it occupies only a limited region of
the x, y, z space, and that it is zero at the boundaries of this region. The basic
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‘M IX E S’
domain, in these cases, is evidently the domain D of variation of the
variables. The entities whose existence is trying to be determined are the
functions that are solutions of the problem, but the structure of the domain
is not directly adapted to the identification of these solutions. The schemas
of genesis of the previous chapter are thus powerless to describe the passage to existence for equations of this type.
These equations are of considerable importance because their solution
presents as a remarkable synthesis of continuous notions and discontinuous notions. They are in effect amenable to solutions u1 . . . un . . . , continuous over the whole domain of variables and zero at the limits for
certain discrete values of the parameter, m1, m2 f mn , which forms the spectrum of eigenvalues of the equation. The solutions u1 . . . un . . . corresponding to these eigenvalues have an essential property for the whole theory
that we will present. They form a complete system of continuous functions
in the basic domain D. This means that any continuous function in this
domain, twice differentiable and zero at the limits, can be represented as a
linear combination of these basic functions. It can again be said that any
arbitrary function f(x, y, z) on the domain can be expanded into a convergent series in the system of basic functions in the following form
f(x, y, z) = c1u1 + c2u2 . . . cnun . . .
An analogous situation was met in the study of integral functions.2 Consider first of all the linear and homogeneous integral equation (that is,
without a second member)
{ (s) - m # K (s, t) { (t) dt = 0
1
(I)
0
The variable s varying from 0 to 1, the unknown function sought after is
the function { (s) . The function K(s, t) is a given continuous function of
two variables. It is the kernel of the integral equation, which we will
assume in the symmetrical series in s and in t: K(s, t) = K(t, s). As in the
previous problems, it is shown that for 0 1 s 1 1 , the equation (I) only
has solutions for certain discontinuous values of the parameter, the
eigenvalues m1, m2 f . The solutions {1, {2 f that correspond to them are
again called solutions or eigenfunctions of the homogeneous equation.
We shall see later that the functions and the eigenvalues depend essentially on the kernel K(s, t). What is of interests to us here is that in the
case in which a countable infinity of eigenfunctions exists, they form a
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complete basic system for the set of functions f(s) that satisfy an equation
of the type:
#
f (s) =
1
K (s, t) x (t) dt
0
x(s) being an arbitrary continuous function.
We therefore have
n=3
f (s) =
/c {
n
n
n=1
and this theorem of expansion in uniformly convergent series of eigenfunctions of the kernel K(s, t) applies particularly to functions that are solutions
of the non-homogeneous integral equation (with second member)
{ (s) - m # K (s, t) { (t) dt = f (s)
1
0
Any solution of the non-homogeneous equation is a limit of the finite
linear combinations of eigenfunctions of the corresponding homogeneous
equation. The central idea that guided Hilbert in his general theory of
linear integral equations can then be seen to appear: on the domain
0 1 s 1 1 , construct a space of functions that have global characteristics
such that, by an appropriate decomposition of this space, a system of basic
functions can be distinguished that are at the same time eigenfunctions of
the proposed equation. Insofar as they form a basic system for the functions of the functional space, they are connected to the structure of this
space. Insofar as they are eigenfunctions of the proposed equation, they
are detached from the set of other functions of the space in order to appear
with their new sense of solutions. There may be no clearer case in which
the structural decomposition of a space is equivalent to the assertion of the
existence of the functions sought after, and we will see that the functional
space of this theory lends itself to similar geneses, because, on the one
hand, it is endowed with a topology as a set of points, and, on the other
hand, its elements are already homogeneous to the solutions sought after,
which are thus found included in this space before being recognized there.
We will first examine the schemas of genesis in the Hilbert space H,
which is a vector space, and then see the application of the results thus
found to the functional spaces of the theory of integral equations.
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‘M IX E S’
Here is the axiomatic definition of the Hilbert space H according to Von
Neumann:3
A. H is a vector space. It is a set of elements x, y . . . , in which the addition
x + y of two vectors and multiplication a. x of a vector by a complex
number a are defined;
B. H is Hermitian. This means that given any two vectors x and y of H, a
‘scalar product’ (x, y) of two vectors exists having the following
properties:
1) (x, y) is linear in
x (a1 x 1 + a2 x 2, y) = a1 (x 1, y) + a2 (x 2, y)
2) (x, y) possesses the hermitienne symmetry4
(x, y) = (y, x )
3) the scalar product (x, y) generates the existence of a positive
Hermitian form, defined
(x, x) = x 2 2 0 unless x = 0.
This form x 2 defines a metric in the space H.
C. H is complete as defined by the metric.
D. H is to a countable infinity of dimensions.
In a similar space, an infinity of complete systems of base vectors
{1, {2 ff exist, such that for any vector x of H we have:
x = / c i {i .
We shall now consider in H, an n-dimensional closed linear manifold,
that is, a subspace of the Hilbert space. ‘Hermitian forms’ can be attached
to this manifold, that is, the algebraic expressions formed from the vectors
of the manifold and amenable to being decomposed in a particularly
remarkable way when a choice is made to express the coordinates of the
vectors from a ‘proper’ system of base vectors. It is the connection of this
choice of the basic system to the decomposition of Hermitian forms that
is the essential stage of schemas of genesis that we claim to describe in this
chapter.
Consider a linear operator A acting on the elements x of the manifold,
that is, such that Ax is also an element of the manifold. The Hermitian
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forms that will be envisaged are the scalar products (x, Ax). What results
from the axioms of Hilbert space is that
(x, Ax) = A (x, x) = A (x, x) .
If any basic system in the manifold is chosen, the vectors x on the axes of
the chosen coordinates have components xi, and we have:
i, k = n
A (x, x) =
/a
ik
x i x k; a ik = a ik
i, k = i
.
In the adopted basic system, the Hermitian form A(x, x) is therefore characterized by the matrix of coefficients
a 11 fffa 1n
A *=
fffff
fffff
a n1 fffa nn
This matrix is comprised of rectangular terms, for which both indices are
different, and square terms (i = k), the diagonal terms. It is possible then,
and this is the central point of the theory, to find a system of privileged
base vectors {1, {2 f {n , such that the components of a vector x having
become in this system x´1, x´2 . . . x´n, the Hermitian form is transformed into
a sum
i=n
/ a xl xl
i
i
i
i=1
containing only the square terms. The change in the basic system therefore
has the effect of transforming the matrix A* into a matrix containing only
diagonal terms:
a1
A *l = 0 j
0
0
an
The ai are the ‘eigenvalues’ of the operator A, to which correspond the
eigenvectors {i .
These results which are due to Hermite (2009) in the case of an
n-dimensional manifold, were extended by Hilbert to the infinitely
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‘M IX E S’
dimensional space H. We have not entered here into the study of the
extremely delicate conditions of convergence that allow the passage to
infinity, we only import the final aspect of the theory. In sum, the space H
is decomposed into subspaces orthogonal to one another H1, H2 . . . Each of
these subspaces is underpinned by the eigenvectors corresponding to an
eigenvalue ai . The Hermitian form A(x, x) then becomes equal to the sum
of its projections
A (x, x) = / ai xli xli
in each of these different subspaces. This shows the dual role of eigenvectors: they have a structural role in the space whose orthogonal parts they
underpin; they also have a sense of existence for the operator A, since they
constitute the system of privileged vectors that allow the reduction of
the Hermitian form into a sum of squares.
These considerations apply immediately the functional space of the
theory of integral equations. Let there be a fundamental space E. This will
be the basic domain of our schemas of genesis, the functional space being
a mix interposed between E and functions sought after. Let n be a
Lebesgue measure defined on E. Let us consider the set X of functions f(p)
of complex value such that
#
E
#
f (p) dn exists and that
E
2
fn (p) dn
has a finite value. This gives the space of summable square functions,
which is, according to the work of Riesz (1907) and Fischer (1907), isomorphic to the Hilbert space. This space therefore possesses a topology
identical to that of space H. Let,
#
E
2
fm (p) - fn (p) dn
be the expression of the distance between two functions of the space. The
space is complete as defined by the metric, and a countable infinity of basic
systems exist such that any function of X can be expanded into a convergent series (as defined by the metric defined in X ),
i=3
f (s) =
/c{
i
i=1
i
.
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All that remains is to see the role of this mix, which is the space X of the
summable square functions, in the genesis of unique solutions of an integral equation with symmetric kernel. We only envisage the study of eigenfunctions of the kernel K(s, t) which are situated in space X , their
determination will be made by a privileged decomposition of this space,
which, in regards to the kernel K(s, t), plays the role that, in the Hilbert
space, is played by the decomposition into eigenspaces corresponding to a
Hermitian form K(x, x). To do this, and this is the central point, Hilbert
associates to the space of summable square functions a space H such that
to kernel K(s, t) there corresponds a Hermitian form
K (x, x) = / k pq x p x q .
The space H is decomposed, as we have just seen, into subspaces each
defined by the eigenvectors corresponding to an eigenvalue of the operator
K. We next go immediately from eigenvectors and eigenvalues of the operator K in Hilbert space to the eigenfunctions and eigenvalues of the kernel
K(s, t) in space X , and thus obtain the solutions of the integral equation.
The genesis of these solutions therefore happens as follows: the basic
domain E is the domain of variation of the variable. On this basic domain,
a functional space is superimposed which, by its topology, is, to some
extent, comparable to a space of points whose elements are functions
homogeneous to the functions sought after. A decomposition of this space
into eigenspaces then brings to the fore the existence of eigenfunctions of
the equation. The essential moment of this genesis is therefore contained
in the decomposition of the space, or what is equivalent, in the reduction
of a quadratic form to a sum of squares. And here again, a text of Hilbert’s
in 1909 characterizes in the clearest way the connection of a structural
decomposition and the existence of solutions:
A quadratic form Q(x), when it is continuous, can always be transformed
by an orthogonal transformation into a sum of squares of new variables
x´ such that we have: Q(x) = k1x´12 + k2x´22 . . . From this theorem the
ensuing important theorem follows: a system of linear equations to an
infinity of unknowns – thus an integral equation – as concerns the
number and existence of its solutions, possesses all the properties of a
system to a finite number of equations possessing the same unknowns.
(Hilbert 1909, 59)
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‘M IX E S’
The importance of the theory of Hilbert space developed by Hilbert and
his students, in particular Weyl and von Neumann, is considerable in the
new physics. It happens to be that this theory, constituted between 1904
and 1906, was the tool the most adapted to the interpretation of quantum
mechanics. Let us take, for example, the Schrödinger equation:
Ta 8r 2m 6E - V (x, y, z)@ a = 0
h
2
The variable parameter being energy E, this equation only has a solution
for discontinuous levels of energy which are the eigenvalues of the equation. The physics of the continuous has often been opposed to the physics
of the discontinuous. We see that the problem of the genesis of the discontinuous on the continuous is not a problem of physics, the drama is played
out much higher, at the level of mathematics. The continuous is the domain
of variation of the variable, the discontinuous are the solutions, and the
mix that is the Hilbert space is homogeneous to the continuous by the
nature and topology of its elements, and to the discontinuous by its structural decompositions. The theory of mathematical relations between the
continuous and the discontinuous therefore seems to receive its entire
meaning from the fact that it is incarnated in the more abstract schema of
genesis, in which the passage from the continuous to the discontinuous
happens through the intermediary of mixes whose fecundity results from
the properties of their dual nature.
2. NORMAL FAMILIES OF ANALYTIC FUNCTIONS
We will now examine another model of mixes, also intermediate between
a basic domain and a sought after function. These are the ‘normal’ families
of analytic functions whose consideration led Montel to a theory of extreme
richness.5 We’ll give an example of this mediating function of normal families by relying on the presentation that Montel made of a theorem of Caratheodory (1932). It is a matter of proving that any simply connected
domain D of the plane of the complex variable Z can be conformally represented (see Chapter 3) on a circle d of the plane of the complex variable z
when the boundary of the domain does not reduce to a point. The basic
domain, as defined by our schemas of genesis, is obviously the simply
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
connected domain in question, the sought after function is the function
z = G(Z), which ensures the conformal representation of the domain on the
circle, and to which corresponds the inverse function Z = f(z), which
ensures the conformal representation of the circle on the domain.
What is interesting for us in this problem is the moment in which the
necessity of a mix intervenes, that is, the moment in which the modes of
passage to existence offered by the simplest theories do not agree with
the envisaged conditions. We therefore admit the classical theorems that
establish the existence of a conformal representation of limited domains
by rectilinear polygons on a circle, and depart from the fact that these
theorems cannot be applied to the case of a domain D, bounded, simply
connected and whose boundary does not reduce to a point. Caratheodory’s
method consists then in interposing between the domain D and the function z = G(Z) a family of functions such that the internal structure of this
family immediately entails the existence within this family of the function
sought after. The condition of structure that is employed is that of compactness. It is a topological notion that, when it is relative to a family of
functions, allows comparing this family to a space of points and ensures a
certain homogeneity between the nature of the family of functions and the
nature of the basic domain. On the other hand, as the family of functions
is composed of functions and not of points, its elements are, in another
sense, homogeneous to the function sought after, so that it presents the
aspect of a mixed intermediate between two heterogeneous realities. Here’s
the definition of compactness: given an infinite sequence of points, this
sequence is said to be compact, that is, it is always possible to extract from
this sequence a subsequence which converges uniformly to a limit point.
As concerns the sequences of analytic functions, the property of compactness is no longer general. In order for it to be applied, it is necessary that
the function satisfies certain restrictive conditions, that of being bounded
as a whole for example. It is said that the functions f(z) holomorphic in a
domain D are bounded as a whole in this domain if we have f (z) 1 M , M
being a fixed number regardless of the function f(z) and regardless of z
interior to d. The following theorem due to Montel then establishes the
existence of compact, or normal, families of analytic functions: let a family
of functions be holomorphic and bounded as a whole in a domain d, from
any infinite sequence of this family a subsequence can be extracted which
converges uniformly in each domain interior to d towards a limit function
(Cf. Montel 1927, 21).
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‘M IX E S’
In the problem of conformal representation that we are occupied with
here (Cf. Montel 1927, 98–102) the solution is going to be obtained by
building a compact family on the domain D of a kind such that a limit function, whose existence is implied by the compactness of the family, is the
function sought after. The structure of domain D is therefore not directly
adapted to bringing to the fore the function sought after z = G(Z), but we’re
going to see how this structure is adapted to the genesis of the mixed intermediary: we cover the plane Z with a grid formed by squares with sides
equal to 1. All the edges are considered to be completely interior to the
domain. They form a simply connected domain D0 whose boundary is a
rectilinear polygon. The grid with 1/2 sides obtained by conserving the
previous gridlines is then envisaged, which allows a domain D1 to be
defined in the same way, containing all points of D0. With the grid with
1/2 n sides, a domain of Dn is likewise obtained. The infinite sequence of
domains D0, D1, . . . Dn . . . is such that: 1) each of these domains is completely interior to D; 2) Dn contains all points of Dn-1; 3) any point interior
to D is interior to the domain Dn starting from a certain rank.
The domain D is thus structured by an increasing infinity of separate
limited domains of rectilinear polygons: D0 . . . Dn . . . By virtue of the
theorems admitted at the beginning of the previous paragraph, there therefore exist functions Z0 = f0(z), Z1 = f1(z) . . . that ensure the conformal representation of the circle d on each of the domains Dn thus defined. The
functions fn(z) are holomorphic in d, they are bounded since the values of
Zn are interior to D for any n, they therefore form a normal family in d.
The equations defining the correspondence can be solved with respect to
z, we obtain:
z = G0(Z0), z = G1(Z1) . . . z = Gn(Zn).
The functions Gn(Z) are holomorphic and bounded in D0 since the corresponding values are the affixes of the points interior to d. They thus form
a normal family in D0, which is truly the essential mix of the theory.
Montel then proves that a subsequence can be extracted from this family
which converges uniformly in every domain Dn. Let G(Z) be the limit function of this sequence and f(z) the inverse limit function of the corresponding sequence in the family of functions fn(z). The function G(Z) ensures the
conformal representation of the domain D on the circle d, and its existence
is thus determined within the normal family defined on the basic domain.
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We thus see how the structural decomposition of D can be interpreted in
terms of existence, not of the entity sought after, but of a mix, and how it
is the structure of this mix that results in the existence of this entity sought
after. One remark is required. In the case of Hilbert space, the structure of
the mix gave rise to the values and to the eigenfunctions sought after by an
internal decomposition analogous to the cuts and the divisions of the basic
domains examined in Chapter 4. We are here dealing with a slightly different schema of genesis. The normal family proceeds from a sort of decomposition of the basic domain, but the limit function is no longer related to
a mode of decomposition of the mix. It results from a selection that allows,
within the normal family, the property of infinite subsequences to be
convergent. This point of view leads to those that we will place in the
next chapter to examine as a whole the schemas of genesis in which the
existence of an entity results from this entity having the benefit of exceptional properties that allow it to be distinguished from among others. It is
no less true that in the case of normal families, the selection of the limit
function is determined by the compact structure of these normal families
with the same logical rigor as the existence of eigenfunctions in the Hilbert
space were connected to the possibilities of decomposition of this space.
The two problems that have just been briefly explained show that the
discovery of the existence of an entity often presupposes the existence of a
set that contains the entity being sought after even before one knows to
see it there. The construction of this set is in effect easier because its nature
is adapted directly to the domain on which it is superimposed. The problem
of mixes in mathematics is therefore divided into a problem of imitation,
the mixes imitating a structure of the domain, and a problem of selection.
The mixes are thus situated well between two realities in the nature of
each of which they participate.
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CHAPTER 6
On the Exceptional Character of Existence
In most of the problems examined in Chapter 4 and Chapter 5, the passage
from the structure of the domain to the existence of the entities defined
on this domain results from a structural decomposition of the domain
interpretable in terms of existence with respect to other entities, those that
are created on this domain. We will now consider a new mode of connection between structure and existence, that in which the sought after entity
results from the selection within a domain of a element distinguished
among all others by its unique properties. Maurice Janet published an
extremely valuable article in Research philosophiques (Janet 1933) on problems of this sort, the most important from the philosophical point of view:
the exceptional properties that characterize the existence of the entity
sought after are in effect extremal properties of maximum or minimum.
This determination of existence by the research of maximum or minimum
suggested to Janet essential connections with the thought of Euler and
Leibniz. Janet cites the text of Euler:
As the construction of the world is perfect and due to an infinitely wise
creator, nothing happens in the world that doesn’t have some property
of maximum or minimum. Also there is no doubt that it is possible to
determine all effects of the universe by their final cause, with the aid of
the method of maxima and minima, with as much success as by their
efficient causes. (Janet 1933, 1)
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This text expresses the same ideas as Leibniz’s famous treatise On the
Ultimate Origination of Things, 1697:
From this it is obvious that of the infinite combinations of possibilities . . .
the one that exists is the one through which the most essence or possibility is brought into existence. In practical affairs one always follows the
decision rule in accordance with which one ought to seek the maximum
or the minimum: namely, one prefers the maximum effect at the minimum cost, so to speak. (Leibniz 1989, 150)
There is in this text as a dual program the application of the calculus of
variations to mathematics and physics, and all the success of variational
considerations in physics is known: Janet briefly redoes the history of
Fermat’s principle for light, Maupertius’s principle for mechanics, and
shows that the analogy of these two principles of the minimum dominate
the flow of ideas ranging from Hamilton to Louis de Broglie.
We intend in this chapter to confine ourselves exclusively to mathematics and would like to show that, in the determination of the existence
of a mathematical entity by considerations of extremum the logical schema
of a novel solution to the problem of the passage from essence to existence
is realized, in which, as in the schemas of previous chapters, essence and
existence are concerned with distinct mathematical entities. When an
entity is determined by the properties of maximum or minimum, it is necessary in effect to consider it as embedded in a whole and then to show
that the structure of the whole is such that it allows the entity sought after
to be distinguished. Insofar as the properties that render the selection
possible are the properties of maximum or minimum, they confer on the
entity obtained an advantage of simplicity and an appearance of finality,
but this appearance disappears when an account is given of that which
ensures the passage to existence. It is not the fact that the properties in
question are extremal properties, it is that the selection they determine is
implied by the structure of the given group. We shall see moreover how it
is of other exceptional properties, very different from extremal properties
and that also allow an entity within a whole to be distinguished. In all
these cases, we will focus on showing how the structure of the whole was
prepared to bring to the fore the distinguished element. The logical problem which dominates these theories is therefore still here that of the connections to be established between the structure of a domain and the
existence of entities defined on this domain, and it is essential for us to be
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O N T H E E XC E P T I O N A L C H AR A C T E R OF E X IST E NC E
able to observe in mathematics the different solutions to this same problem.
One of the main theses of this essay asserts in effect the necessity to separate the supra-mathematical conception of the problem of the connections
that support certain notions and the mathematical discoveries of these
effective connections within a theory. The schemas of genesis by selection
are distinct from the schemas of genesis by decomposition. Both nevertheless carry out the passage from the essence of the domain to the existence
of functions defined over this domain.
First, the two problems described by Janet will be envisaged: that of the
Dirichlet problem, and that of the eigenvalues of an operator in Hilbert
space. The Dirichlet problem consists in proving, given a domain D and a
continuous function on the boundary of D, that a continuous function V
in D exists just as its partial derivatives to two prime orders taking on the
boundary of D the given values and satisfying the Laplace equation:
22 V + 22 V = 0
2x 2
2y 2
Riemann had used the fact that such a function gives the minimum
expression:
I (U) =
# # 8` 22Ux j B + ;c 22Uy m Edxdy
2
2
D
to assert the existence of this function, supposing without proof that the
integral I(U) effectively attains its minimum. Weierstrass has shown that it
is not at all obvious that a given expression effectively attains its minimum
so that if the sought after function exists it rightly has the benefit of the
minimum property in question, but the fact that it has it is not sufficient to
ensure its existence. It is necessary to have recourse to other processes. The
principle of the method discovered by Hilbert, as in the last example presented in Chapter 5, consistsis in interposing between the domain D and
the solution V a compact family of functions, which here again plays the
role of an intermediate mix between the structure of the domain and the
existence of the function. Recall that a family of holomorphic functions
bounded as a whole in a domain D is compact if, for any infinite sequence
of this family, a subsequence can be extracted converging uniformly in
each domain interior to D towards a limit function. The theorem that
asserts the compactness of a sequence of functions is therefore both an
existence theorem for the limit function of a convergent sequence extracted
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
from the family and the fact that the limit function can have in addition
the extremal properties playing no part at all in its determination. This is
the existence theorem whose statement is borrowed from Lebesgue: suppose as given the continuous and monotonic functions u1, u2 . . . un . . .
(which means they have both an upper and a lower limit in a closed
domain and on its boundary, and this irrespective of the domain considered) in a domain D, all equal to each other on the boundary of D, having
at any point interior to the domain finite and continuous first order partial
derivatives, such that finally, for any i, the expression:
I (u i) =
# # 8` 22ux j B + ;c 22uy m Edxdy
2
2
i
i
D
has a meaning and is lower than a finite number H, independent of i. Then,
among u1 . . . un . . . , a sequence can be chosen that converges uniformly
towards a continuous limit (Lebesgue 1907, 386). The existence of the
function sought after is therefore connected to the compact structure of
the family of functions u1 . . . un . . . , and the exceptional property that
distinguishes it is no longer a property of extremum but the property of
being the limit of a convergent sequence.
Analogous considerations emerge from the study of eigenvalues of an
operator. Recall that given an expression:
H (X, X) = / h ik x i x k ,
the determination of eigenvalues of H amounts to the transformation of
the expression
/h
ik
xi xk
into a sum of squares
/m y y
i
i
i
by a transformation of the base vectors of the Hilbert space. One of the
methods for determining the eigenvalues mi is a maximum method. It
relies on the following reasoning: Every real and continuous function with
k real variables in a bounded and closed domain is uniformly continuous.
Therefore it is bounded, therefore it has a maximum attained for a point at
least of the domain of variables or its boundary. The expression H(X, X) is
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O N T H E E XC E P T I O N A L C H AR A C T E R OF E X IST E NC E
then considered as a function of X, in which X is any vector satisfying
X2 = 1 .
The domain of variation of the coordinates xi of X is bounded and closed,
therefore H(X, X) is a real and continuous function of xi in this domain and
presents a maximum mi attained for X = e1. Then all vectors X, of length
equal to 1, orthogonal to e1 are considered. The X variables still describe a
bounded and closed domain in which H attains a maximum m2 # m1 for
X = e2 and so on.1 The essential point of the reasoning is one that ties the
fact that the domain of variation of xi is bounded and closed to the existence of a maximum of the function
H (X, X) = / h ik x i x k
on this domain. The extremal value only appears each time because the
structure of the domain is adapted to bringing it to the fore. The schema of
genesis therefore still carries out the passage from the essence of the
domain to the existence of the maximum.
1. THE METHODS OF POINCARÉ
We will now envisage the exceptional properties that are going to allow us
to operate on a domain of very different selections. These are the ones that
date back to the methods pioneered by Poincaré in his paper Memoire sur les
courbes definies par une equation differentielle (1881). This paper has a capital
importance for the logical questions that we examine because it is here
that Poincaré characterized the structural aspect that in his eyes the modern problem of the integration of differential equations was bound to deal
with. In 1881, Poincaré only envisaged the substitution of the global point
of view for the local point of view, but we will show how these new methods later led to determine the existence of certain solutions by their exceptional properties. Here are the texts to which we refer:
Fuchs, Briot, Bouquet and Kovalevsky . . . instead of studying the
manner of being of the integrals of differential equations or of partial
differential equations for all the values of the variable, that is, in the
whole plane, they were at first occupied with determining the properties
of the integrals in the neighborhood of a given point . . . (Poincaré 1915,
38; 1928, iii)
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To this quantitative method, Poincaré opposes a qualitative method:
To study an algebraic equation, begin with the aid of Sturm’s theorem by
trying to find the number of real roots, this is the qualitative part. Then
calculate the numerical value of these roots, which constitutes the quantitative study of the equation. Likewise, to study an algebraic curve,
begin by constructing this curve, as they say in the special Mathematics
courses, that is, try to find the branches of the closed curve, the infinite
branches, etc. After this qualitative study . . . a certain number of points
can then be exactly determined. (Poincaré 1881, 376; 1928, 4)
This qualitative study is therefore essentially a topological study of curves
in which one tries to determine their saddle points, their nodes, their
focus, their centers. The most important curves are those that are presented in the form of closed cycles around a center. Their mathematical
importance comes from what their knowledge determines at the same
time, the knowledge of the curves that are their neighbors. In the case of
first order differential equations with real variables, they are for example
spirals that approach this limit asymptotically. The physical significance of
closed cycles in celestial mechanics is obvious: they correspond to periodic
trajectories, that is, to celestial bodies that are stable in their orbit. Generally
one tries to determine these periodic solutions by analytical methods, but
the genius of Poincaré was able to discover infinitely simpler methods of
selection, borrowed from topology. We will briefly summarize this research
according to the presentation by Bieberbach (1923, 202).
Poincaré’s new methods that are of interest here are concerned with the
solutions of differential equations in relation with a problem of the calculus of variations. All these solutions are therefore extremal with respect to
this problem, and it is a matter for Poincaré to determine the closed extremals. It is possible, by fairly simple transformations to represent these
extremals as curves of a 3-dimensional space, some of which are closed,
those that correspond to periodic solutions, and the others which are
wound infinitely many times around these closed curves. The whole forms
like a sheaf of curves whose intersection is envisaged with a 2-dimensional
surface. The closed curves only meet this surface at a finite number of
points. That the closed curves only meet the surface at a point can even be
arrived at by an appropriate choice of coordinates, but all the other curves
wind infinitely many times around the closed cycles meeting the surface
of intersection at an infinite number of points. That being so, if a curve
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O N T H E E XC E P T I O N A L C H AR A C T E R OF E X IST E NC E
interior to the sheaf is followed from one of its points of contact with the
surface until it meets a second point on the surface, and if this is carried out
in the same way for all the curves of the sheaf, a transformation is carried
out of all the points interior to the surface of intersection, which to each
point a makes the nearest point of intersection al of the curve passing
through a with the surface correspond. Only the corresponding points to
the closed curves remain fixed in this transformation. A characteristic
property of this transformation is to leave invariant a certain integral
expression. The problem of determining the periodic solution is therefore
reduced to the determination of points of a domain that remains fixed
when a transformation is carried out on all interior points of this domain.
We thus see appear a new exceptional property, that of the fixed points
whose existence is immediately interpretable in terms of existence of periodic solutions of differential equations. It must now be shown how the
existence of these fixed points is related to the structure of the domain on
which the internal transformation takes place.
Poincaré had stated a theorem shortly before his death, the proof of
which was given by Birkhoff a few months later. Here’s this ‘last theorem’
of Poincaré: if a biunivocal continuous transformation conserving the areas
transforms a ring in itself, by realising two rotations in opposite directions
for the two bounded circumferences of the ring, at the interior of the ring
two points always exist that are not affected by the transformation (see
Poincaré 1912; Husson 1932, 50; Bieberbach 1923, 205).
The conditions of existence of fixed points in this statement do not
depend solely on the structure of the basic domain, they also depend on
the nature of the internal transformation carried out since this transformation must conserve an invariant surface integral. The recent development
of topology has permitted the determination, for most topological domains,
by the sole consideration of the topological structure of these domains,
whether or not an internal transformation of their points admits fixed
points. It is known that among the structural characteristics of a complex
figure, above all the Betti numbers of this complex (see Chapter 3), that
is, the maximum number of linearly independent cycles of 0 dimension,
1 dimension . . . n dimension, n being the dimension of the complex. It is
then possible, through the formulas obtained by Hopf and Lefschetz, to
determine the existence or absence of fixed points of an internal transformation solely by the consideration of Betti numbers. Here are some results
that are significant in this regard: let P be a contiguous polyhedron all
the Betti numbers of which are zero except that of dimension 0 (p0 = 1).
