Mathematical Epistemology and Psychology

342 Pages·1974·10.91 MB·English

by  Evert W. Beth

MATHEMATICAL EPISTEMOLOGY AND PSYCHOLOGY
SYNTHESE LIBRARY
MONOGRAPHS ON EPISTEMOLOGY,
LOGIC, METHODOLOGY, PHILOSOPHY OF SCIENCE,
SOCIOLOGY OF SCIENCE AND OF KNOWLEDGE,
AND ON THE MATHEMATICAL METHODS OF
SOCIAL AND BEHA VIORAL SCIENCES
Editors:
DONALD DAVIDSON, Stanford University
J AAKKO HINTIKKA, University of Helsinki and Stanford University
GABRIEL NUCHELMANS, University of Leyden
WESLEY C. SALMON, Indiana University
EVERT W. BETH / JEAN PIAGET
MATHEMATICAL
EPISTEMOLOGY AND
PSYCHOLOGY
Translated from the French by W. Mays
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
EPISTEMOLOGIE MATHEMATIQUE ET PSYCHOLOGIE
First published by Presses Universitaires de France, Paris
as Volume XIV of the' Etudes d' Epistem%gie Genbique'
ISBN 978-90-481-8328-9 ISBN 978-94-017-2193-6 (eBook)
DOI 10.1007/978-94-017-2193-6
All rights reserved
No part of this book may be reproduced in any form, by print, photoprint,
microfilm, or any other means without permission from the publisher
Softcover reprint of the hardcover 1st edition 1966
TABLE OF CONTENTS
IN MEMORY OF E. W. BETH (1908-1964) XI
TRANSLATOR'S INTRODUCTION XIII
FOREWORD XXI
EVERTW. BETH/PART ONE
PRELIMINARY 3
CHAPTER I. MATHEMATICAL REASONING CANNOT BE
ANALYSED BY TRADITIONAL SYLLOGISTICS 6
1. Descartes 6
2. The Locke-Berkeley problem 7
3. Solutions of Berkeley, Hume and Kant 8
4. Analytic and synthetic judgments 12
5. The intuitionism of Descartes and Kant 14
6. Non-Euclidean geometry 17
7. Recent forms of intuitionism: F. A. Lange,
L. Brunschvicg, E. Goblot, H. Poincare, L. E. J. Brouwer 18
CHAPTER II. THE PSYCHOLOGICAL INTERPRETATION OF
MA THEMA TICAL REASONING 24
8. J. Stuart Mill 24
9. W. Stanley Jevons' critique 25
10. E. Mach, Th. Ziehen, G. Storring and G. Heymans 26
11. The supposed anti-psychologism of E. HusserI 30
12. F. Enriques and G. Mannoury 33
v
MATHEMATICAL EPISTEMOLOGY AND PSYCHOLOGY
CHAPTER III. THE LOGICIST TRADITION
13. Aristotle's views: agreement with the practice of Greek
mathematics 36
14. Pascal 38
15. Leibniz: demonstration of axioms 39
16. Frege: influence on HusserI and Heymans 41
17. Russell: the crisis of foundations 42
18. The Set Theorists: Cantor and Zermelo 44
19. Other reactions: the intuitionism of Brouwer, the
psycho1ogism of Mannoury and Enriques, the radical
formalism of Hilbert 46
20. The GOde1ian crisis 53
21. Natural deduction: Gentzen, Curry, Lorenzen 59
22. Syntax and semantics 68
23. The method of semantic tableaux 71
24. Algebraic and topological concepts 81
CHAPTER IV. STRICT DEMONSTRATION AND HEURISTIC
PROCEDURES 86
25. The typology of mathematicians 86
26. Views of Poincare, Hadamard, Po1ya 87
27. Search for a method which is both heuristic and
demonstrative: Descartes and the analysis of the Ancients 93
28. Leibniz and the decision problem 95
29. Persistence of more primitive levels: Archimedes' method 96
30. Original thought: creation or invention, construction or
discovery? The Platonist reply: Frege, Cantor and
Hermite 98
CHAPTER V. INTUITIVE STRUCTURES AND FORMALISED
MATHEMATICS 101
31. Spatial intuition: Kant, Helmholtz, F. Klein, Nicod,
Whitehead and Tarski 101
32. Temporal intuition: Kant, Bergson, Brouwer and
De Groot 105
VI
T ABLE OF CONTENTS
33. Finitist intuition according to Hilbert and the intuition of
the infinite 108
34. Platonism as a real or illusory intuitive vision: the
nominalist critique 111
CHAPTER VI. "THINKING MACHINES" AND
MATHEMATICAL THOUGHT 114
35. Formalisation and the construction of a "thinking
machine" 114
36. The construction of a "thinking machine" presupposes the