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Any internal transformation of the polyhedron admits at least one fixed
point. Let N be the Euler characteristic of a complex, that is, a structural
invariant of this complex, which is equal to the algebraic sum of the Betti
numbers of the complex
;- N =
/ (- 1) p E
k=n
k
k
k=0
So that an internal transformation of a complex, reducible to identity by
continuous deformation, admits fixed points,2 it is sufficient that the Euler
characteristic of the complex is different to 0, etc. All these theorems show
how the existence of fixed points of an internal transformation is implied
by the structural invariants of the complex. There is a close analogy
between the assertion of existence of the maximum of a function on a
domain from the fact that the domain is bounded and closed, and the
assertion of the existence of fixed points from the fact that the Euler constant of the complex is different to 0. If, from these theorems, the logical
schema that they lay out is abstracted, a schema of genesis is obtained,
from the essence of the domain to the existence of the entity distinguished
on this domain, which is as characteristic as it could be of the schemas of
genesis that establish a connection between decomposition and existence.
2. THE SINGULARITIES OF ANALYTIC FUNCTIONS
Poincaré considered, we have just seen, that the qualitative study of the
shape of integral curves is opposed to the processes of his predecessors who
were preoccupied primarily with studying the nature of the solution in the
neighborhood of a given point. One can however show that the knowledge
of the value of a solution around certain points also allows the global characterization of the function that is the integral of a proposed equation,
because the points chosen are not arbitrary; they correspond in effect to
the singularities of the solutions. The singular points are, insofar as they
are points, points in the basic domain as any other point in this domain,
but, in modern theories they increasingly have a dominant and exceptional role. We will see briefly that in them a large number of logical points
of view that we have characterized separately in the course of this work
come to coincide.
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Let the differential equation be w˝ + p1(z)w´ + p2(z)w = 0, whose coefficients p1(z) and p2(z) have an isolated singularity at point a . It can be
proven that any solution of the equation can be obtained at any regular
point of the domain of the variable z by linear combination of two particularly independent solutions w1(z) and w2(z), forming what is called a fundamental system. It is therefore a matter above all of determining two
fundamental solutions of the equation, and this is going to be done first in
the neighborhood of the singular point. To determine these two fundamental solutions in the simplest cases, let us suppose that the coefficients
p1(z) and p2(z) had in a only the poles that are respectively of the first and
second order, that is, that we have:
p 1 (z) =
P (z - a)
P1 (z - a)
p 2 (z) = 2
z - a and
(z - a) 2
In these conditions, the equation:
w m + wl
P1 (z - a)
P (z - a)
+w 2
=0
z-a
(z - a) 2
always admits, in the neighborhood of the point z = a , a fundamental system of the form:
w 1 = (z - a) l P (z - a)
1
w 2 = (z - a) l P * (z - a) + possibly a logarithmic term.
2
The numbers l1 and l2 are roots of an equation called the fundamental
equation, formed with the prime coefficients of the series expansion of P1
(z – a) and P2(z – a). If an equation admits several singular points, the same
method applies: it is necessary to determine in the neighborhood of each of
them a system of two fundamental solutions. The passage of the solutions
w1 and w2 defined in the neighborhood of singular points to the solutions
defined at each regular point is done according to the ideas of Weierstrass
by analytic continuation. We are going to see how these local solutions are
nevertheless amenable to characterizing a global function. The cases that
we are going to envisage are those of the functions admitting 3 singular
points: the points 0, 1 and 3 . It can still in effect be obtained by a linear
transformation that the singularities are produced at these points here.
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
Consider the plane of the complex variable z cut on the real axis from 0 to
- 3 and l to + 3 . The extension from a local solution can never surround
more than one singular point at a time. Let w1 and w2 be a fundamental
system corresponding to the singular point z = 0. Let w be the quotient
w1/w2. It can be proven that the function w(z) ensures the conformal representation of the upper half-plane of the complex variable z on a curvilinear triangle of the plane of the variable w. If we then try to extend the
function w(z) on the lower half-plane of the variable z, a curvilinear triangle of the plane of w is obtained which has a common side with the first,
and by indefinitely repeating the extension of the function w(z) from one
half-plane to the other of the variable z, an infinite curvilinear triangle is
obtained which in certain cases tends to completely cover all the points
situated at the interior of a fundamental circle of the plane of w. It is thus
that, for example, on the plane w the figure below said to be a modular
figure can be obtained.
The function w(z) therefore carries out the representation of the plane z
on this modular figure,3 and its inverse function, z(w), called the modular
function, carries out conversely the representation of the modular figure
on the complex plane z.
If we recall that the global point of view in the theory of analytic functions is that which regards a function g = f (z) as carrying out the representation of the domain of the variable z on the domain of the variable g , we
see that the functions w(z) and z(w) defined from the local solutions of a
differential equation take no less an immediate place in Riemann’s theory.
Figure 1. (Hurwitz and Courant 1925, figure 104)
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Indeed one of their properties accentuates the global character of the result
obtained: in Chapter 3 we showed that the universal covering surface
corresponding to a given algebraic function g = f (z) was complete, in the
sense that a function t always existed on this surface such that the functions g = { (t) and z = } (t) are uniform. Now these functions are automorphic functions, that is, they conserve the same value for certain
transformations of the variable t:
{e
at + b
at + b
o = { (t) ; } e
o = } (t) .
ct + d
ct + d
It then happens to be that the modular function, obtained from a system of
purely local solutions, is the type of automorphic function that globally
uniformizes the algebraic functions.4
We will therefore see the multiple aspects under which singular points
are presented in this theory. (1) They allow the determination of a fundamental system of solutions, analytically extendable on any path encountering no singularities: it is in this sense that their singularity is expressed
in terms of the existence of solutions. (2) They allow a structural cutting of
the complex plane such that a function can exist that represents the similarly cut plane on the modular figure. In this sense their role is to decompose a domain in a way that the function which ensures the representation
is definable on this domain. (3) They allow the passage from the local integration of differential equations to the global characterization of analytic
functions which are solutions of these equations. We announced above
that a large number of logical notions that we’ve studied separately are
found to be mixed in the problems related to singular points: we can in
effect see the role they play in terms of the synthesis of the local and the
global (Chapter 1), in terms of the connection between the structural
decomposition of a domain and the existence of functions on this domain
(Chapter 4), and finally how the nature of singular points on a domain
determines, at each point of the domain of the variable z, the existence of
solutions of the proposed equation, in the same way that the maxima or
minima studied at the beginning of this chapter entail, in other cases, the
existence of specific solutions.
This encounter of distinct logical points of view, within the same problem, has been noted often in the previous chapters: it shows, as paradoxical as the reconciliation of the terms might be, both the intimate union
and the complete independence of the dialectical logic, as we conceive it,
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and mathematics. Mathematical theories are developed by their own force,
in close reciprocal solidarity and without any reference to the Ideas that
their movement reconciles. The logical schemas that the philosopher
then discovers in this movement cannot have the sharpness of contours
that rules given anterior to the experience would have. They only have
existence united to these theories which are made at the same time as
them, and they all impinge upon one another, to be better able to support
the magnificent and universal network of mathematical relations.
182
Conclusion
The preceding studies allow us to propose an answer to the problem that
was considered in the introduction. Trying to define the nature of real
mathematics, we are going to first examine the possible solutions to the
problem. The nature of mathematical reality can be defined from four different points of view: the real, it is sometimes the mathematical facts,
sometimes the mathematical entities, sometimes the theories, and sometimes the Ideas that govern these theories. Far from being opposed these
four conceptions fit naturally together: the facts consist in the discovery of
new entities, these entities are organized in theories, and the movement of
these theories incarnate the schema of connections of certain Ideas.
The first two points of view are adopted in the book by Boutroux: L’Idéal
scientifique des mathématiciens (1920). In his chapter on the mathematical
analysis of the nineteenth century, Boutroux situates mathematical reality
in the resistance that mathematical matter opposes to the logic with which
one tries to describe it, and in the chance discovery of a new path that
allows overcoming encountered obstacles. Thus, for example, examining
the problem of integration, Boutroux shows that the discovery of elliptic
functions, abelian functions, automorphic functions is due to the fact that
certain definite integrals or certain differential equations are not able to be
integrated with the methods that succeeded in the simpler cases. Winter
(1911) develops analogous ideas: the expression
#
dx
R (x)
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can be integrated by means of elementary functions if R(x) is a polynomial
of the first or second degree. If R(x) is of the third or fourth degree, a new
transcendent has to be dealt with:
It is sufficient that the degree of the polynomial contained under the
radical of the denominator increases by one unit, so that suddenly we
find ourselves in the presence of a new transcendent that requires considerable special study . . . Here is a fact that, it seems, formal logic
cannot account for. (Winter 1911, 85)
This definition of mathematical facts is thus intimately bound up with the
discovery of new entities, and Boutroux, in his chapter on the objectivity
of mathematical facts, adopts a much more conceptualist point of view.
Here is the very sentence in which the two points of view are recognizable,
without Boutroux perhaps having claimed to establish the differences:
In order to account for this resistance opposed by mathematical matter
to the will of the scientist, we are obliged to assume the existence of
independent mathematical facts of scientific construction, we are forced to
attribute a true objectivity to mathematical notions. (Boutroux 1920, 203)1
Facts are thus organized under the unity of the notion that generalizes
them. The real ceases to be the pure discovery of the new and unforeseeable fact, in order to depend on the global intuition of a supra-sensible
entity. Boutroux takes as an example the reality of the ellipse. The ellipse
is for him neither the locus of points such that the sum of their distances to
the foci is constant, nor a curve defined by its algebraic equation, nor a
curve to the projective properties of conics. It is all that and much more. It
is, he says,
a whole that does not include parts, . . . a sort of Leibnizian monad. This
monad is pregnant with the properties of the ellipse; I mean that these
properties, even though they have not been explicitly formulated (and
they cannot be since they are infinite in number) are contained in the
notion of ellipse. (Boutroux 1920, 208)
It must be admitted that the thought of Boutroux in the rest of the
chapter is rather difficult to specify. The example of the ellipse tends to
show that the reality of mathematical entities exist for him in the intuition,
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independently of the way the properties of these entities are presented
logically, and yet he refuses to admit that the difficulties of the logical
exposition are exterior to the specific reality of mathematics. These are the
two attitudes that seem, once again, to be opposed and yet here is a passage
in which both can be found:
That the idea of pure intuition, separated from logical reasoning, poses
difficulties is undeniable and it would be highly desirable to be able to
remove these difficulties by eradicating their root. But the distinction
between opposing tendencies in the work of mathematics, appears to us
to require being maintained in one form or another; and we cannot
believe that it has been devised solely for the purposes of the discussion
engaged in by the logicists. (Boutroux 1920, 228)
We cannot understand why it would be desirable to separate the
intuition from reasoning if this duality is inherent in the very nature of
mathematics.
The source of these difficulties seems to reside in two of Boutroux’s
opinions that we would like to discuss. The first is relative to the power
that the intuition would have to reveal to us the existence and objectivity
of mathematical entities. If the example of the ellipse in his thought has a
general bearing (and all indications in the text are that this is so), Boutroux
seems to assert, as a matter of course, a solidarity between the local properties and the global properties of an entity, while bringing this solidarity
to the fore is only possible, in most cases, as we have shown in Chapter 1,
when the entities in question have certain properties of closure or completion. Take, for example, the case of a closed surface with constant curvature, the sign of this curvature is determinable by the global topological
properties of the surface (its Euler characteristic). There is no theorem of
this kind for open surfaces. If therefore the reality of mathematical entities
resided in this inner harmony, a distinction is made between surfaces, sets,
groups. Some would exist ‘more really’ than others, and distinctions would
be seen to reappear in topology analogous to those which prevailed for a
long time in the history of mathematics, between good and bad functions,
good and bad roots.
One point of view however exists from which the multiplicities that are
either completed, complete, closed, enclosed or compact, have the benefit
of a privilege over those that are not. It is that the examination of the latter
is most often reduced to the study of the former. We are going to give two
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examples of this: we have shown in Chapter 2 how the structural properties of a closed complex Q were in duality with those of its complementary
space Rn – Q. But if Q is closed, Rn – Q is open, in such a way that the duality
theorem allows the study of certain open complexes to be partially reduced
to that of closed complexes. Here’s another example: Cartan, constituting
the symmetrical, irreducible and non-Euclidean theory of Riemann spaces,
initially examined enclosed spaces, and then associated to each open space
an enclosed space, so the study of open spaces is completely determined by
the enclosed space (Cartan 1937, 51). These processes can be compared to
those employed in the resolution of systems of linear equations. Initially,
the homogeneous system is resolved and the study of the non-homogeneous system is reduced to that of the corresponding homogeneous system.
The mathematical reality therefore does not reside in the differences that
separate the completed entities from the incomplete entities, perfect entities from imperfect entities. It resides rather in the possibility of determining one from the other, that is, in the mathematical theory that asserts
these connections. Thus we see that the reality in question is not that of
static entities, objects of pure contemplation. If qualitative distinctions exist
in mathematics, they characterize the theories and not the entities.
We end up at the same conclusion by discussing another of Boutroux’s
ideas, that of the independence of mathematical entities with respect to the
theories in which they are defined. Boutroux writes: ‘The mathematical
fact is independent of the logical or algebraic clothes in which we seek to
represent it,’ and later, ‘mathematical facts are totally indifferent to the
order in which they are obtained’ (Boutroux 1920, 203–205). Boutroux
has above all in view the analysis or geometry of the nineteenth century.
It is evident that in Euclidean geometry, an ellipse is or is not, and that it
possesses all its properties as soon as it is defined in any way. On the other
hand, in modern algebraic theories, entities can be envisaged that are amenable to belong to distinct basic domains, and whose properties vary with
the domain in which they are considered. It is impossible then to consider
such ‘mathematical facts’, as defined by Boutroux, independently of the
axioms that define the basic domains in question, and these axioms nevertheless constitute what Boutroux only claims to recognize as the supplementary character of ‘logical or algebraic clothes’. We find here instead an
essential dependence between the properties of a mathematical entity and
the axiomatic of the domain to which it belongs. We won’t describe it at
length because we have examined, in our secondary thesis (Lautman 1938a),
the principal aspects of this Abhängigkeit vom Grundkörper [Dependence on
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the base field] of the German authors, and we will be content to show this
in an example. Let us examine the properties of divisibility of the number
21. If the field K of rational numbers is considered as a base field, 21 is only
decomposable in one way into a product of prime factors: 21 = 3 x 7. If the
field K - 5 obtained by the adjunction of - 5 to the field K of rational
numbers is considered as a base field, two different decompositions of 21
into a product of prime factors are obtained:
21 = 3 x 7 and 21 = (1 + 2 - 5 ) (1 - 2 - 5 )
The properties of the number 21 are therefore not all given with the simple
construction of this number. They can only be studied within the field in
which the number is embedded, and this involves the whole axiomatic of
the theory of algebraic fields and of their successive extensions.
Whether therefore by studying the relations that unite certain entities to
other entities or certain entities to certain axioms, we see that the problem
of mathematical reality is posed neither at the level of facts, nor at that of
entities, but at that of theories. At this level, the nature of the real divides
into two. We have shown during the preceding chapters in effect how
mathematical theories are amenable to a dual characterization, one that
focuses on the unique movement of these theories, the other on the connections of ideas that are incarnated in this movement. These are the two
distinct elements whose reunion constitutes in our opinion the reality
inherent to mathematics and we are going to show how this reunion seems
possible.
The reading of Étapes de la philosophie mathématique (Brunschvicg 1912)
[The Stages in the Philosophy of Mathematics] teaches the philosopher to
associate in an indissoluble way the elaboration, or even only the comprehension of mathematical theories, and the experience that the intelligence
has of its own power. Any logical attempt that would profess to dominate
a priori the development of mathematics therefore disregards the essential
nature of mathematical truth, because it is connected to the creative activity of the mind, and participates in its temporal character. It is on the other
hand indisputable that, since the development of non-Euclidean geometries,
mathematics is not presented as an indefinitely progressive and unifying
extension. The theories are rather figures of an organic unity, and are
suitable for these global metamathematical considerations announced in
the work of Hilbert. The point of view of a new mathematics gradually
asserts itself, which substitutes for the infinitist process of the analysis of
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the nineteenth century, the structural schemas of algebra or of topology.
Elsewhere we have described this evolution of modern mathematics and
the gradual penetration of the methods of the finite in the infinite. We
shall only mention here the idea of a ‘qualitative’ mathematics as defined
by Poincaré or an ‘integral’ mathematics as defined by Severi (see Severi
1931), being developed around structural logical schemas.
Our task therefore is to reconcile the irreducibility of mathematics to an
a priori logic and its organization around similar logical schemas. To do this,
we will therefore attempt to show that it is possible to conceive what we
will call the exigency of a logical problem, without the consciousness of
this exigency implying in any way an attempt at solution. One can even
say that a dialectic which would be engaged in the determination of solutions that these logical problems can bring about, would be involved in
constituting an entire set of subtle distinctions and artifices of reasoning
that imitate mathematics to this point, that it would be conflated with the
mathematics itself. Such is the fate of mathematical logic in its most recent
development. It is possible to conceive the problem of consistency in arithmetic without redoing all of arithmetic, but as soon as you try to establish
an effective proof of consistency of arithmetic, you are obliged to employ
in this proof mathematical means that exceed in richness those of the theory whose validity you are trying to guarantee. These results, due to Gödel,
show definitively that the consistency of arithmetic is not able to be reduced
to the consistency of a simpler theory, and, in the present state of science,
any metamathematical proof of the consistency of arithmetic necessarily
uses transfinite means. It seemed therefore that this problem had lost all
logical interest until Gentzen managed to envisage it under another aspect:
It remains quite conceivable that the consistency of arithmetic2 can in
fact be verified by means of techniques which, in part, no longer belong
to arithmetic, but which can nevertheless be considered to be more reliable than the doubtful components of arithmetic itself. (Gentzen 1936,
500 [1969, 139])3
We see in this way how the problem of consistency makes sense, even
though we are unaware of the mathematical means necessary to resolve it.
This seems to us to be the case for all the logical problems that we have
successively considered. The logical schemas that we have described are
not anterior to their realization within a theory. They lack in effect what
we call above the extra-mathematical intuition of the exigency of a logical
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problem, a matter to dominate so that the idea of possible relations gives
rise to the schema of true relations. The fate of the problem of the relations
between the whole and the part, of the reduction of extrinsic properties to
intrinsic properties, of the ascent towards completion, the constitution of
new schemas of genesis, depends on the progress of mathematics itself.
The philosopher has neither to extract the laws, nor to envisage a future
evolution, his role only consists in becoming aware of the logical drama
which is played out within the theories. The only a priori element that
we conceive is given in the experience of the exigency of the problems,
anterior to the discovery of their solutions.
It is necessary to now clarify the nature of the a priori that we introduce
into the philosophy of mathematics. We do not at all pretend to support
that the ideas of logical problems that mathematics resolves can be deduced
systematically according to the requirements of a rationalist idealism. We
understand this a priori in a purely relative sense, and with respect to
mathematics. It is uniquely the possibility of experiencing the concern of
a mode of connection between two ideas and describing this concern
phenomenologically independent of the fact that the connection sought
after may, or may not be carried out. Some of these logical ‘concerns’ are
found in the history of philosophy: as, for example, the concern of the connections between the same and the other, the whole and the part, the
continuous and the discontinuous, essence and existence. But mathematical theories can conversely give rise to the idea of new problems that have
not previously been formulated abstractly. Mathematical philosophy, as
we conceive it, therefore consists not so much in retrieving a logical problem of classical metaphysics within a mathematical theory, than in grasping the structure of this theory globally in order to identify the logical
problem that happens to be both defined and resolved by the very existence of this theory. A spiritual experience is thus attached anew to the
effort of the intelligence to create or understand. But this experience has a
content different to that of mathematics which is made at the same time as
it. Nor is it the consciousness of the infinite power of thought. Beyond the
temporal conditions of mathematical activity, but within the very bosom of
this activity, appear the contours of an ideal reality that is governing with
respect to a mathematical matter which it animates, and which however,
without that matter, could not reveal all the richness of its formative
power.
Before concluding, we would like to show how this conception of an
ideal reality, superior to mathematics and yet so willing to be incarnated in
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its movement, comes to be integrated into the most authoritative interpretations of Platonism. Certain historical explications are indispensible on
this subject, given the sense that the expression of Platonism in mathematics generally receives. In the open debate between formalist and intuitionist, since the discovery of the transfinite, mathematicians have become
accustomed to summarily designate under the name Platonism any philosophy for which the existence of a mathematical entity is taken as
assured, even though this entity could not be built in a finite number of
steps. It goes without saying that this is a superficial knowledge of Platonism, and that we do not believe ourselves to be referring to it. All modern
Plato commentators on the contrary insists on the fact that Ideas are not
immobile and irreducible essences of an intelligible world, but that they
are related to each other according to the schemas of a superior dialectic
that presides over their arrival. The work of Robin, Stenzel and Becker
has in this regard brought considerable clarity to the governing role of
Ideas–numbers which concerns as much the becoming of numbers as that
of Ideas. The One and the Dyad generate Ideas–numbers by a successively
repeated process of division of the Unit into two new units. The Ideas–
numbers are thus presented as geometric schemas of the combinations of
units,4 amenable to constituting arithmetic numbers as well as Ideas in the
ordinary sense.
The Ideas–numbers therefore constitute, as Stenzel says, the principles
which serve both to ‘dialectically order the arithmetic units to the place
that suits them in the system, and to explicate the different degrees of
progressive division of Ideas’ (Stenzel 1923, 117). The schemas of division
of Ideas in the Sophist are organized according to the same planes as the
schemas of generation of numbers, both can be traced to a ‘metamathematics’5 that is superior to both the Ideas and the numbers.
The existence of Ideas – numbers, governing in regards to arithmetic
numbers, therefore has the consequence of ordering a generation of numbers as of Ideas, which, though not being in the time of the created world,
are produced no less according to an order of the anterior and the posterior. Robin shows how the constitution of bodies in the Timaeus assumes a
matter which, before the existence of the world, has already been the
receptacle of a geometric qualification. ‘There is therefore a generation and
becoming anterior to the generation and the becoming of the world’ (Robin
1935, 235).
The introduction of becoming within Ideas, in Stenzel’s work, takes all
its value from the genius of the text of the Nicomachean Ethics (EN 1.4, cited
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by Stenzel 1923, 118), where Aristotle says that Platonists did not admit the
ideas of numbers because they did not admit as ideas things in which there
exists the before and the after. Stenzel gives an explanation of this text
the importance of which cannot be exaggerated: ideal-numbers being the
principle of determination of essences according to the order of the before
and the after, it is not possible that there are ‘numbers of numbers’, that is,
‘a principle of the division of essences that is superior to this numerical
division itself’ (Stenzel 1923, 119).6 Metamathematics which is incarnated
in the generation of ideas and numbers does not give rise in turn to a metametamathematics. The regression stops as soon as the mind has identified
the schemas according to which the dialectic is constituted. We can thus
see how well our reference to Platonism is justified, as regards the relations
that exist between mathematical theories and the Ideas that govern them.
These relations also appear comparable to those that could be established between mathematics and physics. A simple empiricism tends
sometimes, currently, to be installed in the philosophy of physics, according to which a profound dissociation should be established between the
experimental findings of fact and the mathematical theory that links them
to each other. Any criticism of contemporary science shows the philosophical weakness of such an attitude7 and the impossibility of considering an
experimental result outside of the mathematical framework in which it
makes sense. On the other hand, critical thinking sometimes leads, conversely, to an idealistic dogmatism, from the fact that mathematics increasingly penetrates the domain of physics, reality has become so abstract that
the scientist has the impression of no longer ever happening to be in front
of their own mind. Such is the idea that seems to emerge from Eddington’s
celebrated sentence:
We have found a strange foot-print on the shores of the unknown. We
have devised profound theories, one after another, to account for its
origin. At last, we have succeeded in reconstructing the creature that
made the foot-print. And Lo! it is our own. (Eddington 1920, 201)
This idealism of mathematical physics does not moreover exclude, for
Eddington, the notion of reality, but it is no longer physics that will have
the mission to make it known, it is a direct contact with the supernatural.
In any case, the neo-positivism of the Vienna Circle, like the idealism
of the English metaphysical physicists, separate mathematics and reality
quite distinctly, while the philosophy of physics essentially has as its task
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the problem of their union. We do not pretend to treat this problem here,
which is quite different from those we have envisaged throughout the
course of the preceding pages. We are simply going to show how, to some
extent, and to take up the expressions that Robin made use of in regards
to Plato, the process of connecting theory and experience symbolizes the
connection of Ideas and mathematical theories.
Just as we had recognized that mathematical reality was not at the level
of mathematical entities, but at that of the theories, the problem of physical reality does not intervene at the level of an isolated experience, but at
the level of what might be called a physical system. The notion of system
in physics implies that the set of phenomena that occur within a certain
process are considered globally, and that a series of measurements are carried out relating to magnitudes playing a part in this process. In addition, it
is not necessary that the measurements are concerned with different magnitudes; they can be relative to only a same magnitude. For there to be a
system, it is sufficient that the same magnitude is measured several times.
It is perhaps possible to suppose that the finding of a single measurement
is anterior to any mathematical elaboration, but whatever the sentence or
the relation by which two measurements are coordinated, the expression
obtained is already situated within mathematics. To speak of simultaneous
phenomena is to adopt the language of special relativity, to speak of successively measurable magnitudes is to speak the language of permutable
operators in quantum mechanics. To measure the intervals between two
levels is to set out the results obtained according to a matrix array. To speak
in classical mechanics of the constancy of a certain magnitude in time is
to take a derivative with respect to time. To speak of the invariance of a
certain magnitude with respect to certain variations that other magnitudes
undergo is to use the language of group theory. To note the periodicity of
a phenomenon is to make use of trigonometric functions to represent this
phenomenon. To suppose that measurements converge towards a limit is
to adopt the point of view of the calculus of probabilities. Just as the movement of a mathematical theory was at the same time a logical schema of
relations between ideas, the description of the state of a system at any given
moment or of the evolution of this system over time, amounts to noting
that the magnitudes of the system are orderable according to a structural
law of mathematics. Physical reality is therefore not indifferent to this
mathematics which describes it. Experimental findings call for a mathematics whose outline they already imitate (Cf. Juvet 1933, passim), sometimes even before an adequate mathematics has been developed for them.
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What results from the preceding is that the structure of experience is
not detachable from the experience itself, and that in understanding by
experience the global experience relative to a system, this mathematical
structure coincides with the system of effectuated or possible experimental
measurements. Here again reality resides in the discovery of a structure
which is organizing with respect to a matter that it animates with its
relations. The philosophy of physics would therefore also amount to the
Platonic conclusion to which we have led mathematical philosophy, as we
conceive it. The nature of the real, its structure and the conditions of its
genesis are only knowable by ascending again to the Ideas whose connections are incarnated by Science.
193
BOOK III
New Research on the Dialectical Structure
of Mathematics*
The essays of this series will be devoted to dogmatic philosophy without restriction
of any kind, neither for the objects considered, nor for the processes employed. The
unity, if it appears, – it will in any case never be sighted consciously – could only
be original – influences undergone –, or also aspirational, a certain concern by
which we have a sense of community. To be more specific about what was only a
method would be altogether in vain: any requirement for clarification that does
not satisfy the current instruments of scientific techniques, or their normal development, is philosophical. It is possible that these change and render our problems
devoid of meaning: as was the case for rational evidence after the crisis of mathematical infinity. In the contemporary system of notions and processes of authentic
thought, that is, confirmed by the least experience, the philosopher has their own
activity that recognizes its contents, by discovering the true consequences and
the relations. It is not a matter of subordination to science or renunciation of the
* This essay, published in 1939, was the first of a series called Essais philosophiques,
created by Jean Cavaillès and Raymond Aron at Hermann publishers. It is known
from a letter by Cavailles to Lautman that this essay was closely associated with
the drafting of this Introduction that is worthy of a manifesto. In the series there
were only four volumes in total that appeared. In 1939, in addition to this one,
The Emotions: Outline of a Theory by Jean-Paul Sartre (1939), and in 1946 two that
were posthumous: Transfini et Continu by Jean Cavaillès (1947); Symmetry and
dissymmetry in mathematics and physics by Albert Lautman (1946, included in this
volume).
M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
fundamental questions of metaphysics or morality, but an effort to address these
issues in an effective way. At the level of specific difficulties, arising from effective
research, subject to the sanctions of failure, reflection for us begins. Whether they
are of an original nature, we become conscious of them by exercising them: neither pure intuition nor abstract dialectic, it is initially critical ordeals, doubt and
care for the other. What we are trying to say has the ambition to be true and, as
such, must participate in the destiny of all knowledge, informed by a conceptual
architecture whose structural rigor as well as power of change lend themselves to
communication: philosophia perennis sed in actione hominis manifesta.1
Jean Cavaillès and Raymond Aron
196
Foreword
This essay consists of two distinct parts: in the first, developing the ideas
of our principal thesis (Lautman 1938b), relative to the participation of
Mathematics in a Dialectic that governs it, we try to show in an abstract
way how the understanding of the Ideas of this Dialectic is necessarily
extended in the genesis of effective mathematical theories. We rely for this
on certain essential distinctions in the philosophy of Heidegger who seems
to agree in a remarkable way with the metaphysical problem envisaged.
In the second part, instead of descending from the abstract to the concrete,
we work backwards: we examine a particular mathematical theory, the
analytic theory of numbers, in which it is possible to grasp in a concrete
way the necessity, in order to understand the reason of certain results, of
relating them to the structural Ideas of a higher dialectic.
It may seem strange to those who are used to separating the ‘moral’ sciences from the ‘exact’ sciences, to see, reunited in the same work, reflections on Plato and Heidegger, and remarks on the law of quadratic
reciprocity or the distribution of prime numbers. We would like to have
shown that this rapprochement of metaphysics and mathematics is not
contingent but necessary.
197
CHAPTER 1
The Genesis of the Entity from the Idea
We tried in a previous work (Lautman 1938b) to show in a few examples
how in mathematics the ideal relations of a dialectic abstract and superior
to mathematics is realized in concrete ways. It is in this sense that the
intrinsic reality of mathematics appeared to us to reside in its participation
in the Ideas of this dialectic which governs them. We do not understand by
Ideas the models whose mathematical entities would merely be copies, but
in the true Platonic sense of the term, the structural schemas according to
which the effective theories are organized. This distinction between dialectic and mathematics leads us to a more precise analysis of the nature of
the ‘governing’ (domination) relation that exists between dialectical Ideas
on the one hand, and mathematical theories on the other.
The most habitual sense of a governing relation between abstract Ideas
and their concrete realization is the cosmological sense, and a cosmological
interpretation of such a relation is based almost entirely on a theory of
creation. The existence of a matter that is the receptacle of the Ideas is not
implied by the knowledge of the Ideas. It is a sensible fact, known by some
bastard reasoning, as Plato said, or by a kind of natural revelation, as
Malebranche thought. The Ideas are then like the laws according to which
this matter is organized to constitute a World, but it isn’t necessary that this
World exist to realize in a concrete way the perfection of the Ideas.
Such an epistemology can make sense in regard to physical reality; it
certainly does not in regards to mathematical reality. The cut between the
dialectic and mathematics cannot in effect be envisaged. It is necessary
on the contrary to clarify a mode of emanation from one to the other, a
kind of procession that connects them closely and does not presuppose the
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contingent interposition of a Matter heterogeneous to the Ideas. This
engagement of the abstract in the genesis of the concrete is a ‘transcendental’1 interpretation of the governing relation that can be better accounted
for. We intend to show by this that an effort of understanding adequate to
the dialectical Ideas, by the very fact that it applies to knowing the internal
connections of this dialectic, is creative of systems of more concrete notions
in which these connections are asserted. The genesis is then no longer
conceived as the material creation of the concrete from the Idea, but as
the advent of notions relative to the concrete within an analysis of the
Idea. This transcendental conception of the relation of governing that is
narrowly applied to the case of the relation between the dialectic and
mathematics is not however limited to this case. It happens to be in effect
that independently of any reference to mathematical philosophy, Heidegger
has presented analogous views to explain how the production of notions
relative to concrete existence arise from an effort to understand more
abstract concepts.2
Heidegger’s analysis is based on the distinction between being and entity.
The truth of being is, in the vocabulary of phenomenology, an ontological
truth, relative to the essence. The truth of what exists is ontic, and relative
to the effective situations of concrete existence. The distinguishing feature
of the entity is to manifest itself, to be revealed, but this revelation is only
possible ‘guided and clarified by an understanding of the being (the constitution of being: what something is and how it is) of entities’ (Heidegger
1969, 21, 23 [1938, 56]). In this way therefore, if it belongs to the ontic
truth to be the ‘manifestness of the entity’ (21), the ontological truth is
quite different, it is ‘disclosure’ (117) understood ‘as the truth about being’
(23). We are going to see how, in the analysis of the disclosure of being, a
general theory of these acts is constituted which, for us, are geneses, and
that Heidegger calls acts of transcendence or of surpassing.
Here are the principal moments of the disclosure of being: it comes primarily from the act of asking a question about something, this does not necessarily mean that the thing to which a question is thus posed is conceived
in its essence. This prior delimitation to that about which knowledge is
possible, but which is not yet knowledge of the concept of being, Heidegger
calls the pre-ontological understanding. It precedes the formation of the concept of being, an act by which a structure is disclosed to the intelligence
that thus becomes capable of outlining the set of concrete problems relating to the being in question. What then happens, and this for us is the
fundamental point, is that this disclosure of the ontological truth of being
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cannot be done without the concrete aspects of ontic existence taking
shape at the same time:
One characteristic stage is the project of the constitution of the Being
of the entity whereby a determinate field of being (perhaps nature or
history) is, at the same time, marked off as an area that can be objectivized through scientific knowledge. (Heidegger 1969, 23 [1938, 57])
According to this text, a same activity is therefore seen to divide in two, or
rather act on two different planes: the constitution of the being of the
entity, on the ontological plane, is inseparable from the determination, on
the ontic plane, of the factual existence of a domain in which the objects of
a scientific knowledge receive life and matter. The concern to know the
meaning of the essence of certain concepts is perhaps not primarily oriented towards the realizations of these concepts, but it turns out that the
conceptual analysis necessarily succeeds in projecting, as an anticipation of
the concept, the concrete notions in which it is realized or historicized.
The distinction between essence and existence, and in particular the
extension of an analysis of the essence in genesis of notions relating to the
entity, are sometimes masked in the philosophy of Heidegger by the importance of existential considerations, relating to being-in-the-World, as they
appear in Being and Time. But in the second part of The Essence of Reason
(1969 [1929]), Heidegger relies precisely on the distinction between the
ontological point of view and ontic point of view to explain the link that
exists between human reality and existence-in-the-World. The concept of
World for Heidegger does not signify the simple totality of entities in general. It is an ontic notion, exclusively related to human reality, to effective
situations of man in the world. Heidegger does not support the thesis
according to which being-the-World would properly belong to the essence
of human reality, ‘for it is not necessary that the sort of being we call
human . . . exists factically. It can also not be’ (43, 45 [67]). On the other
hand, it is necessary to recognize that being-in-the-world cannot be attributed to human reality. Whether it is a matter of the finite world in which
the religious destiny of man is hanging in the balance, the world organized
into subjective knowledge of phenomena by the human mind, or the world
in which it is given to man to be man, the World always reveals itself, on
the ontic plane of existence, as ‘that for the sake of which [human reality]
exists’ (85 [88]). Conversely, by considering the idea or essence of human
reality this time on the ontological plane, it is necessary to recognize that its
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‘basic constitutive feature’ is ‘being-in-the-world’ (81, n. 55). If the concept of the World is revealed as that ‘for the sake of’, it is of the essence
of human reality to be the ‘intention’ directed towards this purpose. In
‘the essence of its being’, human reality ‘is such that it forms the world’
(89 [90]). The purpose returns to the intention, just as the ontological
analysis of intention outlines the notions that constitute the purpose in
which this intention is realized.
Heidegger thus describes, as we have just seen, a genesis of the ontical
concept of the World from the idea of human reality. His primordial interest is therefore concerned with the problem of the Self, but this primacy
of anthropological preoccupations in his philosophy should not prevent
his conception of the genesis of notions relating to the Entity, within the
analysis of Ideas relating to Being, from having a very general bearing: he
himself applies them moreover to physical concepts:
The preliminary definition of the being (understood here as what something is and how it is) of nature is established in the ‘basic concepts’ of
natural science. . . . space, locus, time, movement, mass, force, and
velocity are defined in these concepts . . . (23, 25 [57])
These scientific concepts concern the entity, and not being. They do ‘not
include authentic ontological concepts of the being of the entity’ (25
[57–58]) that is put into question by this science.
Whether it is therefore about the Self, or about Nature, Heidegger identifies the grounding, rational activity of founding (Begründung), which is
knowledge by Ideas, and the creative activity of foundation (Gründung)
which constitutes, in the complexity of its internal relations, the world
of the entity. Grounding intervenes as soon as the Leibnizian question of
knowing why something exists is posed. ‘A preconceptual, prior understanding of what something is, of how it is, and even of being (nothing) lies
implicit in the Why, no matter how it is expressed’ (115 [103]). The Why is
therefore not a pure interrogation. The notion of being that it implies
is already an answer and grounding, disclosure of ontological truth. The
passage from this disclosure of the essence to the different possible forms of
existence appears as soon as it is realized that the enquiry into the why is
inseparable from the consideration of the possibles implied in rather than
(why something exists rather than another thing, or rather than nothing).
Grounding is in effect naturally extended in outline of potential entities:
‘By its very essence, ontological founding opens marginal realms of the
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possible’ (125 [108]). But there is more, and the principle of sufficient
reason is only a logical principle, destined to legitimize the real by removing what the real is not. For Heidegger it is a transcendental principle that
the determination of the entity necessarily relies on a creative freedom
rooted in the ontological constitution of the being that determines. It is
thus that grounding gives rise to the formation of a project of the World, in
which the creative freedom of the founding power is asserted (in the dual
sense of founding and foundation). If to this is added, about human reality
at least, that there is reciprocity between the constitution of the World
by the project that descends towards it and the fact that existence in the
world only makes sense by ascending again to human reality whose world
constitutes the purpose, one understands ‘the threefold transcendental dispersion of grounding in the project of world, preoccupation with the entity,
and the ontological founding of the entity’ (121 [106]).
It is possible, in the light of these conceptions of Heidegger, to see the
utility of mathematical philosophy for metaphysics in general. Whereas for
all the questions that do not come out of the anthropology, Heidegger’s
indications remain, despite everything, very brief, one can, in regards to
the relations between the Dialectic and Mathematics, follow the mechanism of this operation closely in which the analysis of Ideas is extended
in effective creation, in which the virtual is transformed into the real.
Mathematics thus plays with respect to the other domains of incarnation,
physical reality, social reality, human reality, the role of model in which
the way that things come into existence is observed.
1. THE GENESIS OF MATHEMATICS FROM THE DIALECTIC
The notion of genesis implying a certain order of the before and the after,
it is first necessary to specify the type of anteriority of the Dialectic with
respect to Mathematics. It cannot be, we have already seen, a matter of the
chronological order of creation. This is not the order of knowledge, because
the method of mathematical philosophy is analytical and regressive. It arises
from the global apprehension of a mathematical theory to the dialectical
relations that this theory incarnates, and it is not a question of determining
an a priori, prior knowledge of which would be necessary to understanding
mathematics. The order implied by the notion of genesis is not about the
order of the logical reconstruction of mathematics, in the sense in which
from the initial axioms of a theory follow all the propositions of the theory,
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because the dialectic is not part of mathematics, and its notions are unrelated to the primitive notions of a theory. We have already defined, in our
thesis (Lautman 1938b), the priority of the dialectic as that of ‘concern’ or
the ‘question’ with respect to the response. It is a matter of an ‘ontological’
anteriority, to use the words of Heidegger, exactly comparable to that of
the ‘intention’ in regards to that ‘for the sake of’. Just as the notion of ‘for
the sake of’ necessarily refers to an intention oriented towards this purpose, it is of the nature of the response to be an answer to a question
already posed, and this, even if the idea of the question comes to mind only
after having seen the response. The existence of mathematical relations
therefore necessarily refers back to the positive Idea of the search of similar
relations in general.
Having established this point allows us to specify that which constitutes
the essence of the Ideas of the Dialectic. First of all let us point out that we
distinguish notions and dialectical Ideas. The Ideas envisage possible relations between dialectical notions. Thus we examined in our thesis (Latman
1938b) the Ideas of the possible relations between pairs of notions such as
whole and part, situational properties and intrinsic properties, basic
domains and the entities defined on these domains, formal systems and
their realization, etc. The essential difference between the nature of
mathematics and the nature of the Dialectic can be inferred from these
definitions.
While the mathematical relations describe the connections that in fact
exist between distinct mathematical entities, the Ideas of dialectical relations are not assertive of any connection whatsoever that in fact exists
between notions. Insofar as ‘posed questions’, they only constitute a problematic relative to the possible situations of entities. It then happens to be
once again exactly as in Heidegger’s analysis, that the Ideas that constitute
this problematic are characterized by an essential insufficiency, and it is
yet once again in this effort to complete the understanding of the Idea, that
more concrete notions are seen to appear relative to the entity, that is, true
mathematical theories. First of all, insofar as Ideas of possible relations,
these Ideas are free from all constraints that always bring, in an effective
realization, the matter upon which one works. They do not participate as
either more or less, and ignore all determinations of sense, sign, degree,
without which nothing would exist. Moreover, being only outlines of possible positions, they do not necessarily bring about the existence of particular entities capable of supporting the relations they sketch. To think them
fully, it is necessary then to rely on some example, perhaps foreign to their
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very nature, but that thus gives shape, at least for thought, to the necessary
matter. So that the example supports the idea, it is then necessary to provide a whole series of precisions, limitations, exceptions, in which mathematical theories are asserted and constructed. Heidegger seems moreover
to have meant something like this in writing about scientific concepts:
Original ontological concepts must instead be obtained prior to any
scientific definition of ‘basic concepts’, so that only by proceeding from
them will we be in a position to evaluate the manner in which the basic
concepts of the sciences apply to being as graspable in purely ontological
concepts. (Heidegger 1969, 25 [1938, 58])
The restrictions and delimitations in question in this text should not be
conceived as an impoverishment, but on the contrary as an enrichment of
knowledge, due to the increase in precision and the certainty provided.
The more the individual character and the very structure of particular
mathematical theories is asserted, the more fertile the Ideas thereby appear
which, as defined by a non-lived history, the philosopher recognizes at the
origin of theories.
All that remains now is the elucidation of one point, that of the transcendence of Ideas. Note first the special sense that Heidegger gives to this
term in his philosophy. When it is of the essential nature of a thing to go
beyond itself in order to go towards an entity exterior to it, without which
this thing would no longer be conceived as existing, this going beyond
of the subject towards the entity, this is transcendence. It follows that transcendence properly belongs to human reality, which could not be conceived otherwise than as oriented towards the world. In thus describing
transcendence as an act of bringing together, and not as a state of separation, Heidegger does not mitigate the ontological distinction that separates
the disclosure of being and the manifestation of the entity, but he insisted
on the fact that the genesis and the development of the entity was the
necessary extension of an effort of disclosure of being. In regards to the
relation of the Ideas of the Dialectic to mathematics an analogous situation
can be described. Insofar as posed problems, relating to connections that
are likely to support certain dialectical notions, the Ideas of this Dialectic
are certainly transcendent (in the usual sense) with respect to mathematics.
On the other hand, as any effort to provide a response to the problem of
these connections is, by the very nature of things, constitution of effective
mathematical theories, it is justified to interpret the overall structure of
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these theories in terms of immanence for the logical schema of the solution
sought after. An intimate link thus exists between the transcendence of
Ideas and the immanence of the logical structure of the solution to a dialectical problem within mathematics. This link is the notion of genesis
which we give it, at least as we have tried to grasp it, by describing the
genesis of mathematics from the Dialectic.
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CHAPTER 2
The Analytic Theory of Numbers
It follows from the preceding pages that, while it is necessary that mathematics exists, as examples in which the ideal structure of the dialectic can
be realized, it is not necessary that the examples which correspond to a
particular dialectical structure are of a particular kind. What most often
happens on the contrary is that the organizing power of a same structure
is asserted in different theories; they then present the affinities of specific
mathematical structures that reflect this common dialectical structure in
which they participate. It is in this regard that we propose to envisage
certain results of analytic number theory.
It is a matter of the results relative to whole numbers, therefore to an
essentially discontinuous set, and whose proof calls for the continuous
functions of analysis. Some of these results, like those on quadratic reciprocity, are also capable of being proved in a purely algebraic way and
without any analytic means. Others on the contrary, for example those that
are related to the distribution of prime numbers, have never been proved
other than using the famous Riemann function g (s) . Mathematicians are
sometimes concerned, for ‘aesthetic’ reasons, to eliminate analysis from
arithmetic. On the other hand, it is for reasons that depend on the very
nature of their conception of the problem of the foundation of mathematics that most theorists of contemporary mathematical logic judge this
‘purification’ of arithmetic of any analytic elements as extremely desirable.
Whether it is an undertaking to reconstruct all of mathematics from the
single notion of the whole number, or the requirement to at least reduce
the consistency of analysis to that of arithmetic, it seems that it will always
rely on the idea that arithmetic is metaphysically anterior to analysis, and
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that calling upon analysis to prove arithmetic results is consequently contrary to the natural order of things. In any case, whatever the effort dispensed on this route, it does not currently appear that it will ever be
possible, for example in the theory of prime numbers, to eliminate analysis
from arithmetic. If this negative result is then compared with the proof by
Gödel of the impossibility of formalizing a supposedly consistent proof of
the consistency of arithmetic without appeal to means that exceed arithmetic, it can lead to thinking that it is incorrect to consider arithmetic as
fundamentally simpler than analysis.
The relations that support these two mathematical disciplines will perhaps appear more clearly if, instead of trying to eliminate analysis from
arithmetic, we wonder why it is possible to prove arithmetic results by
analysis. At least in regards to the facts that will be considered later, it is
possible to provide an extremely precise response to this problem: the
demonstration of certain results related to whole numbers relies on the
properties of certain analytic functions, because the structure of the analytic means employed is already accorded to the structure of the arithmetic
results sought after. More precisely, from a close structural imitation
between analysis and arithmetic, the idea of certain dialectical structures
can be identified, anterior to the diversification of distinct theories in mathematics, and such that arithmetic and analysis, far from being respectively
simpler and more complex than one another, are on the same plane and of
equal status, the realizations of this dialectic which governs them.
1. THE LAW OF RECIPROCITY
Here is what the quadratic reciprocity consists of for ordinary whole numbers (that is, in the field of rational numbers): m is said to be a quadratic
residue modulo p (m and p are relatively prime) if an integer x exists such
that m / x 2 (mod . p) , that is, such that m – x2 is a multiple of p. Legendre
introduced into arithmetic a symbol of quadratic residue (m/p) which is +1
if m is quadratic residue mod. p and –1 in the contrary case. Consider now
two positive odd integers a and b. They satisfy the fundamental law of
reciprocity
(I)
208
a-1 b-1
a
b
` b j = ` a j (- 1) 2 . 2
T H E A N A LYT IC T HE ORY OF NUM B E R S
Assuming for example a / b / 1 (mod .4) , we have, by virtue of I:
(II)
a
b
a
b
` b j = ` a j or ` b j = ` a j = 1
In this case, if a is quadratic residue modulo b, b is quadratic residue mod.
a . The relations (I) and (II) therefore allow those cases to be recognized in
which two whole numbers are likely to exchange their respective roles in
a same algebraic relation. By considering the symbols
a
b
` b j and ` a j
as the inverse of each other, a mathematical situation can therefore
be seen in the relations established by the law of quadratic reciprocity
responding to the following structural problem: given two terms that are
the inverse of one another, in what way can an exchanging between their
respective roles be conceived?
It has been possible to generalize the relations of reciprocity in two
different ways: first by considering the algebraic integers of any field and
not only the ordinary integers of the field of rational numbers. And then
by defining symbols for the integers of this field, analogous to those of
Legendre, but that are attached to more general properties: it is in this way
that Hilbert envisaged not only the numbers congruent to a square, but the
numbers congruent to a power of degree l 2 2 or the numbers congruent
to the norm of a number of the field. The particular case of quadratic reciprocity is thus related to the general law of reciprocity of which there currently exist, through the work of Hilbert, Takagi, Artin, Hasse, Herbrand
and Chevalley, purely algebraic proofs. There also exists, as regards quadratic reciprocity in the case of any field, transcendental methods, due to
Hecke, that have considerable philosophical interest for us, in regards to
a problem that is just as important, to show that the use of continuous
functions allows arithmetic results to be obtained, because these functions
satisfy the structural relations which preform those that are sought after to
be obtained between whole numbers.
Here is the main part of Hecke’s method, according to the summary that
he himself gives:1 let ~ be a number of the considered field, whole or fractional, but distinct from 0; to ~ a certain sum of terms are coordinated, this
sum defines a new number called a Gauss sum C (~) . The Gauss sums
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C (~) attached to ~ are then considered, and C (- 1 4~) attached, not
directly to the inverse 1 ~ , but the number - 1 4~ formed from this
inverse. Between C (~) and C (- 1 4~) there exists a reciprocal relation
that unites the value of the first to that of the second, and it is from this
relation of reciprocity between Gauss sums that the law of quadratic reciprocity between whole numbers is proved.
The crucial point of the method therefore lies in the establishment of the
reciprocal relation between C (~) and C (- 1 4~) . For this (by taking
the field of rational numbers as the base field) the function of a complex
variable i (x) is envisaged defined as follows:
m =+ 3
i (x) =
(III)
/
e - rxm
2
m =- 3
This theta series converges for all values of x of positive real parts, and
the function i (x) admits as a singular line the imaginary axis. It then happens to be that in the neighborhood of a singular point x = 2ir (r being a
rational number) the function becomes infinite, but the expression
lim x i (x + 2ir)
(IV)
x=0
(V)
takes a finite value. This limit is precisely, to a number of factors of no great
importance, the Gauss sum of C(– r) corresponding to the number – r. It
will be further demonstrated that the function i (x) obeys the functional
equation:
i ` 1 j = x i (x)
x
so that a relation can in particular be obtained between the comportment
of i (x) at the point x = 2ir and the comportment of i (xl ) at the point
xl = 1 =- 2i
2ir
4r
However, if for x = 2ir the expression (IV) tends towards a limit which is
of the order of C(– r), for xl = 1 x =- 2i 4r , it tends towards a limit which
is of the order of C(1/4r). A reciprocal relation between C(r) and C(–1/4r)
can therefore be deduced from the functional equation (V) from which the
law of ordinary quadratic reciprocity follows.
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T H E A N A LYT IC T HE ORY OF NUM B E R S
The transcendent proof of the law of quadratic reciprocity is essentially
built on the functional equation (V). This equation represents a ‘transformation formula’ of theta functions, absolutely independent of arithmetic
theorems of reciprocity. However a possible exchange of roles between
notions is also observed, which to a certain extent can be considered as the
‘inverse’ of one another: the value of a function at a point x and the value
of this function at point 1 x . It is in this sense that we could say above that
the analytical tool, that is, the functions, serves to prove an arithmetic
result, because the structure of the tool and that of the result both participate in a same dialectical structure, one that poses the problem of the
reciprocity of roles between elements inverse to one another.
This dialectical idea of reciprocity between inverse elements can be so
clearly distinguished from its realizations in arithmetic or in analysis that it
is possible to find a certain number of other mathematical theories in which
it is similarly realized. There is at least one, which not only thus presents the
close affinities of structure with the previous theories, but is directly involved
in the proof of the functional equation corresponding to (V), in the more
general case of an arbitrary field k. This is the theory of the ‘difference’ of
algebraic fields. In this theory two series of numbers belonging respectively
to two distinct ideals of the field k are brought together, such that: (1) the
ideals in question can be conceived as the inverse of one another (in a broad
sense, distinct from the usual narrow sense); (2) The numbers belonging
respectively to these inverse ideals, in the broad sense, can exchange the
roles of coefficients and unknowns in a set of linear equations.
Here’s how this theory plays a part in the proof of the general formula of
transformation of theta functions. Let n be the relative degree of the field
k, all the conjugates k (p) of which are assumed to be real. Consider in place
of the function defined in III, the function
(VI)
i (t, a) = / e - r(t n
1
(1) 2
2
+ ft n n(n) )
in which t1 . . . tn are complex variables with real positive parts, and in
which n(1) f n(n) represent the n conjugated values of a number n that
goes through all the numbers of an ideal a of the field once and once only.
The theta function thus defined is therefore attached to this ideal a. The
transformation equation that it satisfies is the following:
(VII)
i (t, a) =
N (a)
1
i` 1 , bj
t
d t 1 ft n
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
N(a) is the norm of the ideal a, d the discriminant of the field and b an ideal
that may be characterized as both the inverse and as the reciprocal of a.
It is defined in effect as follows: let a be any number of a. The ideal b is
formed of all the numbers b of the field that satisfies the equation:
(VIII)
b(1) a(1) + f b(n) a(n) = whole number
and that for all a of a. It is proven that the product of two ideals ab forms
an ideal independent of a and only depends on the field k. This ideal being
the inverse (in the usual sense) of an ideal integer of the field, the different
d is represented by l/d. We therefore have
ab = 1 ; a = 1 ; b = 1
d
bd
ad
It is in this broad sense that a and b can be considered as the inverse of one
another. They are in addition reciprocal since in equation (VIII) the a can
also be taken as given and the b as unknown which reverses their respective roles. This reciprocity of the two ideals is therefore defined independently of its application to the proof of quadratic reciprocity, and nevertheless
plays a part in this proof as shown in formula (VII). Just as it passes from a
function of t to a function of l/t, it passes from a function attached to the
ideal a, to a function attached to the inverse and reciprocal ideal b = l/ad.
Hecke’s transcendent proof therefore shows the convergence of three
orders of mathematical facts: the facts of analysis – the transformation formulas of i functions; the facts of algebra – the definition of the different of
a body; of facts of arithmetic – quadratic reciprocity; and this convergence
is explained only by the common dialectical structure in which these three
theories participate.
It is impossible to conceive this problem of the reciprocity of the roles of
inverse elements without also seeing its link with the duality theorem in
topology. We have already, in our thesis (Lautman 1938b), insisted on the
logical framework of this theory. It defines invariants called Betti numbers
for an n- dimensional ‘manifold’ of dimension 0, of dimension 1 . . . of
dimension n, and Poincaré’s theorem asserts, in the case of an orientable
manifold, the identity of the Betti numbers of dimension n – m and those
of dimension m (0 # m # n) .
The numbers n – m and m, whose sum is constant, can still be conceived,
in a new sense it is true, as the inverse of one another, and the identity of
Betti numbers corresponding to the dimension n – m and m can be conceived
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as an exchange of roles between these dimensions. The structural affinity
of this theory with the theories of reciprocity which have been discussed
above is manifest, and until very recently, however, it seemed that there
was no link between them. A recent indication by Andre Weil (1938, 86)
nevertheless shows that in certain cases, an approximation can be carried
out between the laws of reciprocity and the duality theorems. Here again,
the convergence of different mathematical theories results from the affinity of their dialectical structure.
2. THE DISTRIBUTION OF PRIMES AND THE MEASUREMENT
OF THE INCREASE TO INFINITY
The existence of primes in the series of whole numbers has always seemed
to present the type of mathematical facts as objective, as independent of
any prior intellectual construction, as the most manifest physical facts. The
passage from 15 to 16 and that from 16 to 17, for example, are done by
the same act: the addition of unity to the preceding number, and yet the
second operation gives a very different result from the first, since 17 is
prime and 16 is not. What thus confers on the primes their objective character is the unpredictability of their coming. Arithmeticians since Euclid,
but especially since Euler, have sought to progressively reduce this unpredictability in the occurrence of primes, and have tackled the problem from
several sides. What was the nth prime, was sought to be determined a priori
for any n; what was the interval separating two consecutive primes; how
are primes distributed within the different geometric progression of reason
k; what was the number of primes below a given number, etc. Some of this
research provided extremely rich results, some was less advanced. It is
thus, for example, that ‘there are strong indications . . . that there exists an
infinity of pairs of primes differing only by 2 (such as 17, 19 or 10 006 427,
10 006 429)’2 but there is still no proof of the hypothesis in question.
We intend to examine briefly here the results related to the best approximate determination of the number of primes below a given number x, and
this for two reasons. First, in accordance with the purpose of this essay,
because the theory relative to this number r (x) is essentially an analytic
theory, it even constitutes the most classic part of analytic number theory.
And second, because we can verify, at least in the particular case of this
theory, one of the claims of our principal thesis (Lautman 1938b): mathematical reality does not lie in the greater or lesser degree of curiosity that
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can present isolated mathematical facts, but only in the dependence of a
mathematical theory with respect to a dialectical structure that it incarnates.
Research relating to the number r (x) proceeds from a first concern of curiosity about primes, but its success is due to the fact that it was possible to free
oneself from this easy and sensible aspect of things, and to express the problem of determining r (x) in abstract terms that puts a more hidden structure
into play, and of which the theory of analytic functions already offered a
mathematical realization. It is in this connection of the arithmetic problem
to a dialectical problem already resolved in an analytic theory that the reality inherent to the arithmetic theory lies, and by this the problem of knowing why the results of this theory are proven analytically is also resolved.
It is easily noticed that r (x) , which represents the number of primes that
do not exceed x, increases to infinity with x, and the problem arises of
expressing, for x tending towards infinity, r (x) as a function of x. It is this
problem that has been resolved, each time in a more precise way, by the
work of Legendre, Gauss, Tchebicheff, Riemann, Hadamard, La Vallée
Poussin, Landau, etc. A first result gives,
for x " 3 , r (x) "
x
log x .
This arithmetic theorem is closely connected to the properties of Riemann’s
continuous function g (s)
1
;g (s) = / 8 E
n
n goes through the whole numbers from 1 to infinity, and s = v + it with
v 2 1 . The previous formula can in effect be proven by relying on the fact
that g (s) has no zero on the straight line v = 1 of the complex plane, and
conversely, from the proposition
rx "(x3 ) "
x
log x
assumed to be true, the proposition related to the absence of zeros of g (s) ,
on the straight line v = 1 , can be proven. These two propositions are
therefore equivalent, but scarcely any intuitive logical reason for this
equivalence can be seen. We are going to see on the contrary a reason of
this kind appear in a more advanced study of r (x) .
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T H E A N A LYT IC T HE ORY OF NUM B E R S
(IX)
If the nature of the link sought after between r (x) and x is examined, it
is realized that it is a matter of comparing the respective rapidity of two
quantities increasing to infinity, constantly increasing, and one of which,
x, is always the upper possible limit of the values of the other. Instead of
envisaging two increasing quantities, one of which is the upper limit of the
other, two quantities can still be considered, one of which is an independent variable increasing to infinity, and the other is a constantly increasing
function of the first. Thus, for example, the radius r and the area of the
circle for r tending towards infinity. The comparison of their mode of
increase is a problem logically analogous to the one that arises in connection with r (x) and x. Just as it is always r (x) # x , the determination of the
circumference of any circle of radius r, ipso facto sets the maximum area of
any circle of radius t , such that 0 # t # r . Instead of envisaging the area
of the circle, any function f(r) can still be considered, provided that it is
constantly increasing as r tends towards infinity, and it is thus precisely
the point of view of the measurement of their rapidity of increasing
with respect to the radius of their definitional circle, that certain analytic
functions connected to the function g (s) are introduced in the theory of
primes.
It is a matter of analytic integral functions, that is, which only have singularities at infinity. Let f(z) be such a function and M the maximum of its
modulus in the circle z # r . It is proved that the function reaches the
maximum of its modulus M = M(r) for z = r and that consequently this
function M(r) is a constantly increasing function of r when r tends towards
infinity. Consider then the expression:
log M (r)
for r " 3
rb
.
If a real value of b exists, such that the expression (IX) always remains
lower than a constant K, there is a unique number ~ $ 0 , such that it also
has place for b 2 ~ for any b . The number is called the order of the function f(z). The research of the order of an integral function f(z) therefore
comes back to comparing the rapidity of increase of an increasing function,
dependant on the integral function [of the kind log M(r)], with the rapidity
of the increase of successive powers of the radius. We find here again the
structural idea, identified above, in regards to any magnitude f(r), increasing function of an independent variable r.
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
The integral function, attached to g (s) , which plays a part in the theory
of primes is the following:3
p (s) = 1 s (s - 1) r- C ` 1 s j g (s)
2
2
1
2s
By calling M(r) the maximum modulus of g (s) for s = r we have:
log M (r) ~ 1 r log r
2
(X)
log M ( r)
"1
1
or again 2 r log r
for r " 3
This fundamental relation allows the comparison of the rapidity of increase
of M(r) and of r. From this relation another relation is deduced concerning
the increase of
gl (s)
g (s)
in a certain region of the complex plane s = v + it , when t tends
towards !3 . This region is such that it always has v $ 1 - ah ^ t h, with
0 # a # 1 and in which h (t) is a decreasing function of t satisfying certain
conditions, which it is unnecessary to present here. It is sufficient to have
indicated that for a function h (t) satisfying these very specific conditions,
we have
gl (s)
v $ 1 - ah ( t )
= 0 (log 2 t )
g (s)
t "!3
with
(XI)
which means that
gl (s)
g (s)
log 2 t
always remains below a constant quantity.
gl (s)
( g (s) is of the order of log 2 t ).
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T H E A N A LYT IC T HE ORY OF NUM B E R S
All that we now have to show is how the transposition of these measures
of increase in arithmetic is done. Just as we compared M(r) to r, we claim
to compare r (x) to x. Instead of taking r (x) on directly, arithmeticians first
envisage another arithmetic function } (x) , whose increase with respect
to x is closely connected to that of r (x) . This function is in addition, like
r (x) , a discontinuous arithmetic function. It is defined as
} (x) =
/ log p
pm # x
;
} (x) is therefore the sum of logarithms of primes p, such that any power
pm, for positive integer m, is less than x. The essential point of the theory
consists therefore in passing from the consideration of the increase of the
continuous function
gl (s)
g (s)
with respect to the increase of the variable t, to the consideration of the
increase in the discontinuous function } (x) with respect to that of the
variable x. The passage from the continuous to the discontinuous is carried
out through the formulas that allow } (x) to be expressed according to
gl (s)
g (s)
For this the continuous function
}1 (x) =
#
0
x
} (u) du
can be considered and the essential relation:
}1 (x) = 1
2ri
#
c + 3i
0 - 3i
x20
x s + 1 e - gl (s) o ds
s (s + 1)
g (s)
c21
is proved. The link between the continuous and the discontinuous being
established, here are the results:
We deduce from (XI), for x tending to infinity
} (x) = x + 0 (xe - a
log x
)
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
r (x) =
#
2
x
du + 0 (xe - a
log u
log x
)
in which a is a positive constant absolute.4
We thus see in what sense it has been possible to speak of imitation or
structural affinity between analysis and arithmetic, the continuous and the
discontinuous. The idea of comparing the rapidity of increase to infinity of
two constantly increasing quantities, and connect one to the other, is
rationally anterior to the distinction between the continuous and the discontinuous. It can therefore be realized in theories as distinct as the theory
of the increase of integral functions and the theory of the distribution of
primes, just as we have seen above, the search for the possible exchange of
roles between inverse notions realized in arithmetic, in algebra, in analysis,
in topology. Each time it is a common participation in the same dialectical
Idea, which explains the use of the analytical tool in arithmetic.
The convergence of theories which manifest the same structure thus
confirms, in a concrete way and as a posteriori, the results of the deduction
that we attempted in the first part: thought necessarily becomes involved
in the elaboration of a mathematical theory as soon as it claims to resolve
in a precise way a problem that could be raised in a purely dialectical way,
but it is not necessary that the examples are taken from a particular domain,
and in this sense, the various theories in which the same Idea is incarnated,
similarly find in it the reason for their structure and the cause of their
existence, principle and origin. As in the philosophy of Heidegger, in the
philosophy of mathematics, as we conceive it, the rational activity of foundation can be seen transformed into the genesis of notions relating to
the reel.
3. CONCLUSION
If the geneses that have been described in the preceding pages are compared with those that have been studied in the second part of our principal
thesis (Lautman 1938b), one cannot fail to notice the differences between
these two kinds of genesis. Those described here concern the coming of
mathematics from the Dialectic, whereas the schemas of genesis of our
thesis concern the different ways, within mathematics proper, in which
the structure of certain basic domains are adapted to bringing to the fore
certain new entities defined on these basic domains. To understand the
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T H E A N A LYT IC T HE ORY OF NUM B E R S
reasons which led us to thus distinguish in mathematical philosophy at
least two kinds of genesis, it is necessary to recall the two ways in which
the reunion, in a same synthetic theory, of notions related to the continuous and notions related to the discontinuous can be interpreted.
Following a study on the penetration of algebraic methods in analysis
(See Lautman 1938a) we distinguish two types of possible relations
between the theories of the continuous and those of the discontinuous:
relations of imitation and the relations of expression. There is imitation
when the structure of an arithmetic or algebraic theory and that of an analytic theory present affinities comparable with those observed above in
analytic number theory. We have seen in this case how the analysis of
certain dialectical Ideas are extended in the genesis of a reality distinct
from the Ideas: the theories in which these Ideas are realized. There is on
the contrary a relation of expression between the discontinuous and the
continuous when the algebraic or topological structure envelops the
existence of analytic entities definable on this structure. It is thus that the
topological properties of Riemann surfaces envelop the existence of abelian
integrals attached to the surface, or more generally that a correspondence
is established between algebraic entities and transcendental entities, for
example between discontinuous groups and continuous functions. The
structure of the former envelops the existence of the latter and inversely
the existence of the former expresses or represents the structure of the
latter. It is then in accordance with the general theory of geneses to also
define the passage from the structure of the domain to the existence of
representations as a genesis, since this passage from essence to existence
takes place from the structure of an entity to the assertion of the existence
of other entities than the one whose structure was originally in play.
The result of this comparison is that no such difference separates the
genesis of mathematical theories as a result of the Dialectic, from the geneses that are carried out, within mathematics, from structures to existence.
One could say in Platonic terms that the participation of Ideas among
themselves obey the same laws as the participation of Ideas in the supreme
genus and ideal numbers; in both cases, mathematical philosophy essentially offers its services as the object to witness the eternally recommencing
act of the genesis of a Universe.
219
Letter to Mathematician Maurice Fréchet
1st February 1939
Léon Brunschvicg, president of the Société française de philosophie, had made the
rare choice on Saturday, 4 February 1939, to have two presentations that dealt with
mathematical thought presented respectively by Jean Cavaillès and Albert Lautman.
The account of the meeting was published in the Bulletin of the Société in 1946
(Cavaillès and Lautman 1946). It has been included in the volume Oeuvres complètes of Jean Cavaillès by Hermann (1994). Among the interventions are those
notably of Elie Cartan (who had been a member of the thesis jury for each of them
the previous year), Maurice Fréchet, Claude Chevalley, Charles Ehresmann, and
Jean Hyppolite.
Below is an exchange, previously unpublished, between the great mathematician
Maurice Fréchet and Albert Lautman that took place on the eve of this meeting. The
comparison between the response of Albert Lautman and his presentation three days
later allows this letter to be read as a first version of the presentation.
Maurice Fréchet to Albert Lautman, 30 January 1939:
My dear colleague,
Brunschvicg invited me to participate next Saturday in the discussion of
your thesis. The summary of your communication concerns in addition
perhaps also your other publications.
Not being accustomed to the philosophical language, I have some
difficulty grasping the precise meaning of your summary. Thus in the
first paragraph I understand the mathematical examples that you give,
but I don’t quite grasp exactly the general idea they are charged with
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L E T T E R TO M AT H E M AT I C IA N M AUR IC E FR É C HE T
illustrating and that is indicated at the beginning of the last sentence of
this paragraph.
In the second paragraph, I understand that you have gone from certain material realizations to a very formal system, but I cannot conceive
the reverse (except of course after the fact).
I understand the genesis of the Idea from the real but not vice versa.
I do not assimilate the second and third sentence from the end of
the summary. If you had the time to explain in one or two pages
these different points in ordinary language, I would draw more profit
from it.
I’ll send you the only copy I have left (while asking you to return it
to me on Saturday) of my report to the conference in Zurich (December
1938) on the foundations of math.
Please accept my dear colleague, the expression of my highest
consideration.
Albert Lautman responded on the 1st of February, by announcing the
forthcoming publication of the essay New Research on the Dialectical Structure
of Mathematics (Lautman 1939). He continues:
Your first question concerns the way in which particular mathematical
theories, for example those I cite in my summary, seem to receive all of
their meaning from the fact that they provide examples of solution to
the problems that are not strictly mathematical but dialectical (as defined
by Plato).
I call notions the Whole, the part, the container, structure in the topological or algebraic sense, existence etc. I call Ideas the problem of the
elaboration of relations between notions thus defined. So I conceive the
Idea of a dialectical problem of the relations between the Whole and part
as knowing if global properties can be inscribed in local properties.
I even conceive of the Idea or the problem of knowing if the situational
properties can be expressed as a function of structural properties, and it
is to the extent that a mathematical theory provides a response to a
dialectical problem definable but not solvable independently of the
mathematics that the theory seems to me to participate, in Plato’s sense,
in the Idea, in comparison to which it is in the same situation as the
Response with respect to the Question, Existence with respect to essence.
Even if, historically or psychologically, it is the existence of the response which
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
suggests the Idea of the question (the existence of mathematical theories allow the
identification of the dialectical problem to which they respond), it is in the nature
of a question to be rationally and logically anterior to the response. Thus the
problem of knowing whether it is possible to determine the existence of
an Entity by the indication of the exceptional properties that it would
have the use of if it existed seems to me to dominate mathematical
theories as different as the calculus of variations or the topological methods of Poincaré-Berkhoff, determining the existence of periodic solutions by the determination of fixed points in an internal transformation
of the points of a surface. To the exceptional properties of extremum or
fixed points, assertions of existence are, in certain cases, attached. The
two theories are different responses to the same problem. Your second
question concerns the passage from a formal system to material realizations. Let us call the properties of a system of axioms, definable independently of any realization whatsoever in a domain of objects, formal
or structural; for example, consistency, saturation. The concern of knowing whether a system of axioms is consistent has a purely formal meaning and does not require the existence of a realization. But experience
shows that the proofs of consistency nonetheless rely most often on the
consideration in extension of domains of objects in which the hypotheses of the theory are realizable (cf. Herbrand’s theorem).
The consideration of the possibility of realisation (Erfüllbarkeit) can
therefore be conceived as led necessarily by a primarily oriented concern
towards the study of the structural properties of a formal system. Domain
theory is the necessary detour to arrive at conclusions related to formal
consistency. It is in this sense that I see the existence of a realization as
the manifestation of an internal consistency. Formal, it is made explicit
by the material existence of realization. Similarly in the representation
of abstract groups, if one were to consider as the formal or structural
property of an abstract group the number of its classes, this number
determines at the same time the number of irreducible and non-equivalent representations of the group. One can therefore envisage the
number of classes of the group as carrying out the passage from the
structure of the abstract group to the existence of its representations.
The linear representations of the group constitute a material realization.
There is here in group theory a passage from the formal to existence
that I compare in my thesis to the passage that the notion of genus of
a Riemann surface allows to be carried out between the topological
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L E T T E R TO M AT H E M AT I C IA N M AUR IC E FR É C HE T
structure of this surface and the existence of abelian integrals of the first
kind on the said surface. These analogies have led me to substitute for
the usual terminology Form and Matter another terminology in which
the systems of axioms, as well as the abstract group, constitute a basic
domain (by analogy with the Riemann surface) while the representations or realizations constitute the entities defined on a basic domain.
This does not alter the fact that there exists a passage in the usual sense
from the formal to the material.
Your third question is concerned with the genesis of the real from the
Idea. Just as you admitted the passage from the material to the formal
but not the inverse, you admit the passage from the reel to the idea, by
abstraction evidently, and not the inverse. In understanding by Idea, the
idea of a possible dialectical problem, one can envisage abstractly the Idea
of knowing whether relations between abstract notions exist, for example the
container and the content, but it happens that any effort whatsoever to outline a
response to this problem is ipso facto the fashioning of mathematical theories. The
question of knowing whether forms of solidarity between space and matter exist is
in itself a philosophical problem, which is at the center of Cartesian metaphysics.
But any effort to resolve this problem leads the mind necessarily to construct
an analytic mechanics in which a connection between the geometric and
dynamic can in fact be asserted. Here again logical anteriority, the philosophers would even say ontological, of the Idea with respect to real
mathematics. The interest for me of General Relativity taken as a pure
mathematical theory and not physical comes from what appears to me
to be a response to a problem that is able to be formulated independently
of mathematics: to what extent do the properties of space determine
those of matter? Einstein’s theory is not the only possible response: it is
a model of a possible solution among others, but what is necessary is the
constitution of a mathematical theory as soon as the dialectical question
is raised. That’s why I wrote that mathematics is an example of incarnation, in the sense in which mathematical concepts constitute for example a matter on which relations envisaged as possible by the dialectic are
effectively drawn. The comprehension of a mathematical theory or its
elaboration when it is under development has a dual meaning: mathematical from the point of view of the results it provides; philosophical
from the point of view of the constitution, in the process of a schema
of response to a dialectical problem being carried out. It is the spectacle
of the constitution of these structural schemas that seemed to me to found the
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
philosophical interest in mathematical thought. In sum, while Cavaillès searches
in mathematics itself for the philosophical sense of mathematical thought, this
sense appears to me rather in the connection of mathematics to a metaphysics (or
Dialectic) of which it is the necessary extension. It constitutes the matter closest to
the Ideas. It seems to me that this is not a diminution for mathematics. It confers
on it, on the contrary, an exemplary role.
On 4 February 1939, Cavaillès finished his presentation: ‘Clear, rigorous,
mathematical knowledge prevents us from posing objects as existing independently of the system performed on these objects and even independently of a necessary sequence from the beginning of human activity’
(Cavaillès and Lautman 1946, 12).
Albert Lautman: ‘Cavaillès seems to me, in what he calls mathematical
experience, to attribute a considerable role to the activity of the mind,
determining in time the object of its experience. General characteristics
constitutive of mathematical reality would therefore not exist. The latter
would be asserted at every moment like a simultaneously necessary and
singular event. . . . I admit the impossibility of an immutable Universe of
ideal mathematical entities. The properties of a mathematical entity depend
upon the axioms of the theory, that which deprives them of the immutability of an intelligible universe.
‘I consider no less numbers and figures as possessing an objectivity as
certain as that which the mind comes up against in the observation of
physical nature, but this objectivity of mathematical entities, that is manifested in a sensible way in the complexity of their nature, only reveals its
true meaning in a theory of the participation of Mathematics in a higher
and hidden reality which constitutes a true world of Ideas’ (13, abridged).
‘That this experience is the sine qua non of mathematical thought is certain,
but I think that there is in experience more than experience . . . to grasp
beyond the temporal circumstances of the discovery, the ideal reality,
independent of the activity of the mind.’ (p. 39, abridged)
Cavaillès: ‘Personally I am reluctant to posit something else that would
govern the actual thinking of the mathematician, I see the exigency in
the problems themselves. Perhaps this is what he calls the Dialectic
that governs; if not I think that, by this Dialectic, one would only arrive
at very general relations. . . . The future will show which of us is right.’
(p. 36)
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This is surely not a purely verbal comment since a few weeks before, he
had written to Lautman: ‘Basically you may be right. I am for myself so
engrossed in (basically the same) problem of mathematical experience that
I cannot see the connection with any other way of positing it. But maybe
we will concur in the end . . . I’d very much like to’ (Benis-Sinaceur 1987,
123–4).
225
BOOK IV
Symmetry and Dissymmetry in Mathematics
and Physics*
* The original edition (Lautman 1946) reprinted as an overture the biographical
notice written by Suzanne Lautman, also a qualified philosophy teacher, for the
Annuaire des anciens eleves de l’École normale supérieure (Obituaries edition published
in 1945). This moving text, written for the Normalien community even before the
end of the war, is not reprinted here, the author already having withheld consent
to reproduce it in the 1977 edition (Lautman 1977).
CHAPTER 1
Physical Space
The important role played by the so-called paradox of symmetric objects in
Kant’s philosophy is well known (Kant 1768). The difference in orientation
of symmetric figures with respect to a plane in ordinary space appeared to
Kant as a sensible intuition, irreducible to any conceptual determination,
and this necessary intervention of sensibility in the knowledge of left and
right is at the origin of the Kantian distinction between sensibility and
understanding. The specificity of the sensible is therefore marked by the
incongruence of symmetric figures, and it is truly a remarkable achievement that Kant had characterized the sensible as early as 1770 by a property that contemporary science has found at the centre of all investigations
of the structure of phenomena.
In the language of crystallographers, two isomeric crystals, symmetric to
one another with respect to a plane, and not superimposable, are said to be
enantiomorphs. For two symmetrical crystals to be enantiomorphs, it is
necessary that each presents in isolation a certain internal dissymmetry,
as for example the absence of a centre of symmetry. The importance of
dissymmetry by enantiomorphy appeared with the pioneering work of
Pasteur, when he recognised the connection that exists between the difference of geometric orientation of two enantiomorphic (or hemihedral)
crystals, and the inversion of their effects on the plane of polarization of
light. Following these discoveries, Pasteur conceived the theory of molecular dissymmetry, manifesting itself essentially in the hemihedry of two
isomers, and characteristic of living phenomena:
Artificial products therefore have no molecular dissymmetry and I cannot indicate the existence of a more profound separation between the
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
products formed under the influence of life and the others. (Pasteur
1861 [1922, 333])
Even if the conceptions of Pasteur on the connection of life and dissymmetry by enantiomorphy can no longer currently be defended, they nevertheless gave rise to the theory of asymmetric carbon, which is at the origin
of all the structural theories of modern stereochemistry.
The idea of enantiomorphy presents itself to analysis, we have seen, as a
close union of symmetry and dissymmetry. This same requirement for the
mixture of symmetry and dissymmetry is found in the work of Pierre Curie
on symmetry in physical phenomena. Curie no longer understands this
to characterize only biological phenomena as opposed to physical phenomena. The mixture of symmetry and dissymmetry becomes for him a necessary condition of physical phenomena in general. The determination of
the elements of symmetry of a physical phenomenon is carried out, as in
crystallography, by the searching for the center, the axes and the planes
of internal symmetry that present the phenomenon. To any physical phenomenon is tied the idea of a saturation of the symmetry, of a maximal
symmetry compatible with the existence of this phenomenon and which
characterizes it. A phenomenon can only exist in an environment possessing its characteristic symmetry or a lesser symmetry. Therefore, if the
absence of an element of symmetry is called an element of dissymmetry, it
is conceivable how Pierre Curie could write:
Certain elements of symmetry can coexist in certain phenomena, but
they are not necessary. What is necessary is that certain elements of
symmetry do not exist. It is the dissymmetry that creates the phenomenon. (Curie 1894; 1908, 126 [1982, 21])
Thus the presence of an electric field is incompatible with the existence of
a centre of symmetry and of a plane of symmetry normal to the axis of the
field, and the presence of a magnetic field excludes the existence of planes
of symmetry passing through the axis of this field. The dissymmetry constitutive of the physical phenomena is therefore defined by Curie in the
idea of a limited symmetry, of a presence of elements of symmetry to which
the absence of other elements is necessarily conjoined, and the enantiomorphy of Pasteur is only one of these dissymmetries within the symmetry
which generate the sensible world.
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P HY SIC A L SPA C E
In seeing the sensible thus defined by a mixture of symmetry and dissymmetry, of identity and difference, it is impossible not to recall Plato’s
Timaeus (1997). The existence of bodies is based there on the existence of
this receptacle that Plato calls the place and whose function consists, as
Rivaud has shown in the preface to his edition of the Timaeus (Plato 1932),
in making possible the multiplicity of bodies and their alternation in a single place in the sensible world, just as the role of the Idea of the Other in
the intelligible world is to ensure, by its mixture with the Same, both the
connection and the separation of types. This reference to Plato enables the
understanding that the materials of which the universe is formed are not
so much the atoms and molecules of the physical theory as these great
pairs of ideal opposites such as the Same and the Other, the Symmetrical
and Dissymmetrical, related to one another according to the laws of a harmonious mixture. Plato also suggests more. The properties of place and
matter, according to him, are not purely sensible, they are, as Rivaud goes
on to say, the geometric and physical transposition of a dialectical theory.
It is also possible that the distinction between left and right, as observed
in the sensible world, is only the transposition on the plane of experience
of a dissymmetrical symmetry which is equally constitutive of the abstract
reality of mathematics. A common participation in the same dialectical
structure would thus bring to the fore an analogy between the structure of
the sensible world and that of mathematics, and would allow a better
understanding of how these two realities accord with one another.
The development of modern mathematical physics offers in this regard
an extremely suggestive lesson. If we consider the theories elaborated to
account for sensible facts, in which the distinction between the left and
right plays a crucial role, that is, essentially phenomena of rotation, of the
electromagnetic field, of the polarization of light, we realize that they bring
into play abstract mathematical theories developed independently of any
concern for physical application, and which nevertheless present the following aspect: we find there either a duality of opposing elements capable
of being permuted with one another as the left and right are permuted
in a symmetry; or, which is even more characteristic of the mixture of
symmetry and dissymmetry, a division of mathematical entities into two
classes, a rigorously symmetric class as an ambidextrous being would be
and a class that mathematicians call antisymmetric, that is, that changes
orientation by symmetry, as the right hand or the left hand, or the sign by
the permutation of two variables, as a straight line AB in space changes
orientation when it is traversed from B to A.
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Our first example is taken from the theory of spinors, in its relation with
the spin of the electron. The physical requirement that is at the origin of the
theory of electron spin is the necessity of experimental origin to endow the
electron with a moment of rotation, or spin, which could at any time take
two opposing values with two different probabilities. The elaboration of
this conception, in Dirac’s theory, resulted in the attachment to an electron
of a wave of four components, or rather two groups of two components,
constituting what is called a spinor. These two groups of two components,
these two semi-spinors, play a role with respect to one another that can be
considered as the left and the right in the space-time of special relativity.
To any turning of space in space-time, which conserves the sense of time
and changes the sign of the directions of space, an algebraic operation can
be made to correspond which permutes the two semi-spinors, and thus
represents in the abstract space of the spinor components, the sensible
operation of a change in orientation in physical space. These abstract entities, the spinors that are internally divided in two, whose role in quantum
mechanics is so important, had not however been invented by physicists,
since Elie Cartan had discovered them in 1913 in his research on the linear
representations of the rotation group of a space of an arbitrary number of
dimensions. This is a typical case where the physics of the dissymmetric
symmetry refers to an algebra in which there is an exchange of roles
between opposite terms.
The example of the unitary theories of the electromagnetic and gravitational fields offers the analogous case of a physical theory based on the dissymmetry of certain mathematical entities discovered anew by Cartan, well
before their use by Einstein. In his theory of 1928, Einstein associates the
phenomena of electro-magnetism and the phenomena of gravitation by considering spaces endowed with torsion, which were introduced into science by
Cartan in 1922. Here is a simple example, due to Cartan, which may suggest
the notion of torsion. Consider two systems of curves on a surface, such as
the meridians and the parallels on the sphere. Suppose that a ship moves a
distance d along a meridian going from point A to point A´, then a quantity
d along a parallel from A´ to A˝. Now suppose that the ship moves in reverse
order and traverses first d on the parallel passing through A, which leads it
to A n , and d on the meridian passing through Am , which leads it to a point
A n distinct from A˝. If the succession of operations in each case is therefore
designated respectively by dd and dd , then we see that dd ! dd . The existence on the sphere of a vector of torsion A m Am (or of the opposite vector
Am A m ) thus realizes the non-commutativity of the operations d and d . In a
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more general way, the distinction between left and right in this case is only
the sensible expression of an algebra with non-commutative multiplication,
in which the product AB differs from the product BA.
It is necessary to emphasize the fashion in which the distinction between
left and right in the sensible world can symbolize the non-commutativity
of certain operations of abstract algebra. The fundamental property of
symmetry with respect to a plane, applied once, gives a figure distinct in its
orientation from the original figure, and repeated a second time, gives the
original figure again. It is for this reason that symmetry is said to be an
involutive operation. Let us now consider an algebraic operation concerned
with two quantities X and Y, and which can be written (XY), the parentheses denote an ordinary product, or any other operation defined on the two
variables. It is a non-commutative operation if (XY) ! (YX) and the most
fruitful non-commutativity in mathematics is that in which (XY) = –(YX).
The operation (XY) is dissymmetrical, in X and Y, but it is easily verified
that it defines an involutive operation, as does ordinary symmetry. The
expressions (XY) and (YX) are said to be antisymmetric, and this word
reflects well the mixture of symmetry and dissymmetry which is thus
installed deeply in the heart of modern algebra. The whole theory of continuous Lie groups is based on the non-commutativity of the product of
two infinitesimal operations of the group. This theory, which is closely
associated with the theory of Pfaffian forms, expressions with antisymmetric multiplication, allowed Cartan to discover a profound analogy between
the generalized Riemann spaces which play a part in the physic–geometrical theories of relativity and the space of Lie groups.
The examples of mathematical physics that have been cited so far
show how the sensible fact of dissymmetrical symmetry could be conceived
as the equivalent, in the world of phenomena, of the antisymmetry inherent to certain mathematical entities. Recent developments in the wave
mechanics of systems of particles have shown that the distinction between
symmetry and antisymmetry served as the foundation for sensible qualities
of matter, perhaps even more important from the philosophical point of
view than the properties of orientation. It is a question of the constitution
and stability of molecular structures, of the very notion of system, of the
‘whole’, in the sense that a whole possesses the global properties which
characterize it qualitatively, and make something other, and more, than
the sum of its parts.
The wave mechanics of systems of particles considers a system composed
of particles of the same nature, and attaches to this system a wave function
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} (1, 2, f n) , which is a function of n particles of the system. When the
trajectories of these particles of the same nature impinge upon one another,
the particles become indiscernible, and the system must also be described
equally well by any function obtained by permuting in any way the n
variables of the function } as by the function } itself. The result of this
theorem is that for the systems containing only two particles of the
same physical nature, the functions } which describe the evolution must
necessarily be either symmetric or antisymmetric with respect to these two
particles, that is, we have either } (1, 2) = } (2, 1) in the symmetric case, or
} (1, 2) =- } (2, 1) in the antisymmetric case.
As far as systems with an arbitrary number of particles are concerned, an
analogous result follows from experimental data. According to de Broglie,
it is certain that for each type of particle the wave functions are either
symmetric or antisymmetric, and the antisymmetry seems to play a much
more fundamental role in Nature than symmetry. In fact, if the elementary
particles are distinguished from the composite particles, the systems of
elementary particles such as the electron, the proton, the neutron, the
neutrino if it exists, are observed in an antisymmetric state. On the other
hand, composite particules formed by the union of several elementary
particles, like photons or a particles, are in a symmetric or antisymmetric
state depending on whether the number of constituents is even or odd
(See de Broglie 1939). These results have allowed the notion of chemical
valence, and consequently the constitution of molecules, to be linked to the
antisymmetry of the spin of two electrons belonging to two distinct atoms.
Antisymmetry thus plays a crucial role in explaining the molecular bond.
If the mathematical basis of this distinction of wave functions into symmetric functions and antisymmetric functions is examined, it is found in an
internal dissymmetry in the group of permutations of n objects. This group
is decomposed into two families: the subgroup of even permutations and
the family of odd permutations, which does not form a group since it does
not contain the identity permutation. Consider for example the group of
permutations of three variables 1, 2, 3. The subgroup of three even permutations is obtained by circular permutation: 1, 2, 3; 2, 3, 1; 3, 1, 2; the three
odd permutations are 1, 3, 2; 3, 2, 1; 2, 1, 3. Here we have the second type
of dissymmetry discussed above, that of an entity which is divided into
a symmetrical part and a dissymmetrical part. The example of the wave
mechanics of systems of particles, in which we see the saturation of electronic levels in an atom and the formation of chemical molecules related to
a mathematical dissymmetry as essential in its simplicity as that of the
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group of permutations of n objects, shows in the clearest way in what sense
it is possible to speak of the common participation of the sensible world
and of certain mathematical theories in a same dialectical structure, composed of the mixture of symmetry and dissymmetry.
We would like to go further and show the importance of this structure,
not only for those mathematical theories that are applied to the sensible
universe, but, in a general way, for the most abstract domains of mathematics. In order to do this, reconsider for a moment the analysis of the
distinction between left and right. We find there two ideas: 1) the division
of a complete entity into two distinct parts, at least by the inversion of their
orientation, and 2) the existence of an involutive relation between the two
parts, such that A is to B as B is to A, the symmetrical of the symmetrical
restoring the original element. Transposed into a more abstract language,
this situation is equivalent to the possibility of distinguishing within the
same entity two distinct entities X and X´, which are said to be in duality,
first, if an orientation or an order inverse to that of the other can be defined
for each of them, and second, if an involutive relation exists between them,
that is, if X is to X´ as X´ is to X; that is if (X´)´ = X.
A certain number of mathematical theories that are based on this kind of
structure of duality have been known for a long time. The most famous is
the calculation of propositions established by Boole in 1847, which is at the
basis of modern mathematical logic. Let / be the set of all possible propositions of the theory. This set is subdivided into two parts with no common
elements S and S´ which are complementary to one another, that is, their
logical product (in symbols: S + Sl , the set of elements common to S and S´)
is empty, and their logical sum (in symbols: S , Sl , the set of elements
belonging to either S or S´) is equal to the total set / . These two subsets can
be considered, if we wish, as the set of true propositions and the set of false
propositions, but this particular interpretation is not at all necessary. It is
sufficient to consider them as two complementary sets. The involutive
character of this complementation is made evident by the fact that the
complement of the complement of S is equal to S. The global duality
between S and S´ allows a duality to be established between a formula P of
S and a formula P´ of S´, which will be called the negation, the contrary, or
the complement of P, and is such that the logical sum P , Pl is always true
and the logical product P + Pl always false. The essential property of this
duality is to interchange the symbols of logical addition and logical multiplication. Given a formula constructed with elementary propositions, p, q,
. . . etc., and the logical symbols of the sum, the product, and the negation,
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, , + ,´, the negation can be obtained, which is also called, as we have seen,
the contrary or the complement of this formula, by replacing all the elementary propositions of the formula by their negation and by permuting
the signs , and + . Thus we have
(p , q)l = pl + ql and (p + q)l = pl , ql .
If, by using the notion of implication, an order is introduced between two
propositions, it can easily be proved that the duality changes the meaning
of the implication
(p 2 q)l = ql 2 pl
which helps identify the duality of geometric symmetry and the duality of
logical negation.
There is another mathematical theory in which the notion of duality
has played a fundamental role since its discovery by Poncelet in 1822. It is
projective geometry. It is well known that in projective plane geometry, in
any true proposition constructed with the notions of point and line and the
relation of inclusion, a true proposition can be obtained anew by interchanging the words: point and line, and by changing the meaning of the
relation of inclusion. To the points situated on a line there correspond
the lines passing through a point. Thus a curve can be defined by means
of the points which compose it, or by the lines tangent to it at each point.
Here is the algebraic expression of this duality. Consider the equation,
u1x1 + u2x2 + u3x3 = 0. This equation can be interpreted in two ways. If the
three quantities u1, u2, u3 are considered as defining a line in the space of
the projective plane, the coordinates of all the points of this line are defined
by the variables x1, x2, x3 satisfying the equation. Conversely, if we take the
three coordinates x1, x2, x3 of a fixed point, the equation is that of all the
lines u1, u2, u3 passing through this point. The quantities u1, u2, u3 and x1, x2,
x3 can therefore interchange their roles as coefficients and variables in the
proposed equation, and it follows from this fact that the projective plane
can be considered either as a set of points, or as a set of lines, and these two
sets are said to be in correlation with one another. More generally, in an
n-dimensional projective space Sn, the points (elements of 0 dimension)
and the hyperplanes (elements of n – 1 dimension) of this space are in
correlation, and this correlation constitutes a true duality, as defined
above. Modern axiomatic research has, in effect, allowed the sum and the
intersection, or product, of two projective spaces Sp and Sq of respective
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dimensions p and q to be defined rigorously, and to associate in this way,
to every subspace Sm of a space Sn, a dual space Sn–m–1, such that
S m , S n - m - 1 = S n and S m + S n - m - 1 = 0 .
The sum of a subspace and its dual gives the whole space, and their
intersection is empty. It can easily be demonstrated that this duality is
involutive and it reverses the inclusion relation:
if S p 1 S q, S n - p - 1 2 S n - p - 1 .
There is therefore a projective duality as well as a logical negation or a
geometric symmetry.
It seems at first that the aim of the calculus of propositions and that of
projective geometry are different, and yet the logical structure of these two
disciplines, as we have shown, present many analogies. The reason for this
analogy only appeared recently in light of the latest research in the domain
called abstract algebra. Inspired by Dedekind, a large number of contemporary mathematicians, including Birkhoff, von Neumann, Glivenko (see
Glivenko 1938), Ore and others, have constructed a general theory of
structures (English authors call lattices)1 that includes the theory of sets,
number theory, projective geometry, combinatorial topology, probability
theory, mathematical logic, the theory of functional spaces, etc. Here are
the basic notions of the theory. Each time a set S is considered, the parts of
this set that are selected are not the individual elements of this set, but the
subsets of the set. Between any two parts there is defined either an order
(in the case of a set composed of a finite number of parts), or at least a
partial order, through relations such as magnitude, dimension, inclusion,
implication, boundary, and the two operations of sum and product. The set
S* is then considered, obtained by inverting the order between the parts
and the permutation of the symbols for sum and product. The new set thus
obtained is called the dual of the first, and, in the most interesting cases,
the dual set is none other than the original set in which all the order
relations are reversed. Duality thus establishes an anti-isomorphism (or
inverse isomorphism) between S and S itself. It is in addition an involutive
operation, since the dual of a dual set restores the original set. The general
theory of lattices is therefore based on the possibility of structuring the
same set in two mutually inverse ways. To see this internal duality of two
antisymmetric entities, distinguishable within the same entity, is a result
of major philosophical importance, forming the principle generator of an
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immense harvest of mathematical reality. The theories that have been
cited above, according to Glivenko, allow us to consider as lattices, the
totality of rings, the set of convex fields, the set of subgroups of any group,
the set of all positive whole numbers, the set of all the elements of projective geometry, the set of all simplexes subordinate to a topological simplex,
the set of events of probability theory, the set of propositions of propositional calculus, etc.
This development of the theory of lattices has naturally led to the
establishment of distinctions between all the structures satisfying the law
of duality. Thus, for example, in the case of mathematical logic, the complementary formula of a given formula is determined in a unique way,
whereas, given m + 1 points that define a subspace Sm of a projective space
Sn, there exist an infinity of ways to choose n – m other points that determine a complementary subspace Sn–m–1 of Sm. The determination of the complementary element is therefore possible in both cases (this not the case for
all lattices), but the uniqueness of this element, established in the propositional calculus (or Boolean algebra), does not hold for projective geometry.
These represent two different realizations of the same dialectical structure of both complementary and antisymmetric duality, and it is interesting to emphasize for a moment the foundation of this difference.
Boolean algebra satisfies the following laws of distribution for sum and
product
x + (y , z) = (x + y) , (x + z)
x , (y + z) = (x , y) + (x , z)
while projective geometry satisfies only a weaker distributive law, called
the modular identity, discovered by Dedekind:
If x # z,
x , (y + z) = (x , y) + z
By replacing the symbols , and + with the usual symbols of the product
and sum, we see that in the distributive lattice x(y + z) = xy + xz, while in
the modular lattice we have, for x # z , only (x + y)z = x + yz.
In a famous article, ‘The logic of quantum mechanics’, Garret Birkhoff
and von Neumann (1936) relied upon this difference to establish that the
calculus of propositions relative to the observations of classical mechanics
has the structure of a Boolean algebra, while the calculus of propositions
relative to the observable facts of quantum mechanics would satisfy modular
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identity, but not the distributive law, and would thus have the structure of
a projective geometry. Berkhoff and von Neumann provide the following
example in support of this thesis: If proposition a corresponds to the observation of a train of waves al on one side of the plane, proposition } to the
observation of } on the other side of the plane, and b to the observation
of } in a symmetric state with respect to the plane; neither b + a nor b + al
can be observed simultaneously, therefore we have b + a = b + al = 0
(0 represents the identically false proposition). Thus (b + a) , (b + al ) = 0 .
On the other hand, since a , al is an identically true expression, the
conjunction b + (a , al ) is equivalent to b. Since clearly b 2 0 , we conclude
that:
b + (a , al ) 2 (b + a) , (b + al ) .
The distributive law is therefore not satisfied (Birkhoff and von Neumann
1936, 823).
The logical difference between two lattices both satisfying the law of
duality would therefore be translated in the sensible world into the difference between classical mechanics and quantum mechanics. These applications of the theory of lattices to physics cannot substitute at present for
mathematical physics properly speaking. They nevertheless appear to us to
justify the hypothesis of a similar importance of dissymmetrical symmetry
in the sensible universe and antisymmetric duality in the mathematical
world. Moreover, it is remarkable that even in the mathematical theories
that do not at present seem to be related to lattice theory, there are laws of
reciprocity comparable to the duality that we have just studied. In certain
cases, this reciprocity presents itself as a possible exchange of roles between
arguments of a same relation or of a same function, accompanied by a
change in sign of the relation. This reciprocity is therefore comparable in
every way to the antisymmetry of non-commutative products. Just as we
had (XY) = –(YX), we have f(x, y) = –f(y, x). In other cases, the relation of
reciprocity appears as a total symmetry that is not accompanied by any
change of sign, that is, by any dissymmetry. Nevertheless, it seems that the
symmetry that we will call symmetric is only a limiting case of an antisymmetric symmetry which remains the general case. Just as when (XY) =
–(YX), we nevertheless have (XY) = (YX) when (XY) has the particular
value 0, an antisymmetric relation between two elements related to the
numbers p and n – p whose sum n is constant, can become symmetric in
the particular case in which n – p = p, that is to a say if n = 2p.
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Something analogous is found in the extremely important theorems of
quadratic reciprocity in arithmetic. Legendre introduced into arithmetic
the symbol (m/p) which equals +1 if m is a quadratic residue modulo p, that
is, if there exists an integer x such that m – x2 is a multiple of p, and –1 in
the opposite case. Consider now two positive odd integers a and b. They
satisfy the fundamental law of reciprocity:
a-1 b-1
a
b
` b j = ` a j (- 1) 2 . 2
This law therefore includes at once all the cases of reciprocity in the strict
sense of the word, that is, the cases in which a is to b as b is at a , and the
cases in which reciprocity does not hold. The general law contains an element of dissymmetry (the factor –1), which disappears in the particular
cases in which reciprocity actually holds.
Seeking to determine the nature of mathematical reality, we have shown
in a previous work (Lautman 1938b) that mathematical theories can be
interpreted as a matter of choice destined to give substance to a dialectic
ideal. This dialectic seems to be principally constituted by pairs of opposites
and the Ideas of this dialectic present themselves in each case as the problem of connections to be established between opposing notions. The determination of these connections can only be made within the domains in
which the dialectic is incarnated, and it is thus that we have been able to
follow in a great number of mathematical theories the concrete outline
of the edifices whose effective existence is constituted as a response to
problems posed by the Ideas of this dialectic. It seems certain in this regard
that the idea of the mixture of symmetry and dissymmetry plays a dominant role, not only with respect to physics, but as we have tried to show,
with respect to mathematics. The two realities are thus presented in accord
with one another as distinct realizations of a same dialectic that gives rise
to them in comparable acts of genesis.
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CHAPTER 2
The Problem of Time
In the previous chapter, an essential property of physical space, the difference of orientation of symmetrical figures, is interpreted as the sensible
manifestation of a dialectical structure that is as much the generator of
abstract mathematical realities as of the conditions of existence for the
world of phenomena. Such an analysis thus situates at the level of Ideas
what seemed to be one of the characteristics of spatiality, and this is perhaps the most current sense that the notion of intelligible extension can
take today. The success of the spatial problem leads us to pose an analogous problem for time: Is it possible to describe within mathematics a structure that is like a first outline of the temporal form of sensible phenomena?
At first glance, this problem appears much more difficult than the previous
problem, since time, more so than space, seems to be linked to the sensible
existence of the Universe. That it is defined in effect by the movement of
the earth, the order of causality, biological aging, thermodynamic irreversibility, the duration of consciousness, one always makes use of notions
that are meaningful only for a sensible observer in a sensible world. Our
task therefore does not consist in introducing change or becoming into the
unchanging world of mathematical truths, but in distinguishing from sensible time an abstract form of time whose necessity is essential to the intelligible Universe of pure mathematics as to the concrete time of Mechanics
and Physics. The importance of this attempt is easily seen for the problem
of reasons for the application of mathematics to the physical universe.
Such a study is comprised naturally of several moments. First, we
will describe the sensible properties of time that are inscribed in the equations of mathematical Physics. We will then show that the mathematical
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structure of these equations comes to them not from the physical domain
to which they are applied, but from the mathematical domain from which
they proceed. We will then search for the dialectical bearing of these
results. As this regressive analysis proceeds, time is assuredly stripped of its
unstable and lived aspect, but on the other hand we will reach this uncreated germ that contains within it both the elements of a logical deduction
and an ontological genesis of becoming sensible.
1. SENSIBLE TIME AND MATHEMATICAL PHYSICS1
Before studying the relations between sensible time and mathematical
physics, it is first necessary to pose as a preliminary point: all theorists of
modern science are in agreement in recognizing that the notions of space
and time, such as they result from sensible experience only make sense in
classical mechanics. On the other hand, relativistic mechanics and more so
quantum mechanics and wave mechanics require in their domain of validity the development of radically new notions relating to time and space.
Is it still possible in these conditions to try to characterize in a unique and
general way the manner in which sensible properties of time are expressed
in the symbols of a mathematical physics that cannot be considered as one?
This objection however is not of great weight, since if there is a rupture
between the physical sense of classical space and time, space-time relativity
and the uncertainty of quantum relations, there is continuity in the mathematical form of these various mechanics. The same authors who insist on
the fact that modern conceptions are unrepresentable in the classical
framework are still trying to find in the apparent harmony of the classical
theories, the origin of all the mathematical complications of more recent
theories. It is thus that the distinction between opposing points of view in
modern science very often proceed from the strict equivalence of two
modes of presentation of certain classical results. We can therefore infer
the common structural conditions that are essential to the succession of
physical theories, and that, given both the sensible aspects of time and the
exigency of a dialectic ideal, carry out the connection between the sensible
and the intelligible.
The sensible properties of time, which are constituted as experimental
facts that any physical theory must account for, can be expressed in the
following propositions:2
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T HE P R OB LE M OF T IM E
1) Time always flows in the same direction. This property establishes a
dissymmetry between time oriented from the past towards the future,
and space which knows neither direction nor privileged meaning.
2) Material objects persist over time. This property relates the existence
of material objects to the flow and the direction of time. A material
object can in effect be independent of other material objects which
exist simultaneously elsewhere, but its existence at a given time is
inextricably linked to its own past and its own future. The continuity
of time is thus an essential element of the permanence of objects.
3) Magnitudes, other than the time which characterizes physical systems, vary as a function of time.
The first two properties are distinct in the sense that one concerns the irreversibility of what could be called pure time, and the other, the spreading
out in time, according to this irreversible order of the before and the after,
of the physical objects of the universe, but they are nevertheless closely
linked. Joined together, they make of time an oriented direction, necessarily associated with the directions of space for the location of physical phenomena. This direction of time possesses no less than space a special
dissymmetry which comes precisely from its orientation from the past
towards the future. The third property, on the contrary, does not concern
the direction of time. It only indirectly concerns time, since it is relative to
the changes undergone by the other physical magnitudes of the universe.
It happens in effect to be with respect to time that the physicist studies the
changes of position, speed, temperature, density, energy, etc. . . . within
any physical system, but these changes do not at all obey in themselves any
requirement of irreversibility. This was the opinion of Boltzmann, and still
is the opinion of Schrödinger when he wrote:
Even the laws of nature, that are called irreversible, by themselves imply
no temporal direction when they are interpreted statistically. The predictions they allow to be stated dependent in effect on the boundary
conditions in two temporal sections t0 and t1 and are absolutely symmetrical with respect to these two sections, without the order of time
playing any role. (Schrödinger 1931, 152)
Time, which plays a part in the statement of the laws of nature, in
kinematics, in dynamics, in thermodynamics, etc. . . . is therefore not the
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irreversible time of the duration of things. The role it plays is simply that
of a factor of evolution and it would be perfectly possible to study the evolution of physical phenomena in terms of another magnitude taken as an
independent variable. The result of these considerations is that two kinds
of properties of sensible time can be clearly distinguished, those connected
to the notions of dimension and orientation are the geometric properties
of time, and those connected to the notion of evolution determine in particular the dynamic properties of bodies. We will now study the mathematical aspects that these sensible properties of time take within the various
physical theories.
First of all, let us consider the distinction between time and space that
results from the existence of an orientation of time. It is known that the
development of the theory of special relativity has led to the constitution
of a spatio–temporal synthesis in which the time coordinate, in certain
transformations, plays a symmetric enough role to that of the space
coordinates. It is nonetheless true that a fundamental difference subsists
between time and space. While in Euclidean space the square ds2 of the
distance between two points is given by a sum of three squares preceded
by the + sign, ds2 = dx2 + dy2 + dz2, in space–time the square of the distance
between two points is given by a sum of three positive squares and one
negative square ds2 = dx2 + dy2 + dz2 – c2dt2, or conversely by calling ds2, for
reasons of convenience of calculation, the quantity – ds2, ds2 = c2dt2 – dx2 –
dy2 – dz2. Here is a geometric interpretation of this formula, the speed of
light being taken as unity. To each point P of coordinates x, y, z, t of space–
time a cone can be attached that has two nappes of vertex P which has the
time axis as the axis of revolution, is open towards the future, and whose
generators are define by the equation ds2 = dt2 – dx2 – dy2 – dz2 = 0. These
lines of zero length define the luminous trajectories in space–time. All
other lines of the universe are interior to the two nappes of the cone, oriented from the nappe of the past to that of the future and satisfying the
condition ds 2 2 0 . This direction and sense imposed on the lines of the
universe constitute what de Broglie calls the fibrous structure of space–time.
The dissymmetry of time and space is therefore expressed in two equivalent ways. One, algebraic, connected to the idea of default or difference, is
the possibility of attaching to each point in space-time a quadratic form
composed of a positive square and three negative squares. The other, geometric, connected to the idea of orientation, is the possibility of attaching
to each point a set of oriented directions constituting the generators of
the cone of the future relative to this point. With the connected notions
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of difference of sign and oriented direction attached to the points of the
universe, we thus possess the mathematical expression of the dimensional
properties of time in classical mechanics and relativistic mechanics. We
shall see what it is in quantum mechanics later.
Let us now shown how the role of time conceived as a parameter of
evolution is presented in classical mechanics. Let us now place ourselves in
the simple case of a single material point moving in ordinary Euclidean
space whose three coordinates x, y, z are three functions x(t), y(t), z(t) of the
parameter t which represents time. Let T be the kinetic energy of this point,
V its potential energy which is a function of the three coordinates x, y, z
and of time. Consider the Lagrange function L (x, y, z, t) = T – V. Suppose
that at the instant t0 the material point is at A and at the instant t1 at B. Let
C be the curve representing the trajectory of the point. Hamilton’s principle asserts that the real movement that leads
the point from A to B along
t
the curve C is such that the expression # Ldt taken along C is extremal
t
with respect to infinitely near curves connecting
the same point A to the
t
same point B, that is, we have in symbols d # Ldt = 0 . All the laws of clast
sical mechanics can be deduced from this extremum principle.3 Similarly,
all the laws of classical electro-magnetism and the whole of the theory of
special relativity can be deduced from an analogous variational principle. It
is likely that this is also the case for the theory of general relativity, and as
concerns the theory of quanta, the importance of Hamilton’s principle has
been fairly well demonstrated by the work of de Broglie. Suffice it to say
that when there is conservation of energy, from Hamilton’s principle
d # Ldt = 0 , Maupertuis’ principle of least action d # mvds = 0 can be
deduced, in which mv represents the momentum vector, and ds the element of the arc of the curve traversed, and it is through action that quanta
are introduced in physics. It is known in addition that the analogy of the
Maupertuis principle for the dynamics of material points, and Fermat’s
principle for optics, is the basis of wave mechanics.
The Hamilton and Maupertuis principles are therefore currently among
the most general principles of all theoretical physics, and it is important for
us to note that if time figures as a parameter of evolution in the statement
of the first, d # Ldt = 0 , this role is played by the element of the arc of a
curve in the statement of the second, d # mvds = 0 . We thus grasp in a
particularly simple example in what sense the fact for time to be the parameter of evolution, as a function of which the other physical magnitudes
vary, is independent of the geometric properties of time, since this role can
in certain cases suit another variable. In the domain of classical mechanics,
1
0
1
0
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
the difference between the geometric aspect and the dynamic aspect of
time should not however be exaggerated since there is strict equivalence
between them. Hamilton’s dynamic principle can in effect be expressed in
a geometric form in which time figures as a coordinate associated with the
coordinates of space but, due to its characteristic dissymmetry, marked by
a different sign. For this,4 consider a configuration space of n + 1 dimensions defined by the n spatial coordinates ql . . . qn and time t. Each state of
the system defined by a value of these n + 1 coordinates corresponds to a
point in this space. Let L be the function which generalizes the Lagrange
function defined above, L = T – V, and E = T + V, the energy of the system.
dL
Associate to each coordinate qi the canonically conjugated quantity p i = dqo i .
We have seen that with this notation the system energy can be written
E = / p i qo i - L .
Q
t
From this is deduced #t Ldt = #P p 1 dq 1 + fp n dq n - Edt , by calling P the
point of the configuration space of n + 1 dimensions, which corresponds to
the instant t0 and to the state of the system at that instant, and Q the point
corresponding to time tl and to the state of the system at that instant. The
expression p1dq1 + . . . pndqn – Edt, whose integral from P to Q thus presents
a minimum with respect to the infinitely near trajectories of the configuration space–time, can be regarded as the work in this space of n + 1 dimensions q, . . . qn, t of a vector which would have as spatial components the
n ordinary components p1 . . . pn of the momentum, and as components
following time, the energy with sign changed. A spatio–temporal synthesis
is thus found again analogous to that of special relativity in which time
imposes a difference of sign with respect to space.
The equivalence thus shown in classical mechanics between the dynamic
aspect and the geometric aspect of time no longer easily subsists in wave
mechanics in which we will see instead their opposition asserted. We can
then state a fundamental principle of our whole study: when it is a question of two distinct notions, their equivalence or their opposition appears
on the same plane as subsequent to the fact of their simple duality,5 conceived as indifferent again to the assertion of any relation between them.
To reuse terminology which we have made use of elsewhere, we will call
Idea the problem of determining a connection made between distinct
notions of a dialectical ideal. It is in these conditions that we conceive the
existence of a theory of Ideas, of a dialectic common to wave mechanics
and classical mechanics, even though the former throws into opposition
the notions whose agreement is evident in classical mechanics.
1
0
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T HE P R OB LE M OF T IM E
It is necessary to now envisage the relations of geometric time and
dynamic time in wave mechanics. In this presentation of these questions,
we will follow the analysis of de Broglie cited above and in particular that
of the last chapter of his book: L’Électron magnétique (1937c). De Broglie
addresses the problem of the relations of space and time in wave mechanics by starting with the fact that Dirac’s wave mechanics equations are
invariant under a Lorentz transformation operating in the space-time of
special relativity, and noting nevertheless that in wave mechanics time
plays a very different role to that of the coordinates of space. Here is a
summary of the main considerations that he develops in this respect: to
any physical magnitude, coordinates of space, momentum, energy, torque,
etc. . . . there corresponds, in wave mechanics, no longer an accurate
measurement, as in ordinary mechanics, but a mathematical entity called
a ‘Hermitian operator’. The measurement of any physical magnitude A can
only give one of the ‘eigenvalues’ a1 f an of the corresponding operator
A(op), and the probability that an observation attributes the precise value ai
to the magnitude A is equal to the square c i 2 of the module of the coefficient ci of the eigenfunction {i corresponding to ai in the development
} = / c i {i of the wave function } according to the eigenfunctions {1 f {n
of the operator A(op). In general these probabilities are functions of time and
that is why the system evolves, but time thus remains a regular number
and no operator corresponds to it. It therefore appears as a parameter of
evolution in the study of probabilities attached to no matter which physical
magnitude, and not as a coordinate likely to be seen as a proper probability
distribution in space-time. We thus find a distinction analogous to that
already encountered above. De Broglie later envisages the notion of ‘average value’ of a magnitude in wave mechanics. It is a very important notion
since if it is impossible to speak of the precise value of a magnitude A at a
determined moment, we can calculate at each instant an average value A
as the sum of the mathematical expectations of each of the possible values
at the considered time. We thus have A = / ai c i 2 in which again, by a
simple proof A = # } * A op }dxdydz . The integration takes place in an area
of space T for which A is consequently a constant. This definition makes
sense because it is possible to consider a particle as practically isolated from
the rest of the universe. The influence of fields of forces exterior to the
atom on the waveform } is entirely negligible because these waves tend
towards zero when moving away from the atomic domain (De Broglie
1937c, p. 306). Whereas, the integration in a space-time domain dy dx dz dt
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would suppose a static physics from which all evolution would be banished, which is evidently absurd. De Broglie finally considered uncertainty
relations. It is known that Heisenberg’s relations establish the impossibility
of measuring accurately a coordinate of space qi and the corresponding
component of momentum pi: Tq i Tp i $ h . There exists a fourth relation
concerning energy and time TETt $ h , but it has a very different meaning
to the previous ones. It defines, not an error committed in the measurement of E, but the minimum duration of the experiment that would allow
a value to be attached to the energy of the particle marked by a minimum
uncertainty equal to TE .
This difference in nature between time and space in the equations of
wave mechanics is related for de Broglie to the fibrous structure of space–
time that has already been described above. To each particle there corresponds a line of the universe oriented in the direction of time, and this
cleavage of the universe allows the operation of the spatial sections of all
the lines of the universe in order to obtain a coexistence of independent
systems in space. It does not appear however that this reason is sufficient
to also explain the difference between the evolution parameter of time and
the oriented dimension of time. De Broglie also seems to be led to the idea
of two distinct times:6
It is evident that it would be desirable to introduce into quantum theory
the idea that the coordinate t is also linked to a probability distribution,
but it should be done by keeping in the theory a variable of evolution,
and we have said, this does not seem very easy. (De Broglie 1941, 200)
It is interesting to emphasize the mathematical source of this duality of
the roles of time in wave mechanics. The work of de Broglie starts with the
relativist conception in which energy E and the three components of the
momentum of a particle, px, py, pz, constitute the four elements of a fourvector of space–time. If, according to the general principles of the theory of
quanta and wave mechanics, E = hv and m = h m are posed, in which v is
the frequency of the wave and m the wavelength attached to a particle, m
represents its mass and v its speed, then the plane monochromatic wave }
associated with an isolated particle, in the case of a constant external field,
is given by the formula
} (x, y, z, t) = A e) $
248
2r 6Et - p x - p y - p [email protected]
.
x
y
z
h
T HE P R OB LE M OF T IM E
The function Et – pxx – pyy – pzz represents the phase of the wave and
associates time to space with this difference of sign which always characterises the dimensional conception of time. The following relations can be
deduced from the preceding equation, obtained by derivation, and valid in
the conditions indicated:
- h 2} = p } - h 2} = p } - h 2} = p } h 2} = E }
x
y
z
2ri 2x
, 2ri 2y
, 2ri 2z
, 2ri 2t
.
It is these relations which led to replacing the quantities px, py, pz, by the
operators
- h 2} , - h 2} , - h 2}
2ri 2x 2ri 2y 2ri 2z .
The relativist analogy between space and time also suggests replacing the
energy E by the operator
h 2}
2 ri 2 t
but this is only permissible for quantified states of energy. The operator
which corresponds in all cases to energy is the Hamiltonian operator H
obtained as follows:
let E = H (q i, p i, t) = 1 6 p x2 + p y2 + p [email protected] + V (q i, t)
2m
be the expression of classical Hamiltonian energy. In H, replace px, py, pz,
with the corresponding operators
-h 2 f
2ri 2x
we obtain the Hamiltonian operator
H c x , y, z , - h 2 , - h 2 , - h 2 , t m
2ri 2x 2ri 2y 2ri 2z
corresponding to energy, in all cases. It is now essential to note that the
operators thus defined are conceived only as operating on functions
satisfying certain conditions which are made functions of what is called a
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Hilbert space. Thus, in wave mechanics, the operators corresponding to
physical magnitudes are always applied to wave functions } that belong
to a Hilbert space. By then applying the operator H to a function } , and
by drawing on the particular relation
h 2} = E}
2ri 2t
the fundamental equation of propagation of wave mechanics is obtained
by posing:
2}
H (}) h
2 ri 2 t
This equation is an equation of wave propagation, since at each instant the
partial derivations of the function } with respect to time are envisaged.
The evolution of this function can therefore be calculated from its acquaintance with an initial instant t0. It is nevertheless necessary to point out the
very special nature of this equation of evolution. The derivation with
respect to time plays a part there in effect as the formal result of the application of a privileged operator H , the energy operator, to a function }
defined in a Hilbert space. This process therefore allows the evolution of
the function } , which is not a physical magnitude, to be studied, but does
not apply to the study of the evolution of an arbitrary physical magnitude
attached to the particle. In fact, the study of the evolution of a physical
magnitude in wave mechanics always leads to the consideration of certain
relations of the operator corresponding to this magnitude with the energy
operator. It is thus that we have the following very important theorem:
The necessary and sufficient condition so that a physical magnitude A is
constant during the motion defined by the Hamiltonian operator H is that
the operators A and H permute. In classical mechanics, Hamilton’s canonical equations already relate the evolution in time of mechanical magnitudes pi and qi to the consideration of a privileged Hamiltonian function.
We had in effect
dq i
dp i
= 2H ,
= 2H
2p i dt
2q i
dt
without this connection of evolution in time of a mechanical magnitude
and of energy being the result of two different roles of time. Whereas, in
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T HE P R OB LE M OF T IM E
wave mechanics it seems that it is necessary to distinguish between the
operator
h 2 f
2ri 2t
applied uniquely to the wave function } , and the derivation with respect
to time of all magnitudes, mechanical and physical, attached to the particle. The first operator is associated with energy as the operators
-h 2 f
2 ri 2 x
are associated with components px of momentum. Whereas, the time that
plays a part in ordinary derivatives is unrelated to a dimension of spacetime and serves only in the study of the evolution of the magnitudes of the
system.
2. THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS
We have seen the equations of mechanics bring to the fore two aspects of
time, sometimes equivalent, sometimes distinct, but both adapted by their
intrinsic characteristics to the sensible dissymmetries of experience. The
fundamental result of the a priori deduction that we will present is then
the following: this duality of aspects of time, each endowed with its own
dissymmetries, does not appear only in the application of mathematics to
the physical universe, that is, in mechanics, but it already exists at the level
of pure mathematics, independently of any concern for its application to
the universe. Whatever the dialectical origin of this duality of time that is
found inherent in the theory of differential equations and partial differential equations, mechanics, insofar as the problem of time is concerned, does
not provide any arrangement whose schema lets itself be seen in the pure
abstractions of which it is the application. Physical time in all its forms, is
only the sensible realization of a structure which is already manifested in
the intelligible domain of mathematics.
The method that we are going to follow in this proof is a method of
regressive analysis. We will proceed from the concrete to the abstract, from
the composite to the simple, in order to always refer the crux of the problem to a higher level in the hierarchy of Ideas. And it is this incorporation
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of empirical data in an ideal structure that constitutes for us the a priori
deduction of the sensible dissymmetries related to time. This method
presents, in our eyes, a considerable advantage over the deductive syntheses of time that the idealist philosophies of the nineteenth and twentieth
centuries had attempted. We do not carry out an arbitrary a priori deduction of time, we observe, in the order of the universe, the stages constituted by that deduction.
Such a requirement of analysis explains the reasons why we will consider the theory of partial differential equations before the theory of differential equations. The equations of mathematical physics are generally
in effect partial differential equations so that the general theory of these
constitutes for example the first abstract domain in which the pure play of
relations that support the different aspects of physical time is found.
Consider7 the first order partial differential equations, to 2 independent
variables, F(x1, x2, u, p1, p2) = 0, in which u is an unknown function of two
variables x1 and x2 and in which
2 F 2 + 2F 2 ! 0
p 1 = 2u , p 2 = 2u
c
m c
m
2p 2
2x 1
2x 2 . Suppose that 2p 1
.
At each point of the 3-dimensional space, x1, x2, u, this equation defines a
family of possible tangent planes to an integral surface u(x1, x2) passing
through this point. This family of planes envelops a cone, at each point of
the envisaged space the Monge cone attached to this point is defined, so
that the equation F = 0 can be considered to associate, to all points of the
space, a cone or even a sheaf of characteristic directions, the generators of
the Monge cone at that point. To integrate the proposed equation consists
in finding a surface tangent at each point to a characteristic direction passing through this point, and this integration is done by considering the characteristic curves of the equation. Here is what is meant by a this: a
characteristic curve of the equation F = 0 is tangent at each of its points to
a characteristic direction passing through this point. It shows that an integral surface u(x1, x2) of the equation F = 0 is generated by a family of characteristic curves, so that the proposed problem of the integration of partial
differential equations is reduced to the problem of the integration of differential equations that define the characteristic curves of this equation.
It is very important for us to consider the differential equations of these
characteristics. For this in general, the coordinates x1, x2, u, of the points of
a curve in the space envisaged are considered as functions x1(t), x2(t), u(t)
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T HE P R OB LE M OF T IM E
of a parameter t. The characteristic curves of the equation F(x1, x2, u, p1, p2)
= 0 then satisfy a set of three differential equations of which we retain here
only the following:8
(I)
dx 1 = 2F , dx 2 = 2F
2p 1 dt
2p 2 .
dt
A surface integral is generated, we said, by characteristic curves, so that
given a characteristic curve defined by differential equations (I), this curve
can be subject to the supplementary condition of being situated on a surface integral of the equation F = 0. This condition gives rise to two new
equations that we write under the simplified form they take when the
function F does not contain the variable u.
(II)
dp 1
dp 2
=- 2F ,
=- 2F
2x 1 dt
2x 2 .
dt
The determination of surface integrals is therefore reduced to the integration of a system of differential equations in which we will retain only equations (I) and (II) written in the form
(III)
dx i =- 2F ; dp i =- 2F
2p i dt
2x i .
dt
The equations (III) have exactly the form of the canonical equations of the
dynamics of the material point in which F represents the energy of the
particle, and t the time. Here, in the a priori deduction of the laws of the
sensible universe, is a result of considerable importance, that of seeing the
equations of the trajectories of the dynamics result a priori, without special
physical hypothesis, from the problem of the integration of partial differential equations. Geometric space x1, x2, u thus contains, like a state of possibles, with its directions and its characteristic curves of the equation F = 0,
the form of trajectories and the dynamic law of motion that material particles will take when a physical interpretation, transmuting the function F
into energy and the parameter t into temporal variable, thus projects, fully
armed, into sensible existence, a mathematical universe already equipped
with all the necessary richness of organization.
So far we have only shown the possibility of an a priori deduction of the
laws of mechanics in which time plays a part as a parameter of evolution.
We now come to the central point of the proof announced above by
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
showing within the same theory of partial differential equations the genesis of a dimensional conception of physical time, which is in principle
distinct from the parametric conception but serves to resolve the same
problems. Let us now place ourselves in the general case of a space of
n + 1 dimensions, and envisage the following partial differential equations,
in which the function F explicitly does not contain the function u(x1 . . . xn,
xn+1) sought after:9 F (x1 . . . xn+1, p1 . . . pn+1) = 0. Following the presentation
of Hilbert and Courant (1937), we are going to distinguish a variable, for
example xn+1, and resolve the proposed equation with respect to the corresponding derivative pn+l.
We obtain the equation:
pn+l + H (x1 . . . xn, xn+1, p1 . . . pn) = 0
c pn+1 =
2u ; p = 2u ; i = 1 f n
m
2x n + 1 i 2x i
The expression pn +1 + H therefore replaces the expression F of the given
equation. The first group of equations of characteristic curves, dx i = 2F ,
dt
2p i
with i = 1 . . . n + 1 give for i = n + 1, dx n + 1 = 1 .
dt
The variable xn+1 is therefore always equal, to within an additive constant, to the parameter t, and we can replace the parameter t of the preceding theory by the independent variable distinguished in the new theory.
The characteristic equations are then written
dx i = 2H ; 2p i =- 2H ; i = 1 f n
2p i dx n + 1
dx n + 1
2x i
,
and the abstract form of Hamilton’s canonical equations is retrieved. In the
space of n + 1 dimensions defined by the variables x1 . . . xn+1, the fact of
solving the proposed equation with respect to a partial derivative concerning one of these variables makes this variable play the role of the temporal
variable. There is in this case absolute equivalence between the role of the
parameter and the role of the dimensional coordinate of the same distinguished variable, but it remains no less the fact that the purely mathematical theory of partial differential equations thus allows the emergence of
two different conceptions of the same variable that are at the origin of the
duality of the sensible properties of time. In addition, if the second conception presented, which is that of Hamilton and Jacobi, is at the base of the
classical theory of Hamilton’s canonical equations when the distinguished
and variable parameter is being identified, it is also the starting point of
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T HE P R OB LE M OF T IM E
wave mechanics in which the derivation with respect to time, symmetric
with a change of sign of the deviation with respect to the coordinates of
space, has an operational meaning quite distinct from the parametric sense
of time. Let us in effect go back to the equation:
pn+l + H (x1 . . . xn, xn+1, p1 . . . pn) = 0
This equation can be written, with
2u
2u
xn+1 = t and p n + 1 = 2x = 2t .
n+1
H (x 1 f x n, t, p 1 f p n) =- 2u
2t
(IV)
If it is posed that,
2u = E
p i = 2u
2t
2x i ,
and
the function H of equation (IV) gives the classical Hamiltonian expression
of energy:
H (x i, t, p i) = 1 6 p x2 + p y2 + p [email protected] + V (x, y, z, t) = E
2m
.
Whereas, as we have also seen above, by replacing pi in the function H of
equation (IV) by the operators
- h 2
2ri 2x i ,
the Hamiltonian operator H of the wave mechanics is obtained. If this
operator is then applied to the wave function } , playing the role of the
function u of equation (IV), the Schrödinger equation is obtained:
2}
H ({) = h
2 ri 2 t .
The Jacobi equation thus gives the Schrödinger equation directly when the
operator
h 2
2ri 2t ,
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
plays a part, operating on the wave function } , and these are the characteristic equations associated (as defined by the theory of partial differential
equations) to the Jacobi equation which gives the derivatives of magnitudes xi and pi in terms of a parameter t, conceived this time independently
of any relation with the coordinate xn+l. The derivation with respect to
the time coordinate therefore appears in the partial differential equation
itself, and it is only in the characteristic equations of this equation that the
derivation with respect to the time parameter appears. The first of these
derivations is concerned with the wave function sought after, the others
with the mechanical magnitudes attached to a mobile point that describes
the trajectories defined by the characteristic equations. The distinction of
the two times is therefore attached to the distinction of partial differential
equations and characteristic equations. This is a fundamental and ineffable
distinction of which we will find the full meaning by studying the theory
of differential equations.
There remains one point to consider. We have just rediscovered a priori
in the theory of partial differential equations the distinction between the
time parameter and the time coordinate. We must now show a priori how
the time coordinate plays a part in a spatio–temporal synthesis affected by
this special dissymmetry with respect to space which is manifested by a
difference of sign. The same problem will allow us to rediscover both the
distinction between two conceptions of time and the special dissymmetry
of dimensional time. For this, we will envisage the theory of second order
partial differential equations, and we will restrict ourselves to the cases in
which these equations are linear, that is, of the form
/a
(V)
ik
u ik + / b i u i + cu + d = 0
with
2
u ik = 2 u ; u i = 2u
2x i 2x k
2x i ,
u being the unknown function sought after.
It can be shown that a characteristic form can be associated to any equation of this type,
/ a X , (i = 1 . . . n) with a = ! 1 .
i
256
2
i
i
T HE P R OB LE M OF T IM E
The characteristic form associated with the equations that describe propagation is said to be of the hyperbolic type, and is composed of n – 1 positive
squares and one negative square:
X 21 + f + X 2n - 1 - X 2n .
We are occupied here only with the hyperbolic case. As in the case of first
order partial differential equations, a surface integral of equation (V) is
generated by the characteristic manifold { = 0 which here satisfies the
equation
/a { {
i
i
k
=0
and these characteristic manifolds are in turn generated by the characteristic
radius that defines in n-dimensional space the differential equations
=n
dx i = k/
a ik {k
dt
k=1
A variable xn can be distinguish anew and solve the equation { = 0 with
respect to xn to obtain x n = } (x 1 f x n - 1) . Under these conditions the equation of the radius gives:
dx n = 1
dt
and the dimensional variable can be identified anew with the parameter of
evolution. A concrete example nevertheless shows the character of each of
these two aspects of time. Consider the classical wave equation u44 – u11 –
u22 – u33 = 0. In 4-dimensional space-time a distance between two points
can be defined
dv2 = dx 24 - dx 21 - dx 22 - dx 23 .
The lines of zero length ( dv = 0 ) issuing from an arbitrary point define
the characteristic cone attached to this point and any direction situated
at the interior of the cone satisfies the inequality dv2 2 0 . In particular, for
the time axis x4 defined by the relations
dxl = dx2 = dx3 = 0 we have dx 24 2 0 ,
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M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
and this shows the ‘oriented’ character of the variable x4 in its role as
dimensional time. Let us now place ourselves in 3-dimensional space. The
distance between two points is given by the formula
dt2 = dx 21 + dx 22 + dx 23 .
If t designates in this case the length counted on a radius from the origin,
the speed of propagation being taken as unity, the surfaces t = constant
defines the wave surfaces which represent the wave front at each instant.
There is equivalence in the results obtained between the time, as a dimension of 4-dimensional space, and the time as a length in 3-dimensional
space, but each of these two conceptions of time brings with it a specific
element: the conception of the oriented time dimension is linked to the
existence of a difference of sign in terms of a sum of squares; the parametric
conception is linked to the kinematic notions of speed and displacement.
3. THE THEORY OF DIFFERENTIAL EQUATIONS
AND TOPOLOGY10
The relations that support the different notions embraced by the theory of
partial differential equations become much more intuitive in the theory of
differential equations. The geometric meaning of the partial differential
equations that we have envisaged is in effect the following: the given equation defines, at each point of a space, a cone characteristic of this point, and
the integration of the equation consists in finding the curves and the surfaces tangent to each point of the cone characteristic of this point. The
distinction of the time dimension and the time parameter is related to a
certain extent to the following duality: the direction of opening of the
characteristic cone assigns at each point of the space a privileged role to
one direction which can be considered as a temporal direction, and on the
other hand, the curves and surfaces tangent to this field of cones are a
function of a parameter that can also be considered, but in another sense,
as a time variable. In the theory of differential equations, the situation is
even simpler. An equation
dy
= f (x, y)
dx
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T HE P R OB LE M OF T IM E
defines in the x, y plane a field of directions, that is, one directly attached
to each point, and solutions of this equation are the curves tangent to each
point in the direction that passes through this point.
The geometric interpretation of the theory of differential equations
therefore brings to the fore two absolutely distinct realities. There is the
field of directions and the topological accidents that can occur on it, as for
example the existence in plane of singular points to which no direction is
attached, and there are the integral curves with the form they take in the
neighbourhood of the singularities of the field of directions. Consider for
example the equation
Q (x, y)
dy
=
dx
P (x, y)
(I)
in which the functions P and Q are supposed uniform, continuous and
bounded in absolute value. Under these conditions the equation (I) defines
a field of directions. If the variables x and y are related to a parameter t, we
have instead of (I) the following system
(II)
dx = P (x, y)
dt
dy
= Q (x, y)
dt
.
This system no longer defines only a direction at each point but a direction
and a meaning, that is, a vector attached to each point of the plane x, y for
which P and Q do not vanish simultaneously. The points of indetermination in which P = Q = 0, constitute the singularities of the vector field. In
his famous ‘Mémoire sur les courbes définies par une équation différentielle’
(1881), Poincaré established a classification of these singularities according
to the bearing of the integral curves in the neighbourhood of these points.
He distinguishes: saddle points, through which two and only two curves
defined by the equation pass; nodes, in which an infinity of curves come to
be crossed; foci, around which the curves turn by drawing constantly closer
to a logarithmic spiral; centers, around which the curves present themselves in the form of closed loops enveloping each other and surrounding
the center. The existence and distribution of singularities are notions relative to the vector field defined by the differential equations. The form of the
integral curves is relative to the solutions of this equation. The two problems are most certainly complementary because the nature of singularities
of the field is defined by the form of the curves in their neighbourhood.
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It is no less true that the vector field on the one hand, and the integral
curves on the other are two essentially distinct mathematical realities. It
might seem that we are here in purely abstract domains in which any reference to physical problems of time has vanished. In fact the parameter t as a
function of which coordinates x and y are defined can be conceived anew
as a parameter of temporal evolution. As for the notion of vector field, it is
linked in an essential way to the dimensional aspect of time. We will show
this by presenting the results due principally to Ehresmann (1943). We will
see the distinction between time and space within a 4-dimensional manifold, interdependent with the existence in this manifold of a field of directions without singularity and determined by the global topology of this
manifold. In opposition with the parametric properties of time which can
only relate to a limited evolution within a well defined interval of time without reference to any overall structure, the dimensional properties of time
have a cosmogonical meaning and reflect the general form the Universe.
Here are the principal stages of Ehresmann’s reasoning. Consider a differentiable manifold, that is, a topological manifold Vn in which a set of
systems of local coordinates are defined such that every point of Vn is found
in at least one of them, and such that two systems of coordinates which are
defined in the same domain of Vn correspond to one another by a transformation of continuously differentiable coordinates. On this manifold, a precise, positive quadratic differential form can always be defined, that is, a
sum of n squares formed by the differentials of coordinates:
dx 21 + dx 22 + f dx 2n
that is reducible at each point to a sum of
n squares w 21 + w 22 + f w 2n in which w 1, w 2, f w n
refer to n independent linear differential forms. The problem posed is to
know under what conditions there exists at each point of this manifold a
quadratic form reducible to a sum of p positive squares and n – p negative
squares, as in the case of the invariant of the theory of special relativity
ds 2 = dx 24 - dx 21 - dx 22 - dx 23 , in which p = 1 and n = 4.
Ehresmann has proven that the existence at each point of Vn of a quadratic
form composed of p positive squares and n – p negative squares is equivalent
to the existence of a p-dimensional field of contact elements without
260
T HE P R OB LE M OF T IM E
singular points.11 In the case of a 4-dimensional space, the existence of a
relativistic invariant of the universe is therefore equivalent to the existence
in this Universe of a field of directions presenting no singularity at any
point. Now, a fundamental theorem due to Hopf established the fundamental topological conditions under which a continuous vector field without singularities on a closed differentiable manifold can be defined. It is
necessary and sufficient that the Euler characteristic12 of this manifold
is zero, which is the case for all closed manifolds with an odd number of
dimensions, but which on the other hand eliminates a large number of
manifolds with an even number of dimensions. For example, the sphere
S4 and projective plane P4 of 4-dimensional space, having a Euler characteristic different from 0, cannot be provided with a vector field without
singularity (Cf. Ehresmann 1943). We can therefore not define the Lorentz
invariant at each point of these manifolds and they would not constitute
a possible Universe for the theory of relativity. On the other hand, if a
4-dimensional universe is compact and such that a distinction between
past and future at each point can be defined by continuity, it can be concluded that the Euler characteristic of this universe is equal to 0. The determination of the meaning of time in the universe is therefore supportive of
the global structure of the universe.
Let us try to identify the scope of the results thus obtained. We started
with the distinction of the sensible properties of the time reference point,
which is a dimension associated dissymmetrically to space in a 4-dimensional synthesis, and with the time factor of evolution which is a parameter.
The mathematical study of these two aspects of time led us to envisage in
the theory of differential equations the distinction between two kinds of
mathematical entities: vector fields and solution curves. Neither one nor
the other of these two realities is by nature a temporal reality, but the problems posed by the study of each are directly interpretable in terms of time.
The theory of vector fields shows a necessary connection between the
existence of a privileged direction at each point of a geometric universe
and the default of a quadratic form, the difference in sign between terms
of ‘space’ and terms of ‘time’. The same theory shows in addition that the
distinction at each point of a privileged direction is possible only if the
entire Universe has satisfied certain global conditions. The application of
these results to physical theories is, we have seen, obvious: the orientation
of time, the duration of things have mathematical meaning only if there
are no holes, of interruption, in this continuity of openness towards the
future, and this exigency is cosmogonical. The aspect of time that we called
261
M ATH EMATICS , ID EAS A N D T H E P H YS I C A L R E A L
geometric, throughout this chapter, is thus in conjunction with the general
form of the whole Universe.
Now let us envisage the other aspect of time, that to which the study led,
on a field of vectors, of integral curves defined as a function of a parameter.
The principal problems of this theory are immediately interpretable in
terms of time. The problem of determinism can be cited, that of finality,
that of the return of things. In certain cases, the evolution of a magnitude
as a function of time can be described by a differential equation, or partial
differential equation, such that the knowledge of initial data, at a time t0,
determines the subsequent evolution of the magnitude in question. This is
a local determinism, operating step by step, as the propagation of light for
example. In other cases, in which partial differential equations of the elliptic type are encountered, as in the phenomena of thermal equilibrium, the
knowledge of initial data is insufficient to determine the whole evolution
of the phenomena. It is necessary to be given both the initial and final
conditions. The problems of this kind allow a theory of finality to be established in mathematics, as Maurice Janet has shown in his article on finality
in mathematics and physics (1933, 1). As for the problem of the return of
things, it presents itself in all the cases of closed trajectory, in celestial
mechanics for example, where to a same value of position coordinates
correspond several distinct values of the variable t. This demonstrates how
the study of integral curves defined as a function of a parameter led to
consider the laws of evolution of physical systems, which clearly brings to
the fore the dynamic meaning of this parameter.
The mathematical distinction between cosmogonical time and dynamical
time is therefore the expression of a duality inherent in theories as abstract
as the theory of differential equations, and it is consequently highly probable that it corresponds to an intimate structure of things that has its source
in the structure of Ideas. This conclusion leaves us with the feeling of the
limits that the reduction of dynamics to cosmogony must necessary encounter. In a forthcoming book, Tonnelat (1955) shows the history of the unitary
theories of the electromagnetic field and gravitation since the geometrization of gravitation by Einstein in 1916. The impossibility of geometrizing the
electromagnetic field in a physically acceptable way led Tonnelat to the contrary attitude, that is, to give a physical sense, a dynamic sense back to gravitation. There can be no question of going back to a pre-relativist conception
of a geometric container definable independently of its physical contents,
but the identification of the two notions seems equally impossible. Here
again the notion of complementarity seems to play a significant role.
262
Notes
Introduction
1. Testimony of Gilbert Spire, a student at the Ecole Normale Supérieure
in 1936, prisoner with Lautman at Oflag IV D, where he also attended
Lautman’s lectures at the camp university. Spire provided valuable help in
the escape of the group of which Lautman was a part; Spire would later be
Inspector General of Philosophy.
2. Blay 1987, dedicated to Jean Cavaillès and Albert Lautman, with
articles by Catherine Chevalley, Gerhard Heinzmann, Jean Petitot and
some unpublished letters of Jean Cavaillès to Albert Lautman, presented
by Hourya Sinaceur.
4. WWII.
3. Gonseth 1997. A work that brings together various texts and in which
Lautman is cited several times.
5. Jacques Herbrand, 1908–1931, presented a thesis in 1930 for a doctorate of Science entitled Investigations in Proof Theory (1930), which was
considered of premier importance by Hadamard, professor at the Collège
de France, but his colleagues from the Sorbonne found it too philosophical.
It took the insistence of Vessiot, mathematician and director of the ENS, for
the defence to take place. Shortly after, Herbrand made a major contribution to the problem of consistency in arithmetic in which he used the
finitistic methods of Hilbert (Herbrand 1931). In July 1931, this young
mathematical genius was killed in a climbing accident. In the Yearbook of
263
NOTES
the Association of Former Students of the Ecole Normale Supérieure (Annuaire de
l’Association des anciens élèves de l’ENS) in 1931, Claude Chevalley and Albert
Lautman, who co-sign the obituary, wrote: ‘the sublime beauty of the
paths indicated by this adored person’ (Chevalley and Lautman 1931).
6. The Bourbaki group set out provide an encyclopedic treatment of the
whole of mathematics based on a general theory of structure —Tr.
7. The street address of the ENS, Paris —Tr.
8. The nickname given to a sophomore (second year) in the preparatory
literature class —Tr.
9. The philosopher Emile Chartier, also known as Alain, was head of the
Lycée Henri IV from 1909 to 1933 —Tr.
10. See Sirinelli 1992. This is the richest published source on the life of
Albert Lautman, who is mentioned on many occasions.
11. The National Science Fund —Tr.
12. Centre National de la Recherche Scientifique (National Center for Scientific
Research) —Tr.
13. Those studying for the secondary level teaching qualification —Tr.
14. Up until 1969, the Doctorat d’Etat had two theses.
15. Among the witness accounts is that of the poet Patrice Tourdu La
Pine.
16. A former student of the military school at Saint Cyr —Tr.
17. Tunisian soldiers who served in auxiliary units attached to the French
Army —Tr.
18. Société Nationale des Chemins de fer français, or French National Railway
—Tr.
19. The Pat O’Leary network recuperated and evacuated shot-down allied
airmen —Tr.
20. The Spanish Maquis were resistance fighters who held several valleys
and passes throughout the Pyrenees —Tr.
21. Another group of resistance fighters —Tr.
264
NOT E S
Albert Lautman and the Creative Dialectic of
Modern Mathematics, by Fernando Zalamea
1. See the secondary bibliography on Lautman for additional
references.
2. From now on all citations in quotes, without other indication, are
from Lautman.
3. It is, therefore, necessary not to conflate the usual meaning of effective in mathematics (Brouwer’s intuitionism or constructivism as exemplified by Markov) with the use made by Lautman.
4. Since the vast majority of mathematical examples studied by analytic
philosophy can be reduced to trivial arithmetic or set theoretical cases (in
so striking a way in Wittgenstein), one is forced to doubt that this approach
oriented towards the elements can build a faithful image of mathematical
activity.
5. See Deleuze 1994 in the Secondary Bibliography.
6. The Lautmanian ‘structural schemas’ anticipate (in their conception)
the mathematical techniques of his time, and they can only be clearly
defined in the later context of the mathematics of category theory. The
schemas, the dialectic, the Same/Other pair, the ideas and Platonic–
Lautmanian mixes acquire a notable technical exactitude through the
notions of diagram, free object, representable functor and adjoint pair from
category theory.
7. It would suffice to sample a random number of Mathematical Reviews
to explain the very small space in mathematics occupied by the research
into foundations, which contrasts surprisingly with the enormous space
given to discussions about the foundations of mathematics in philosophy.
In our opinion, the Lautmanian spectrum goes much further. In this
regard, it is startling to browse the index of Lautman’s work prepared for
the Spanish edition of his writings (Lautman 2008): closely following the
classification adopted by the Mathematical Reviews (MSC 2000) one observes
the still very strong presence – 60 years later – of Lautmanian texts in an
extraordinary quantity of ramifications in the MSC 2000. [The Mathematics Subject Classification (MSC 2000) is used to categorize items covered by
the two reviewing databases, Mathematical Reviews (MR) and Zentralblatt
MATH (Zbl) —Tr.].
265
NOTES
8. The maps of contemporary mathematics are very similar to the maps
of knowledge included in the Enciclopedia Einaudi (Romano 1977–1984).
It is interesting to note that Jean Petitot has been one of the central experts
of the Enciclopedia, and that his knowledge of Lautman perhaps influenced
the remarkable tables and diagrams elaborated by Renato Betti and his collaborators (volume 15: Sistematica, Betti 1982). If one disreagrds the article
by Bernays (1940) – unfortunately, little known and quickly forgotten –
the text by Petitot (1982) in the Enciclopedia Einaudi was the first to have
made Lautman known beyond the Francophone world. See the Secondary
Bibliography.
9. Francastel’s relay (1965) proportion – for the work of art – another
mix of high value, in which perception is conjugated with the real and the
imaginary. If we compare a definition of the work of art as a form that signifies itself (Focillon 1934), with a definition of mathematical work as a structure that forms itself (our extrapolation, motivated by Lautman), it is not
difficult to point out – once again – the terrain underlying the aesthetic and
mathematics. It happens to be a fundamental terrain, for Herbrand, as for
Lautman, even if they didn’t develop it. See the Notice on Herbrand (by
Chevalley and Lautman 1931) and on Lautman (by his wife, 1946) published in the Annuaire de l’Association amicale de secours des anciens élèves de
l’Ecole Normale Supérieure. For a recovery of the history of art that (unwittingly) follows Lautmanian lines, that combines the complex and the differentiated, and reconstructs them in a stratified and hierarchical dialogue
attentive to the universal and the truth, see Thuillier 2003, 65 (Focillon’s
definition and subsequent discussion).
10. See Wiles 1995.
11. The great advantage that Alain Badiou has derived from his studies of
Cohen’s independence theorems in set theory is a kind of exception
(Badiou declares himself a great admirer of Lautman, Badiou 2005 [1998]).
Nevertheless, Cohen’s forcing is a part of the logical (mathematical) rather
than mathematics properly speaking. The philosophical ideas buried in the
mathematical work of a Grothendieck, of a Langlands, or of a Gromov, to
mention only the major examples, do not yet have the right of existence in
the philosophical city.
12. Lautman does not appear to have been aware of the notes to the
course on the Sophist given by Heidegger, in Marburg, during winter
semester 1924–1925 (see Heidegger 1997). Even if, in 1928, Lautman had
266
NOT E S
attended a Franco–German encounter in Davos, and could, therefore, perhaps have heard talk of Heidegger, the first direct references to Heidegger
only appears in New research on the dialectical structure of mathematics.
Heidegger’s course entangles his nascent hermeneutical method with a
very careful reading of the Sophist (610 manuscript pages in German), and
can thus provide exceptional philosophical support for Lautman’s work,
oriented specifically toward Plato and Heidegger. It is remarkable that
Heidegger defines the fundamental work of the dialectic as that of ‘renewing the relation between ontological structures each time to a unity, so that from this
unity the whole ontological history of a being up to its concreteness can be followed’
(Heidegger 1997, 389, author’s emphasis). If we leave aside the ontological
terms, it is difficult to find so Lautmanian a resonance in another author.
13. It is the prerogative of the heirs of the later Wittgenstein, who converted the study of the world into a study of the linguistic constraints and
blockages of knowledge: certainly a thriving academic industry, but unfortunately far from the reality in which science moves.
14. Lautman could not have know category theory, which emerged at the
time of his death (Eilenberg and MacLane 1942; 1945). It is difficult to know
to what extent the conversations with his friend Ehresmann – creator of the
general theory of fiber bundles in the forties and proponent of category theory in France in the fifties – could have influenced the resources of a conception of mathematics as clearly categorical as that of Lautman. Nevertheless,
in the session of the Société Française de Philosophie (4 February 1939) where
Cavaillès and Lautman defended their theses, Ehresmann already signaled
how the philosophical conceptions of Lautman merited in their turn to be
treated technically and converted into a baggage at the interior of mathematics
itself (‘I think the general problems raised by Lautman can be expressed in
mathematical terms, and, I would add, that one cannot help but express
them in mathematical terms’, Cavailles, 1994, 614). The rapid development
of the mathematical theory of categories proves Ehresmann right.
15. Peter Freyd points out that the lemma didn’t really appear in Nobuo
Yoneda’s original article (Yoneda 1958), but in a lecture given by Saunders
MacLane (1998) on the treatment of Yoneda Ext functors of higher order
(Freyd 2003, xii). It is amusing to know that the lemma, so close to
the structural resources of Lautman’s thought, in fact emerged in a lively
discussion between Yoneda and MacLane in the Gare du Nord in Paris (See
Biss 2003, 581).
267
NOTES
16. See Dumitriu 1991, 28. For the original text in which Avicenna introduces his trimodal interpretation, see Avicenna 2005.
17. Cavaillès betrayed Lautman’s ‘ardor’ (Ferrières 1950, 77) and
Herbrand his ‘exigency’ (Chevalley 1987, 76).
Preface to the 1977 Edition, by Jean Dieudonné
1. The example chosen is Garland’s thesis, ‘A finiteness theorem for K2
of a number field’ (Garland 1971).
2. One can say without exaggeration that there have been more fundamental mathematical problems resolved since 1940 than from Thales to 1940.
Considerations on Mathematical Logic
1. Published by Rudolf Carnap and Hans Reichenbach, Leipzig: Felix
Meiner Verlag, 1931–1940.
2. The number of logical operations is of little importance and varies
with the operators. It is sufficient that one can deduce all other operations
from the ones that have been chosen.
3. The work of Herbrand is inspired by the proofs given in Hilbert and
Ackermann 1922.
Book I: Introduction
1. Some authors, like Helmut Hasse, clearly distinguish the notion of
function ‘as defined by analysis’ from the notion of function ‘as defined by
algebra’ (see Hasse 1933).
Chapter 1
1. The field k ( 2 ) is generated by the ‘adjunction’ of
rational numbers.
2 the field of
2. n is here the measure of the space E, as defined by Lebesgue.
268
NOT E S
3. This means that there are the following relations between the functions:
+r
#
+r
{m (s) {n (s) ds = 0; m ! n
-r
#
{m (s) ds = 1
-r
4. This whole paragraph is from Hellinger 1935, 106.
5. This theorem constitutes, under a particular form, the theorem of
Riemann-Roch which makes the function F(z) depend on n – p + 1 constants. See Picard 1896, 474.
6. The importance presented by ‘the arithmetic theory of algebraic functions’ for the problems encountered in this chapter must be noted here.
The theory of algebraic functions and of their integrals is for Riemann a
theory of analysis and is based on transcendent methods. Dedekind and
Weber explained, in 1882, a purely arithmetic theory of algebraic functions with one variable, in which only finitist notions of arithmetic and
algebra are called upon, even for the definition of a point. Thus, an algebraic fouction h is characterized by a symbolic quotient of two complex
points called polygons:
h= A
B
Two polygons A and B , likely to correspond as numerator and denominator to the same function, belong to the same class of polygons. The
classes have a dimension defined by the number of elements of their base,
and the theorem of Riemann-Roch is presented immediately in this theory
as tied to the study of the dimension of the classes of polygons. The structural and dimensional point of view that only appears in second place in
the ordinary theory thus appears prominently in the ‘arithmetic’ theory.
Chapter 2
1. We indicated in our principal thesis (Lautman 1938b) how, following
the work of Elie Cartan, the metric broadly defined could, in certain cases,
nevertheless be Riemannian.
2. See the introduction to this essay.
3. Poincaré 1928, xi. The summary of his work is by Poincaré himself.
4. For this whole paragraph, see Nevanlinna 1936, ch. 6.3.
269
NOTES
Chapter 3
1. For all this, cf. de Broglie 1932, 199.
2. Recall briefly that a ring is a set of numbers in which two formal procedures of composition are defined: the addition and multiplication of
two elements.
3. Cf. the lectures by von Neumann 1935 as well as van der Waerden
1931, ch. 16.
4. This whole paragraph is from Dubourdieu 1936; see also Kahler 1934.
5. For all that follows cf. Cartan 1937, 1311–1334; 1909, 1335–1384.
6. It is a matter here of infinite continuous groups, see the lecture by
Cartan, ‘Sur la structure des groupes infinis,’ Seminaire de l’Institut H. Poincare,
annee 1936–1937 (Cartan 1909, 1335–1384).
Chapter 4
1. Here’s the simplest definition of the norm: two numbers a and b are
said to be congruent modulo A if their difference a - b is in A ; which is
written a / b (mod .A) . This equivalence relation determines a division
into classes in K and the number of these classes is called the norm of the
ideal A .
2. Recall that the two whole or fractional ideals A and B are equivalent,
A~B , if their ratio A B is a principal ideal (formed from the multiples of
a single whole number). This equivalence relation determines a division
into classes of ideals of the field K.
3. For all this cf. Hecke 1923.
4. The work of Takagi, Artin, Herbrand and Chevalley have generally
eliminated the analysis of existence theorems in class field theory, but in
the case of the imaginary quadratic field, the function J(z) gives more than
the existence of the field of classes, it gives numbers belonging to this field.
Conclusion
1. Cf. the often-repeated adage of André Bloch: ‘Nihil est in infinito quod
non prius fuerit in finito’ (1926, 84). [‘Nothing is in the infinite that was not
first in the finite’ —Tr.]
270
NOT E S
Book II: Introduction
1. Cf. this passage from Russell: ‘They [the mathematical propositions]
all have the characteristics which, a moment ago, we agreed to call tautology. This combined with the fact that they can be expressed wholly in
terms of variables or logical constants . . . will give the definition of logic or
pure mathematics’, in Russell 1919 [1993, 204–205].
2. In our secondary thesis, Essay on the unity of the mathematical sciences
in their present development (1938a), we present some aspects that seem to
us to distinguish modern mathematics from classical mathematics.
Chapter 1
1. In fact, the global study and the local study do not lead to strictly
equivalent results. Borel has in effect proved by the discovery of classes
of quasi analytic functions that the class of Cauchy functions is more
extended than the class of Weierstrass functions.
2. See mainly Cartan 1924, 294; 1925, 1; and especially 1927, 211.
3. Recall that a differential equation establishes a relation between a
function of a single variable and a certain number of its successive derivatives, while a partial differential equation establishes a relation between a
function of several variables and a certain number of its partial derivatives
with respect to all or several of its variables.
4. An analytic function f(z) is holomorphic in a connected region D of
the plane of the complex variable z, if it is continuous in D, and if at any
point z of D there correspond unique values for f(z) and f´(z).
5. For this whole paragraph, see Fehr 1936. This edition reproduces the
lectures on the theory of partial differential equations held at the University of Geneva in June 1935. We are particularly inspired by Hadamard’s
lecture, from which the form of the equations cited in the text are also
borrowed (Hadamard 1936).
6. This distinction, that will be discussed below, is found in Hopf’s article
(1932).
7. This is the case for set-theoretical topology, as well as combinatorial
topology or algebraic topology.
271
NOTES
8. The characteristic of a surface is a property of algebraic topology
which will be returned to later (cf. Chapter 4).
9. Cartan 1930. Let us recall the meaning of some terms: one group
being a set of transformations operating on the points of a space is said to be
linear if the new coordinates xli of a point are expressed algebraically as a function of the old coordinates x j by relations of the type xli = a i x 1 + a i x 2 + f a i x n .
The a ij are the coefficients of the transformation. A quadratic form of two
variables is, for example, the expression Ax2 + Bxy + Cy2. It is defined if its
discriminant is negative.
1
2
n
10. c is the sub-group of the linear adjoint group that corresponds to
the largest sub-group g of G which leaves fixed a point in space.
11. The distance f - g of two functions f(x) and g(x) is defined as the
maximum absolute value, f (x) - g (x) , x belonging to the basic set f .
12. These formulas of representation as well as those given for the
polynomials of representation are valid only for analytic functions, in
their circle of convergence. The formulas of approximate representation
of continuous functions f(x) by the polynomials P(x) are, in the general
case, more complicated.
Chapter 2
1. For this whole section, see Cartan 1924, 297.
2. Instead of envisaging the insertion of a polyhedron in space, topologists often consider the points belonging to the polyhedron to be taken
from space.
3. For these definitions, see the topology texts already mentioned, as
well as Lefschetz’s Topology (1930).
4. Instead of a decomposition into simplices, a decomposition into cells
obtained from the simplicial decomposition can sometimes be envisaged.
5. Given an n-dimensional simplex, the algebraic sum of its (n – 1)dimensional faces is called the boundary of the simplex. An algebraic sum
of k-dimensional simplices, each possibly multiplied by an integer coefficient, is called a k-chain. A closed chain is called a cycle, that is, a chain
whose boundary is zero. Among these cycles, some are found at the same
272
NOT E S
time to be boundaries of (k + 1)-dimensional chains. Thus, for example, a
circumference is not only a 1-dimensional chain, but a cycle, since it is
closed, and a cycle boundary, since it is a boundary of a 2-dimensional
simplex (the surface of the circle, topologically equivalent to the triangle).
A cycle boundary is said to be homologous to zero. These preliminaries
being posed, here is the definition of the independence of cycles: m kdimensional cycles
u1k, u2k, . . . umk
are said to be independent if no linear combination
t1k + t2u2k . . . + tmumk
of these m cycles exists, which is homologous to zero without all the
coefficients ti cancelling each other out. If m independent cycles can be
found on a complex and m + 1 cannot be found on it, the Betti number of
dimension k is m.
6. It is the extrinsic notion of enlacement that allows Alexander’s theorem to be tied to Poincaré’s theorem. It is said that there is enlacement
between two closed curves without common points if there is intersection
between the plane included at the interior of one these curves and the
other curve. In the case which occupies us here, there is enlacement in
Rn between the r-dimensional cycle boundaries belonging to Q and the
(n – r – 1)-dimensional cycle boundaries belonging to Rn – Q. This means
that there is intersection between the cycles boundaries of dimension r of
Q and certain chains of dimension n – r of Rn – Q. These cycles of dimension
r and these chains of dimension n – r can then be considered as belonging
to the complexes in duality in Rn, and thus it is seen how the passage from
the ‘internal’ case to the ‘external’ case is carried out.
7. Torsion groups are structural invariants determined at the same time
as Betti numbers.
Chapter 3
1. Descartes 1976, 156; quoted and commented on by Étienne Gilson,
Descartes 1925, 315.
2. Cf. in particular Winter 1911, 146–185.
273
NOTES
3. Cf. van der Waerden 1930.
4. It goes without saying that we use the expression of imperfection here
without any reference to the algebraic meaning of the term ‘perfect field’.
5. Cf. for this theory Chevalley 1934 and Herbrand 1936.
6. Here are the definitions of notions used in this paragraph. Given an
algebraic number field k, any set of numbers of the field is called the ideal
of this field such that:
a) If a is part of this set, then so is ma , whatever the integer a ;
b) If b is another element of this set, a + b is also included;
c) There is an integer n such that for all a of the set an is an integer.
Given any set of numbers of the field, a1, a2 f , the set of c1 a1 + m2 a2 + f ,
the mi being arbitrary integers of the field, forms an ideal, the ideal
( a1, a2 f ).
The ideal ( a ) given rise to by a single number a ! 0 is called principal.
A product of two ideals can be defined, and an ideal integer a that cannot
be put in the form bc, where b and c are integers different to (1), is called
a prime ideal (Herbrand 1936, ch. 1.5).
7. What is interests us here is not so much the mathematical nature of
this decomposition than the fact of knowing that it is easier for the prime
ideals of the group of ideals H than for the ideals that do not belong to H.
In fact, here is this law: given an ideal A of K, let a be the set of numbers
of A that are in the base field k. Consider the conjugates of A in the Galois
field conjugated of K. The product of all these conjugations is called the
norm of the ideal A; and if A is prime in K, it can be demonstrated that
Norm A = af, f is then said to be the relative degree of A with respect to k.
In these conditions, only the prime ideals of k belonging to H decompose
in K into a product of different prime ideals, of relative degree 1. If, on the
other hand, p is a prime ideal of k not belonging to H, and if pf is the smallest power of p situated in H, p is decomposed in K into a product of prime
ideals of relative degree f with respect to k.
8. For this chapter, see Seifert and Threlfall 1934, ch. 7 and 8; Weyl
1913; Threlfall 1935.
274
NOT E S
9. According to the article by Threlfall (1935).
10. An algebraic function g = f (x) is a function which satisfies an algebraic equation of the type: g 0 (z) gn + g 1 (z) gn + 1 + f g n (z) = 0
11. Recall that a conformal correspondence between two domains is a
biunivocal and bicontinuous representation of the two domains on one
another and which is such that any analytic and regular function at one
point of a domain is transformed into an analytic and regular function at
the corresponding point.
Chapter 4
1. Cf. for this whole section Cavaillès 1938b.
2. For this whole part ,see Hilbert and Bernays 1939, and Bernays 1935,
196 ff.
3. We translate ‘beweistheoretische Method’ by structural method and
‘mengentheoretische Method’ by extensive method.
4. See Herbrand 1930, 118 [1971, 168] and particularly 1931, 3 [1971,
288].
5. It is said that an analytic function g = f (z) has, at a point z0, a pole of
order n, when the series expansion in the neighborhood of this point is
presented in the form:
a1
a2
+
+ f an + an+1 + f a1 ! 0
(z - z 0)
(z - z 0) n (z - z 0) n - 1
g=
A function of the type g = lg z = lg r + i{ has at point r = 0 a logarithmic
singularity, because at this point the expression lg r becomes infinite in
negative values.
6. The first sentence is altered, and Analysis situs is replaced by ‘topology’, and the second sentence omitted in the completely revised third
edition (1955) translated in 1964. See Weyl 1913 [1964, 152] —Tr.
7. Recall that a potential function of two variables follows the equation
d2 u + d2 u = 0
dx 2
dy 2
275
NOTES
Let there be an analytic function g = u + iv . It is proved that the functions u(x, y) and v(x, y) that thus figure in the expression of an analytic
function, when the real part is separated from the imaginary part, are
potential functions.
8. Cf. for this proof Hurwitz and Courant 1925, 368.
9. See Hilbert 1897, reprinted in Hilbert 1935. The theorem in question
is proved on p. 155. See also Hecke 1923, 154 [1981, 134].
10. x i is the complex conjugate variable xi.
11. Let s be an element of the group, U(s) the transformation corresponding to s in a certain system of coordinates of the fundamental space E. Let
A be the matrix of a change of coordinates of this space. The representation
s " U (s) is transformed into an equivalent representation s " AU (s) A - 1 .
12. In the case of finite groups, any representation is equivalent to a unitary representation.
Chapter 5
1. De Broglie 1932, 43.
2. For all this, see Hellinger 1935, 94 ff.
3. We refer to the presentation of Andre Weil (1935).
4. In a general way x is the complex conjugated quantity of x.
5. For this whole section, see Montel 1927.
Chapter 6
1. Summary according to Julia 1938, 191.
2. For all of this cf. Seifert and Threlfall 1934, 290; Alexandroff and Hopf
1935, 532.
3. Or rather of the Riemann surface obtained by the superposition on
the plane of z of an infinity of planes welded in crosses along the cuts
0 " - 3 and 1 " + 3 .
4. For this whole paragraph, see Bieberbach, 227 ff.
276
NOT E S
Conclusion
1. The two terms in italics (the second by us) show that Boutroux brings
together two different conceptions of mathematical reality.
2. Lautman replaces ‘elementary number theory’ with ‘arithmetic’ while
maintaining the gist of the section from which the passage is cited —Tr.
3. For the work of Gödel and Gentzen, see Cavaillès 1938b.
4. We report here on the interpretation that Becker gives of the famous
Aristotelian texts relating to Ideas–numbers (Becker 1931, 464 ff). Stenzel
(1923) had proposed, for the generation of numbers from the one and the
dyad, the following schema:
1
2
4
8
3
5
6
7
9 10 11 12 13 14 15
Figure 9
For Becker, this scheme has several major defects: first, only even numbers are truly generated by division, odd numbers resulting from the addition of a unit to the preceding even. In addition, the distinction of
Ideas–numbers and ordinary numbers is not clearly explained.
Becker associates the following schemas to every Idea–number:
1
2
3
4
5
Figure 10
277
NOTES
Each schema is composed of units. Some of these units are present in
normal numbers (they are represented by black circles), others are hidden
(the white circles) but all result from the dividing in two of a unit that is
situated higher. The ideal number is thus the schema that generates an
arithmetic number by means of present and hidden units (which would
explain, for Becker, the mysterious text of the Metaphysics, M 7, 1081 A
(Aristotle 1998): in the dyad, there is a third unit, in the triad a fourth and
a fifth . . .).
5. For the application of this term to the philosophy of Plato, cf. Robin
1935, 149.
6. By translating Zahlenmässige Gliederung as numerical division, we
believe that we have in no way betrayed the thought of the author.
7. This neo-positivism which thus appears to us to be unsustainable by
excessive empiricism, is associated in an unexpected way to a rigorous
‘tautological’ and deductive conception of mathematics in what is called
the physicalism of the Vienna Circle.
Book III
1. Perennial philosophy that is manifested in human action —Tr.
Chapter 1
1. As defined by Heidegger (cf. next note).
2. We will rely in what follows on Corbin’s 1938 translation of Heidegger’s
Vom Wesen des Grunde, 1929 (Heidegger 1969, see Translator’s Note).
Chapter 2
1. Hecke 1923, all that follows after paragraph 55 of chapter 8.
2. Cf. for these examples and for the whole theory presented in this
paragraph, Ingham 1932, 8.
3. We have C (s) =
278
#
0
3
y s - 1 e - y dy (v 2 0) .
NOT E S
4. As in formula (XI), the expression f(x) = 0(x) for x " 3 , means that
f (x) is always less than a constant quantity.
x
Book IV
Chapter 1
1. From here on, all occurrences of the term ‘structures’, when relating
to the theory of structures (or lattices), will be translated as ‘lattices’ — Tr.
Chapter 2
1. Throughout this chapter, we draw directly from the work of Louis de
Broglie on the relations of relativity and its quanta: 1937a, 223–239; 1941,
183–204; 1937c, 301–307.
2. The statement of the first two properties is in de Broglie 1937a, 226.
3. Let us call qi the coordinates of a particle in any space, qo i the derivatives of the coordinates with respect to time, and pi the quantities:
2L
2qo i .
In these conditions the energy H of the particle is given by the formula
H (q i, p i, t) = / p i q i - L (q i, qo i, t)
and from Hamilton’s principle the canonical equations can be deduced:
dq i
dp i
= 2H ,
=- 2H .
2p i dt
2q i
dt
4. Summary according to de Broglie 1939, 14.
5. This term signifies here simply the state of things which are two,
without reference to the more particular sense that it has in the previous
chapter.
6. Lichnerowicz seems to be the first to have formulated in a precise
mathematical way the necessity of considering two distinct times in physical theories (1939).
7. Summary according to Hilbert and Courant 1937.
279
NOTES
8. The third equation in question is
du = p 2F + p 2F
1
2
dt
2p 1
2p 2
which, together with equations (I) and (II), defines a characteristic band,
that is, a characteristic curve and a plane tangent to the curve at each point
of the curve.
9. It was proved that this case can always be reduced to by introducing
an additional independent variable.
10. Cf. for this section Bieberbach 1923.
11. A p-dimensional contact element in Vn is the set of one point of V and
of a p-dimensional linear manifold tangent to this point Vn.
12. The Euler characteristic of a topological manifold is an invariant of
global topology of which the following is an intuitive example: in the case
of an arbitrary polyhedron of ordinary Euclidean space this invariant characteristic is equal to the sum of vertices plus the sum of sides and minus the
sum of the edges. It is in this case always equal to 2.
280
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Index
a fortiori 7
a priori xxx, xxxi, xxxiii, 22, 89,
109, 158, 187–9, 203, 213,
251–3, 256
abelian 130
integral 29, 58, 149–51, 219, 223
Ackermann, Wilhelm 268, 288
adjoint pair 265
adjunction xxxv, 90, 128, 187, 268
Ahlfors, Lars 65–6, 281
Ajdukiewicz, Kazimierz 21
Alexandroff, Paul S. 116, 122–4,
276, 281
algebra xxvi–ii, xl-ii, 17, 24,
40–1, 48–9, 51, 52–6, 59,
67–8, 70, 72, 74–5, 77, 80–3,
91, 126–8, 142, 188, 212, 218,
232–3, 268
abstract 45, 128, 233, 237
Boolean 238
fundamental theorem of 51–2
modern 10, 47–9, 67–8, 80, 126,
186, 233, 294
non-commutative 46
ambidextrous 231
analytic xxvii, 18, 33, 96, 118, 124,
275
analytic number theory xxviii, 74,
76–7, 79–81, 197, 207, 213, 219
analyticity 101
anharmonic ratio 61, 97
anthropology 202–3
anti–isomorphism 237
anticipation 23
of the concept xxvii, 210
antisymmetric 45, 231, 233–4, 237,
239
see also duality, field, function
antisymmetry 233–4, 239
Antoine, Louis 124, 282
arc 63–4, 245
Jordan 124
area 32, 65, 177, 215
Aristotle xxxvii, 191, 278, 281
asymmetric carbon, see carbon
atom xxviii, 80, 231, 234, 247
Avicenna xxxvi–ii, 268, 281
axiom of choice 4, 28, 73, 89, 142
axiom of infinity 4
axiom of reducibility 3–4, 10
axiomatic 20, 24, 31–2, 41, 45, 47,
148, 186–7, 236
Hilbert xl, 5, 10, 19, 24, 89, 163
axioms of neighborhood 131
IN D EX
Bachelard, Gaston xx, 21, 281–2
Badiou, Alain xv, xx, 266, 281
Becker, Oskar xxxiii, 5, 190, 277–8,
281
being x, xv, xxxiii, 3, 102, 200–5, 267
imperfect 126, 128
perfect 125, 128
unity of 41
Benis-Sinaceur, Hourya xv, xx,
225, 282
Benjamin, Cornelius 20–1, 282
Benjamin, Walter xxviii
Bernays, Paul xiv, xx, xxvi, 17, 29,
34, 89, 142–7, 266, 275, 282,
288
Betti number 119–24, 177–8, 212,
273
Betti, Renato 266, 282
Bieberbach, Ludwig 97, 176–7, 276,
280, 282
Birkhoff, Garret 177, 237–9, 282
Biss, Daniel K. 267, 282
Black, Max xx, 282
Blay, Michel xx, 263, 282
Bloch, André 270, 282
Boole, George 1, 235
Borel, Émile 33, 47, 271
Bouligand, Georges 32, 41–2, 282
Bourbaki xiv, xvi, xxvii, 264
Boutroux, Pierre 183–6, 277, 282
Braithwaite, Richard B. 20–1, 282
bribable 7, 142
Brouwer, Luitzen E. J. xxxi, 4–5,
123, 265
Brunschvicg, Léon xiv-v, xvii, 12,
22, 88–9, 187, 220, 282
Buhl, Adolphe xx, 282
calculus 1, 7, 18–19, 24, 68, 80–1,
237, 286
algebraic 1
differential 36
formal 14, 16
298
non-commutative 72
predicate 146
of probabilities 192, 292
propositional 89, 237–8
transfinite 2
of variations 172, 222
calculus ratiocinator 1
Caratheodory, Constantin 168, 282
carbon, asymmetric 230
Carnap, Rudolf xv, 3, 10–11,
13–20, 87, 144, 268, 283
Cartan, Elie xiv, xli–ii, 68–72, 97–9,
105–6, 113–15, 186, 220,
232–3, 269, 270–2, 283
Castellana, Mario xx, 283
category theory xxxiv-vi, 265, 267
Cavaillès, Jean xiv-v, xx–i, xxxii,
xxxix, 13, 15, 23, 74, 90, 146,
195–6, 220, 224, 263, 267, 268,
275, 277, 283
Chevalley, Catherine xxi, 263, 284
Chevalley, Claude xvi, xxvii, xxxix,
xli, 17, 24, 209, 220, 264, 266,
268, 270, 274, 283–4, 287
class field xxv, 29, 76–7, 79, 126,
128–30, 151–2
theory xxv, 17, 126, 128–9, 151,
270
classes of elements 29, 34–5, 48,
77, 155
closed 98, 106, 185
algebraically 127
chain 272–3
path 63, 132–3, 136
surface 47, 185
cohomology xli–ii
compactness xxxi, 168–9, 173
complementarity 33, 262
complementary 48, 235, 238, 259
aspects 81
concepts xxiv, xxx
sets 235
space 121–2, 124, 186, 238
INDE X
complete determination 6–7
completion xxxi, xli, 55, 90, 104–8,
128, 131, 137, 143, 146–8, 189
of a theory 147
complex variable xxiii, xxvii, 10,
33, 47, 49, 62–3, 65, 75, 77, 81,
134, 167, 180–1, 210–11, 271
complexes 119, 186, 273
complexity xxiv, 33, 133, 202, 224
compossibility 143
Comtism 20
conclusive schema 143
cone 244, 252, 257–8
conformal representation xxvii, 63,
136, 168–9, 282
consistency xxvii, 4, 6, 7, 16–17,
74, 90, 142–3, 145–7, 186,
207–8, 222, 263, 287
constructivist xxiv, 47
continuity xxxi, 46, 80, 82, 89, 160,
242–3, 261
analytic 73, 96, 179, 181
continuous xiii, xxvii, xxx–i, xl–i,
48–9, 56, 69, 72–5, 79–82, 96,
100, 107, 120, 131, 142, 150,
160–1, 166–7, 174, 189,
217–19, 259–61, 271
deformation 58, 63, 104, 116, 120,
122, 131–2, 134–5, 149, 178
correlation xxxiv, 236
Costa de Beauregard, Olivier xxi,
284, 290
Courant, Richard 97, 180, 254, 276,
279, 284, 288
creativity, mathematical xxiii, xxv,
xxix–x, xxxv
crystallography 230
Curie, Pierre 230, 284
curve 58
algebraic 97, 176
closed 29, 123, 131, 273
elliptic xxix
surface 46
cut see canonical cutting
cutting 152, 181
canonical 29, 58, 150
cycles, in topology 119, 122, 176–7,
272
De Broglie, Louis 80–1, 172, 234,
244–5, 247–8, 270, 276, 279,
284
De Broglie’s hypothesis 33
decomposition 29, 50–4, 56–9,
68, 76–7, 82, 106–8, 119,
121, 129–30, 142, 151,
155–7, 162–3, 166–7,
170–1, 173, 178, 181, 187,
272, 274
algebraic 52, 82
dimensional 53, 68, 82
proper 51–3, 56
theorem of 51–2, 56, 58, 68, 142,
156
deductive theory 4, 88
Deleuze, Gilles xv, xxi, xxv, xxviii,
265, 284
Descartes, René 17, 27, 125–6, 273,
284
diagram xxxv-i, 265–6
dialectic xxiv, xxx, xxxi–ii, xxxiv,
xxxvi, 188, 190–1, 196–7,
199–200, 203, 204–6, 208,
218–9, 223, 224, 240, 242, 246,
251, 265, 289
Platonic xxiv–v, 41
dialectical xiv, xxiii, xxxi, xxxv–vi,
28, 125, 181, 203, 204–5, 218,
231
problem xxxvi, 28, 205, 211, 214,
218, 221–3, 246
structure xxvi, xxx, 207–8,
211–14, 221, 231, 235, 238,
241, 267
Dieudonné, Jean xi, xiii–iv, xxxix,
268, 284, 290
299
IN D EX
dimension 50, 51–4, 56–8, 82, 98,
100, 112–15, 117–24, 131, 135,
163–5, 176–7, 212–13, 232,
236–7, 244–6, 248, 252, 254,
257–8, 260–1, 269, 272–3, 280
theories of 10
and time 98, 232, 244–6, 248–9,
251, 254, 256–8, 260–1
dimensional decomposition 49, 53,
68
Dirac, Paul 46
Dirac’s theory 232
Dirac’s wave mechanics
equations 247
Dirichlet, Johann Peter Gustav
Lejeune 74–5
Dirichlet problem 100–1, 173, 190
disclosed 200
disclosure 202
disclosure of being 200, 205
discontinuity xxxi, 58, 77–80, 150,
160
domain of 77, 79, 82
discontinuous xxvi, 4, 48, 49, 72–4,
76, 78–82, 160–1, 167, 189,
207, 217–19
discrete xiii, xxvii, xxx–i, xl–ii, 81,
161
dissymmetry xiii, 195, 227, 229–31,
233–5, 239–40, 243–4, 246,
256, 290
molecular 229
division xxv–i, xlii, 20, 41, 46, 78,
95, 126, 130, 151–2, 155, 170,
190–1, 231, 235, 270, 277–8
method of xxx, 31, 33, 37, 41,
289
dogmatic philosophy 16, 195
domain, basic 29, 36–8, 52, 77, 79,
96, 127, 148, 152, 155, 157–8,
160–1, 165–70, 177–8, 186,
204, 218, 233
concrete 24, 148, 155
300
of individuals 17–18, 29, 145,
147, 158
duality xxxvi, 33, 37, 49, 59, 81,
90, 95, 101, 113, 115, 118,
121–2, 145, 158, 185–6, 231,
235–9, 246, 248, 251, 254, 258,
262, 273
anti–symmetric xxxiv, 238–9
law of 238–9
projective 237
theorem xxvii, xxxi, xxxv, 118,
120, 122–4, 131, 212–13
Dubourdieu, Jules 270, 284
Dumitriu, Anton 268, 285
Dumoncel, Jean-Claude xxi, 285
Eddington, Arthur S. 191, 285
Ehresmann, Charles xvi, xxvii, 220,
260–1, 267, 285
eigenfunction 161–2, 166, 170, 247
eigenvalue 29, 161, 164–7, 173–4,
247
Eilenberg, Samuel 267, 285
Einstein, Albert xiv, 80, 88, 98,
113, 223, 232, 262, 294
electricity 88
electro-magnetism 232, 245
electromagnetic field 231–2, 262
electron 21, 232, 234, 247, 284
enantiomorphy 229
energy 88, 160, 167, 243, 245–51,
253, 255, 279
Enriques, Federico 21–2, 285
entity x, xxxiv, 6, 28–9, 41, 50, 53,
91, 101–2, 110, 113, 118, 121,
128, 130–1, 137, 141, 148,
151–2, 155, 157, 170–2, 178,
184–6, 190, 199–205, 212, 219,
222, 224, 234–5, 237, 247
equality, axioms of 34, 89
equations, algebraic 53, 55–6, 59,
126, 184
analytic theory of 70
INDE X
classical wave 257
differential xxiii, 46, 49, 67, 70–2,
80, 95–6, 99, 160, 175–7, 181,
183, 251–3, 256–9, 261–2
fundamental 179, 250
integral xxvi, 11, 54–6, 161–2,
165–6
partial 54, 72, 98–9, 101, 160,
175, 251–2, 254, 256–8, 262,
271
recurrence 89
Euler characteristic 104, 178, 185,
261, 280
Euler, Leonhard 171, 213
evolution xxvi, xli, 101, 143, 153,
188–9, 192, 234, 244–5, 247–8,
250–1, 253, 257, 260–2, 282,
288
excluded middle 4, 142, 145, 147
exemplary 224
exigency xi, xxxi, xxxvii, 188–9,
224, 242, 261, 268
existential xxiv, 201
experience xxxiv, 2, 4, 11–12, 14,
16, 18–22, 24, 28, 33, 88, 182,
187, 189, 192–3, 195, 222,
224–5, 231, 242, 251, 282
intuitive 12, 22
extremal 176, 245
properties 171–2, 174
value 175
extremum principle 245
extrinsic xxvii, xxx, xxxvi, 30, 110,
113, 115, 117, 123, 189, 273
false 6, 18, 21, 23, 142, 235, 239
Fehr, Henri 271, 285
Fermat’s theorem or principle xxix,
172, 245, 294
Ferrières, Gabrielle 268, 285
field, algebraic 187, 211
antisymmetric 45
extension 40, 79, 127–30, 151
gravitational 232
number xxiii, 39, 41, 126–7, 129,
143, 152, 268, 274, 285, 287
vector 20, 259–62
see also non-ramified abelian field
figure 32, 46, 62, 78, 97, 111–12,
116–22, 132, 135, 177, 180–1,
187, 224, 229, 233, 241, 277,
281
Fischer, Ernst 165, 285
Focillon, Henri 266, 285
force 182, 202, 247
formalism 14, 16–19, 22, 26–7, 90,
141, 143, 283
foundation xxiii, xxxix, 1, 17, 22,
45, 73, 89, 202–3, 207, 221,
233, 238, 265, 285, 288–9,
292–3
founding 202–3
Francastel, Pierre xxviii, 266, 285
Fréchet, Maurice 31–2, 41–2,
220–5, 285
free object xxxv–vi, 265
Frege, Friedrich L. G. 2, 73, 82,
286, 288
Freyd, Peter J. xxvi, xxxvi, 267, 285
Fueter, Rudolf 74, 285
functional, analysis xxiii
properties 10, 95
spaces 54, 82, 162, 165–6, 237
function, abelian 183
algebraic 62, 148–9, 133–7, 181,
269, 275
analytic 33, 49, 56, 61–2, 66, 70,
74–7, 81, 95–7, 133–4, 167–8,
178, 180, 181, 208, 214–15,
271–2, 275–6
see also theory of analytic
functions
antisymmetric 234
arithmetic theory of 269
automorphic 77, 79, 82, 181, 183
complex variable xxiii, 47
301
IN D EX
function, abelian (Cont’d)
continuous 54–5, 72, 77, 79,
82, 88, 96, 106–8, 161–2,
173–5, 207, 209, 214, 217,
219, 272
continuous, without
derivatives 33, 88
differentiable 54, 108, 161
discontinuous 217
elliptic 134, 183
integral 54, 57, 215–16, 218
meromorphic 57, 64–6, 293
orthonormal 54–5
potential 150, 275–6
real variable 47
Riemann 207, 214
space 54–5, 82, 106–7, 155, 162,
165–6, 237
theta 210–1
trigonometric 54, 192
uniform 58, 62–3, 133, 136, 150,
181, 259
functor xxxv, 267, 282
adjoint xxxv
representable xxxv, xl, 265
Furtwängler, Philipp 129, 285
Galois xxix, xxxvi, 274
correspondences xxvii
group xxv, 127
theory xxv, xxxvi, 126–30
Galois, Evariste xiii
Garland, Howard 268, 285
Gauss, Carl F. 46, 62, 112, 214, 285
Gauss sum 209–10
genera 31, 41
genesis, geneses ix, xxvi, xxxv,
29, 75, 77, 79, 137, 139, 141,
143, 147–50, 152–3, 155–7,
161–3, 165–7, 169–70, 173,
175, 178, 189, 193, 197, 199,
200–3, 205–6, 218–19, 221,
223, 240, 242
302
dialectical xxxv, 197, 199–200,
203, 205–6, 218–19, 221, 223,
240, 242
Gentzen, Gerhard 146, 188, 277, 286
geodesic 112, 115
geometry xli–ii, 12, 32, 45, 60–2,
72, 95, 97–9, 106, 110, 112–13,
135, 143, 186, 292
abstract 61
affine 97
algebraic x, xxviii, xxix
analytic x
differential x, xxviii, xlii, 11, 46,
72, 102, 110, 112–13, 115–16,
135
Euclidean 186
Klein 97–8, 106
Lobatchewsky 61, 63
Non-Euclidean 62, 187
projective 45, 61, 97, 236–9
pure 112
Riemann 61, 98, 106
synthetic 47, 62
Glivenko, Valère 237–8, 286
global xiii, xxvi, xxvii, xxx–ii,
xxxvi, xl–i, 46–7, 52, 61, 66,
77, 91, 95–8, 101, 105–9, 129,
162, 175, 178, 180–1, 185, 187,
189, 192–3, 203, 221, 233, 235,
260–1, 271, 280
decomposition 68
function 179
integration 99
intuition 184
space 51, 60, 105
structure 48, 50, 57, 104, 118, 261
uniformization 134, 136
Gödel, Kurt xxxv, 10, 15–16, 74,
141, 146–7, 188, 208, 277, 286,
288
Gonseth, Ferdinand xv, xxi, 19, 24,
263, 286, 287
Granell, Manuel xxi, 286
INDE X
Granger, Gilles-Gaston 286
Grassmann’s exterior algebra xlii, 69
gravitation 11, 232, 262, 285
see also field, gravitational
grounding 202–3
group, adjoint 106, 272
closed 105–6
continuous 71, 80, 105, 270
see also Lie group
discontinuous 63, 77, 80, 219
fundamental xxv, xxxv, 131–3,
136
Klein 106
linear 105–6, 222, 232, 272
modular 78–9
representation of 152–5, 222,
232, 276
rotation 63, 232
theory xxiii, xl, xli–ii, 10, 24–5,
45–7, 62, 80, 91, 97, 102, 105,
192, 222, 294
see also ideals, theory of
transformation 25, 36, 60–1,
63–4, 71–2, 78, 82, 97, 105–6,
127, 153–4, 272, 276
Hadamard, Jacques 100, 214, 263,
271, 286
Hamilton, William 172, 245–6, 254
Hamilton’s principle 245–6, 279
Hamiltonian equations of
mechanics 11, 250, 254
Hamiltonian operator 249–50, 255
Hankel’s principle of the
permanence of formal laws 89
harmonic forms, theory of xlii
Hasse, Helmut xli, 209, 268, 286
Hecke, Erich 74, 209, 212, 270,
276, 278, 286
Heidegger, Martin x, xv, xxxiv, 5,
197, 200–5, 218, 266–7, 278,
286
Heinzman, Gerhard xxi, 263, 287
Heisenberg, Werner 22, 46
Heisenberg’s uncertainty
relations 16, 248
Hellinger, Ernst 269, 276, 287
hemihedral crystals 229
Herbrand domain xxvi, xxix
Herbrand, Jacques xvi, xxvii, xxxix,
6, 7, 14, 17, 141, 146–7,
159–60, 209, 263, 266, 268,
270, 274–5, 284, 287
Herbrand’s theorem xxxv, 147, 222
Hermite, Charles 164, 287
Hermitian form 154, 163–6
Hermitian operator 247
heterogeneous 147, 157, 168, 200
Heyting, Arend 5, 287
hierarchy xxiv, xxix, xxxv, 2, 8, 17,
130, 251
Hilbert, David xiii, xiv, xxvi–ii,
xxxv, xxxvi, xl, 4–7, 10, 17–19,
24, 34, 49, 54–6, 73, 82, 89–90,
128–9, 136, 142, 143–6, 151–2,
162, 164, 166–7, 173, 187, 209,
254, 263, 268, 275–6, 279, 282,
285, 287–8
Hilbert space xxvi–ii, xxix, xxx, xl,
11, 24, 29, 54, 68, 160, 162–7,
170, 173–4, 250
history xv–vi, xxi, xxiv, xxxiv,
xxxvii, 22, 54, 88, 172, 185,
189, 201, 205, 262, 266–7, 285
holomorphic 100, 168–9, 173, 271
homogeneity 97, 114, 157, 168
homographic transformation 97
Hopf, Heinz 102–5, 116, 122–4,
177, 261, 271, 276, 281, 288
Hopf theorem 261
Hurwitz, Adolf 180, 276, 284, 288
Husserl, Edmund 5, 21, 286
Husson, Édouard 177, 288
hypercomplex systems 47, 52, 156
hyperplanes 236
hypothetico-deductive method 20
303
IN D EX
idea xxx, xxxiv, 13, 27, 41–2, 45,
77, 80, 83, 88, 91, 95, 102,
108–9, 111, 125, 172, 179,
182–3, 187–93, 197, 199, 200–6,
208, 211, 215, 218–19, 221–4,
231, 240–1, 246, 251, 262, 265
dialectical x, xxxiii, 30, 91,
199–200, 204, 211, 218–19
ideal norm 75, 212, 270, 274
see also prime ideal
ideals 79, 152, 270, 274
class of 76, 79, 129–30, 151, 270
group of 129–30, 151, 274
theory of xxxv, 10, 47
see also group theory
inclusion ix, 236–7
incongruity 112
incongruity of symmetric
figures 111–12
infinitely small 39, 88, 103
Ingham, Albert E. 278, 288
initial principles 88
intrinsic xxvi–ii, xxx, xxxvi–ii, 28,
41, 89, 91, 110, 112–13, 115,
117–18, 120, 123, 137, 152–3,
189, 199, 204, 251
intuitionism 27, 141, 265
invention 6, 89
mathematical xxiii, xxix
inversion xxvii, xxxv, 229, 235
involutive operation 233, 237
isomeric 229
isomorphism xxxv, 237, 285
Jacobi, Carl Gustav Jacob 254
Jacobi equation 255–6
Janet, Maurice 171–3, 262, 288
Jordan, Pascual 124
Jordan’s theorem 123
Julia, Gaston xvii, 276, 288
jump 58, 150
Jürgensen, Jorgen 1, 288
Juvet, Gustave 25, 192, 288
304
Kähler, Erich 270, 289
Kant, Immanuel x, 110–12, 118,
124, 157–8, 229, 285, 289
Kerszberg, Pierre xxi, 289
Klein, Felix 61, 97–8, 289
Kovalevsky, Sonja 99–101, 175, 289
Lagrange function 245–6
Laplace equation 100, 173
Laplace’s ‘spherical harmonics’ xlii
Latapie, Daniel xviii, 289
lattices xxvi–vii, xxxi, 237–9, 279
see also theory of structures
Lautman, Albert ix–x, xiii–xxix
law 11, 19–20, 24, 48, 52, 70, 76,
98–9, 114, 130, 192, 199, 231,
243, 245, 262
distributive 238–9
mathematical 11
physical 21
reciprocity 197, 208, 210, 213,
239–40
see also duality, laws of
laws of a harmonious
mixture xxviii
Le Roy, Édouard 21
Lebesgue, Henri 38–9, 47, 123, 165,
174, 268, 290
Lefschetz, Solomon 121, 177, 272,
290
left and right 111, 229, 232–3, 235
Leibniz, Gottfried Wilhelm 1–2,
110–11, 113, 117, 119, 124,
143, 171–2, 184, 202, 290
Levi–Civita, Tullio 114–15, 290
Levy-Bruhl, Lucien 12, 25, 290
liberation xxiv, xxv, xxx, xxxi
Lichnerowicz, André xxi, 279, 290,
294
Lie group xli–ii, 71–2, 233
see also group, continuous
light 80, 88, 101, 172, 229, 231,
244, 262
INDE X
line, straight 78, 97, 114, 214, 231
Lobatchewsky, Nikolai 66
local xiii, xxvi, xxvii, xxx–i, xxxvi,
xl–i, 60, 91, 95–102, 104–6,
114, 135, 175, 179, 185, 221,
260, 262, 271
integration 46, 181
solutions 180–1
uniformization 134–5, 149
locus 184, 202
logic, Aristotelian 15, 41
mathematical xv, 1, 9, 19, 28,
73–4, 82, 141, 143, 145, 158,
188, 207, 235, 237–8, 266
logicism, logicist 2, 4–5, 8–11, 14,
16–17, 20–2, 24, 109, 143, 185
Loi, Maurice xii, 290
Lorentz invariant 98, 261
Lorentz transformation 247
Lukasiewicz, Jan 13, 23
Mach, Ernst 14, 22
MacLane, Saunders 267, 285, 290,
294
magnitude (grandeur) 16, 22, 37–9,
41, 45–6, 48–50, 67–8, 72, 128,
149, 192, 215, 237, 243–5, 247,
250–1, 256, 262
non-commutative 67–8
Malebranche, Nicholas xiv, 199
manifold xxxiv–v, xl, 98, 105,
112–15, 118, 131, 132, 158,
163–4, 212, 257, 260–1, 280
characteristic 257
Riemann 113
mass 33, 160, 202, 248
mathematics, effective xxiv, xxvii,
xxxi–iii, xxxvi, 27, 160, 197,
199, 205
modern xiii, xxiii, xxiv, xxviii,
xxxiii, xxxv–i, xlii, 10, 31,
46–7, 49, 61, 188, 271
formal 10, 17–18, 90, 143, 155
matrix 164, 276
array 192
Matter 11, 19, 30, 80–1, 88, 92, 98,
112–13, 148, 154, 183–4,
189–90, 193, 199, 200–1,
204–5, 223–4, 231, 233
Maupertuis’ principle of least
action 172, 245
Maxwell’s theory 20
mechanics 6, 223, 241–2, 247, 251,
253
celestial 176, 262
classical 192, 238–9, 242, 245–6,
250
new 21
old 21
quantum 10, 16, 22, 45–6, 67–8,
160, 167, 192, 232, 238–9, 242,
245, 282, 294
relativistic, 242, 245
statistical xxviii
wave 233–4, 242, 245–8, 250–1,
255
meridian 232
Merker, Joël xxi, 290
metalogic 17–19
metamathematical 17, 29, 90, 141,
143–4, 152, 159, 187, 188
metamathematics 6, 17, 89–90,
143, 191
method, analytical 176, 203
structural/extensive 29, 145–7,
152, 159, 275
metrization xxvi, 103–4, 129
mixed mathematics, mixes,
mix xiv, xxiv–vi, xxviii–x,
xxxii–v, 8, 41, 91, 157–8, 160,
165–70, 265–6
model xxv, xxx–iii, 17, 102, 128,
167, 199, 223
mathematics as 203
theory xxxii, 292
modeling xxxi
305
IN D EX
modular, figure 180–1
function 79, 180–1
identity 238
lattice 238
see also group
molecule xxviii, 231, 234
ortho and para 33
monad 110–11, 113, 119, 184
Monge cone 252
Monge, Gaspard 46, 62, 290
Montel, Paul 81, 167–9, 276, 291
Montel’s normal families 168
Morris, Charles W. 13, 21
movement xiii, xxvi, 27–8, 30, 67,
82–3, 89–91, 108, 111, 113,
126, 128, 130–1, 137, 182–3,
187, 190, 202, 241, 245
nappe 244
neighborhood 31, 33, 57, 60, 62,
96–8, 100–1, 105, 108, 110,
114, 119, 124, 131–2, 134–5,
175, 178–9, 210, 275
axioms of 131
Neurath, Otto 13
Nevanlinna, Rolf 64–6, 269, 291
Nicolas, François xxi, 291
Nitti, Francesco F. xix, 291
non-being 5
non-commutative 69, 72, 233, 239
non-Euclidean metrics 49, 60–2,
64, 80
non-orientable 118
surface 104
non-predicative definition 109
non-ramified abelian field 129
normal families of functions xxix,
167–8, 170
notions, primary xxvii, 4, 10, 42
primitive 3, 11, 24, 87, 204
number, cardinal 3–4
hypercomplex 10
imaginary 20, 61, 79, 143, 210, 270
306
irrational 6, 88
see also analytic number theory
Occam 21
ontic 200–2
ontological xxxiv, 200–5, 223, 242,
267
operations, abstract 24, 147–8, 233
order, partial 237
orientation xiii, xvi, 110, 112, 118,
229, 231–3, 235, 241, 243, 244,
261
Osgood, William F. 58, 291
Other xiii, xxviii, xxxiii, 41, 189,
231, 265
Padoa, Alessandro 20
pairs xii, xiv, xxiv, xl 204
of ideal opposites xxviii, 231, 240
parallelism on any manifold 114
Pasteur, Louis xiii, 229–30, 291
Peano, Giuseppe 2, 148
periodicity 192
permutation 134, 231, 234–5, 237
permute 127, 231–2, 250
Petitot, Jean xiv, xxi, 263, 266, 291
Pfaffian systems theory xlii, 70–1,
233
phenomenalist 22
phenomenology 21, 200
Philebus xxx, 41
philosophy, mathematical xiv, xvi,
xxiii, xxix, xxxii, xxxvii, 4, 24,
27, 73, 87–9, 109, 143, 152, 157,
189, 193, 200, 203, 219, 293
of science 1, 9, 12–13, 20–2, 25–6
photon 33, 234
physical constant 88
physics, atomic 22, 33
mathematical 191, 231, 233, 239,
241–2, 252, 288
theoretical 22–3, 33, 245
Picard, Emile 64–5, 269, 291
INDE X
Plato xxx, xxxiii–iv, xxxvi, 12, 25,
192, 197, 199, 221, 231, 267,
278, 286, 291–3
Platonic dialectic xxiv, xxv, 41, 190
Platonic method of division xxv, 41
Platonism xxxii–iv, 30, 42, 190–1,
193, 199, 219, 265
Poincaré, Henri xiv, xvii, xxi, xxv,
xxxv, 10, 21, 62–3, 66, 82, 88,
120, 123, 131, 136, 141, 175–8,
188, 212, 222, 259, 269–70,
273, 287, 291–2, 294
Poincaré’s fundamental group xxv,
131, 136
point, fixed 177–8, 222, 236
primitive 153
Poirier, René 6, 88, 292
Poizat, Bruno xxvi, 292
polarization of light 229, 231
pole 33–4, 57–9, 111, 149–50, 179,
275
Poncelet, Jean-Victor 236
Poncelet’s principle of
continuity 89,
Pontrjagin, Lev S. 123, 292
Popper, Karl 20, 292
Possel-Deydier, René 38, 292
presheaf xxxv
prime ideal xli, 76, 129–30, 274
primes 75–6, 213–18
primordial 202
Principia Mathematica xv, 2–4, 6, 19,
24, 89, 293
probability 23, 247–8, 292
theory 23, 237–8, 292
projective plane 104, 236, 261
proof xxvi, xxix, 12, 15, 17, 29, 32,
89, 100, 108, 129, 143, 146–7,
150–1, 173, 177, 207, 211–13,
247, 251, 253, 268, 276
algebraic 209
of consistency 4, 6, 17, 188, 208,
222
of existence 152
theory 14, 90, 143, 263, 287
transcendent 74, 211–12
properties, extrinsic 110, 117, 123,
189
internal 110, 112, 116–17, 123
intrinsic 41, 91, 110, 112–13,
117–18, 120, 123, 137, 189,
204
situational 115, 117–18, 120,
122–4, 204, 221
structural 29, 52, 72, 76, 96,
115–17, 122, 124, 144, 147,
152, 160, 186, 221–2
Pythagoras’s Theorem 32
quadratic, differential form 72, 97,
260
form 105, 151, 166, 244, 260–1,
272
reciprocity 197, 208–12, 240
quanta 245, 279, 284
theory of 245, 248
quantity 64–5, 67–8, 72, 82, 104,
216, 232, 244, 246, 265, 276,
279
Ramsey, Frank P. 3, 21
real, mathematical xxxiii–v, 9, 27,
87, 183–7, 192, 199, 238,
240–1, 260, 277
physical xiv, 9, 25, 192, 199, 203
realization 25, 32, 144–8, 153, 156,
159, 188, 199, 201, 204, 208,
211, 214, 221–3, 238, 240, 251
rebound effect 77, 79
reciprocity, laws of 197, 208–11,
213, 239–40
see also quadratic
Reichenbach, Hans 13, 23, 268,
283, 292
relativity, general 98, 223, 245, 285
special 98, 192, 232, 244–7, 260
307
IN D EX
relativity, general (Cont’d)
tensorial representation of the
theory of 11, 19
theory of 19, 33, 97, 112, 233, 242
representation xxvi–vii, xxix,
xxxvi, 11, 29, 53, 63, 102, 106,
119, 136, 152–6, 158, 168–9,
180–1, 219, 222, 232, 272,
275–6, 282
retrosections 58, 134–6, 149–50
Reymond, Aron 1, 24, 292
Riemann, Bernhard xiii, xiv, 47,
58–9, 75, 95–8, 112, 135–6, 149,
173, 180, 214, 269, 290, 292
function 207
see also manifold
space 60, 97–8, 105–6, 114, 186,
233
surface xxx, 29, 47, 58, 62–3,
65–6, 114, 126, 133–7, 149,
151–2, 155, 219, 222–3, 276,
286, 294
Riesz, Frigyes 165, 292
right see left and right
Rinow, Willi 103–4, 288
Robin, Léon xxxiii, 190, 192, 278,
292
Romano, Ruggiero 266, 282, 292
root 51–2, 56, 126–8, 134, 176,
179, 185
square 151
roots of unity, arithmetic theory
of xlii
Rosser, J. Barkley xxi, 293
Russell, Bertrand xv, xxix, 2–6, 8,
10, 14, 16–17, 19, 23, 73, 82,
87, 89, 109, 141, 271, 293–4
Same xiii, xxviii, xxxiii, 41, 189,
231, 265
Satre, Jean-Paul 293
saturation xxiv, xxv, xxvi, 222,
230, 234
308
Scedrov, André 285
Schlick, Moritz 13, 21–2, 283, 293
scholasticism 21
Schrödinger, Erwin 46, 67, 160,
167, 243, 255, 293
Scott, Sir Walter 4
Seifert, Herbert 116, 117, 132, 274,
276, 293
semi–spinors 232
semiotic 21
sequence, fundamental 40, 103–4,
106
set theory 4, 6, 46, 143, 146, 237,
266, 271
Cantorian 2, 82, 141
paradoxes of 1–2, 109
semantics, set-theoretical xxxi
Tarski 19
Severi, Francesco 188, 293
sheaf xxvi, xxviii, xli, 176–7, 252
Shimizu, Tatsujiro 65, 293
ship 232
Sichère, Bernard xxii, 293
simplex, simplices 119, 272
Simpson, Stephen G. xxvi, 293
singularity 98, 150, 179, 181,
260–1
logarithmic 193, 275
Sirinelli, Jean-François 264, 293
snail shells 111
solidarity 9, 51, 58, 91, 102, 106,
110, 129–30, 137, 147, 182,
185, 223
Sophist xxx, 41, 190, 266–7, 286
space, Klein 97, 105
physical 232, 241, 290
space-time 98, 232, 242, 244,
246–8, 257
space-time, fibrous structure of 244
species xlii, 31, 41
spectral lines 33
sphere 63–5, 104, 115, 119, 131,
136, 232, 261
INDE X
spherical triangles 112
spin of electron 232, 234
spinors 232
spinors, theory of 232
spiral 111, 176, 259
Stenzel, Julius xxxiii, 190–1, 277,
293
stereochemistry 230
structural schemas xxvii, xxxiii,
xxxv, 188, 199, 223, 265
structural/extensive, see method
subspace 51, 154, 163, 165–6, 237–8
subspace, complementary 238
substitution 53, 71–2, 77–9, 127,
143–4, 158–9, 175
successor function, axioms of 89
sufficient reason, principle of 2,
203, 248
sum, algebraic 178, 272
summable squares 54, 165–6
superimposable 112, 229
surpassing 200
see also transcendence
symmetric spaces, theory of xlii
symmetry 25, 34, 121, 163, 195,
227, 229–36, 239–40, 284, 290
antisymmetric 239
dissymmetrical xxxiv, 231,
233–4, 239
geometric 236–7
lesser 230
sympathy 19, 111
Takagi, Teiji 129, 209, 270, 293
Tarski, Alfred 17–19, 293
tautology xxv, xxix, 3, 9, 18, 25,
27, 87, 278
temporal evolution 260
tensor, fundamental 11, 19
theory of analytic functions 60, 64
Thirion, Maurice xxii, 293
Threlfall, William 116–17, 131–2,
274–6, 293
Thuillier, Jacques 266, 294
time, cosmogonical 260–2
dimensional 245, 249, 254,
256–8, 260
dynamical 262
parametric 254, 255, 258, 260
plurality of 33
sensible 241–2, 244, 252, 254, 261
Tonnelat, Marie-Antoinette 262,
294
topological structure 10, 101, 103,
135, 149, 177, 219, 222–3
value 150
topology xxvii, xli, 9, 46–9, 62, 66,
91, 96, 102–3, 105–6, 110, 116,
118–19, 122, 124, 160, 162,
165–7, 176–7, 185, 188, 212,
218, 258, 260, 272, 280, 290,
293
algebraic xxiii, xxviii, xxxi, xxxv,
xli, 46, 115–16, 118, 123, 271–2
combinatorial 46–7, 118, 131,
237, 271
set theoretical 131, 271
torsion 123–4, 232, 273
trajectories, luminous 244
transcendence 88, 200, 205–6
see also surpassing
transcendental 47, 158, 200, 203,
209, 219
Transcendental Analytic 157
transfinite 2, 6–7, 28, 73, 88,
142–3, 188, 190
transformation
analytic 71
continuous 46, 71, 116, 177
formula 211
group 25, 36, 60–1, 63–4, 71–2,
78, 82, 97, 105–6, 127, 153–4,
272, 276
internal 63, 127, 177, 222
linear 63, 97, 179, 272
orthogonal 166
309
IN D EX
true 6, 18, 21, 23, 142, 235, 239
types, theory of 2–4, 19
uncoil 111
understanding, pre-ontological 200
preconceptual 202
uniformization 62, 126, 133–7
unitary theories 232, 262
universal covering surface xxv, 62–3,
126, 130–3, 136–7, 151, 181
valence 234
value, algebraic 147
van der Waerden, Bartel L. 39, 47,
156, 270, 274, 294
variable, complex xxiii, xxvii, 10,
33, 47, 49, 62–3, 65, 75, 77, 81,
167, 180, 210, 211, 271, 276
real 47, 174, 176, 211
vector, momentum 245
velocity 202
Venne, Jacques xxii, 294
Vienna Circle xv, 9–10, 12–14,
16–17, 20–1, 191, 278
310
virtual 203, 281
von Neumann, John 5, 68, 163,
167, 237–9, 270, 282, 294
Waverley 4
Weil, André xxvi, xli, 213, 276,
294
Weil, Simone xvii
Weyl, Hermann 4, 10, 45–9, 61,
73, 77, 80, 97, 105–6, 115,
134–5, 149, 154–5, 167,
274–5, 294
Whitehead, Alfred North xv, 2–4,
19, 89, 293
Wiles, Andrew 266, 294
Winter, Maximilien 183–4, 273,
294
Wittgenstein, Ludwig 10, 14, 16,
18, 87, 265, 267, 294
Yoneda, Nobuo xxxv, 267, 290, 294
Zalamea, Fernando ix, xi, xiii, xxii,
xxiii, 1, 265, 290, 294
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