solution of a decision problem 115
37. The irreducibility of the "leap from the end to the means"
according to Brouwer 118
38. Recursive functions: unsolvable problems, absolute
unsolvability 119
39. The two degrees of freedom of mathematical thought:
solving a problem and setting a problem 123
40. Acquired self-evidence according to Bernays 124
NOTE ON THE IDEA OF A "THINKING MACHINE"
by Jean-Blaise Grize 127
JEAN PIAGET/PART TWO
PRELIMINARY 131
CHAPTER VII. LESSONS OF THE HISTORY OF THE
RELATIONS BETWEEN LOGIC AND PSYCHOLOGY 137
41. The three stages of the history of the relations between
logical and psychological investigations 137
42. The need for co-ordination 143
43. The genetic viewpoint and the normative viewpoint 153
CHAPTER VIII. GENERAL PSYCHOLOGICAL PROBLEMS OF
LOGICO-MATHEMATICAL THOUGHT 163
A. The problem of structures 163
44. Bourbaki's "matrix structures" 164
VII
MATHEMATICAL EPISTEMOLOGY AND PSYCHOLOGY
45. The structures of classes and relations in the subject's
actions and operations. The formalisation of a "grouping" 166
46. The two forms of reversibility (inversion and reciprocity)
and their final combination in a group of four
transformations 175
47. The primacy of topology in the child's geometry 183
48. Relations between the three elementary structures and
Bourbaki's matrix structures 186
CHAPTER IX. GENERAL PSYCHOLOGICAL PROBLEMS OF
LOGICO-MATHEMATICAL THOUGHT (Continued) 191
B. Self-evidence, intuition and invention 191
49. Self-evidence, its variations and logical necessity 191
50. Invention and discovery 198
51. The multiple forms of mathematical "intuition" 208
CHAPTER X. THE PSYCHOLOGICAL PROBLEMS OF "PURE"
THOUGHT 226
52. The genetic roots of pure mathematics 226
53. The psychological problem of pure mathematics 242
54. The psychological reasons for formalisation 247
55. How a formalisation of ordinary thought brings together
the genetic and axiomatic methods 256
CHAPTER XI. SOME CONVERGENCES BETWEEN FORMAL AND
GENETIC ANALYSES 259
56. The construction of natural numbers 259
57. The difficulties of logical reductionism 272
58. The limits of formalisation 276
CHAPTER XII. EPISTEMOLOGICAL PROBLEMS WITH LOGICAL
AND PSYCHOLOGICAL RELEVANCE 281
59. Empiricist interpretation and apriorism 281
VIII
TABLE OF CONTENTS
60. The nominalist or linguistic interpretation of
mathematics 285
61. The Platonist interpretation of mathematics 290
62. The interpretation of mathematics by the laws of the
general co-ordination of actions 296
GENERAL CONCLUSIONS 305
by Evert W. Beth and Jean Piaget
BIBLIOGRAPHY 313
NAME INDEX 318
SUBJECT INDEX 321
IX
IN MEMORY OF E. W. BETH (1908-1964)
It is with great regret that I begin this volume by recalling to the reader
the premature death of E. W. Beth. He had taken great pleasure in this
English translation, the excellent work of W. Mays, and had looked
through the text with his usual thoroughness. E. W. Beth was a great
logician, who was also extremely well acquainted with all aspects of the
history of logic and its connections with other disciplines, and this
essentially determined his epistemological position. At the beginning of
the first part of this book he relates how, starting from a position close to
Kantianism and from reflections on Mannoury's psycho-linguistics and
the psychological factors involved in Brouwer's intuitionism, he arrived -
by a kind of "intellectual conversion" - at an attitude more exclusively
formalist and logicist. When I made his acquaintance around 1950 he
was, in fact, mistrustful of everything connected with, or even in any way
evocative of, psychology.
It is thus well worth remembering the origin of the present work, for
it is to the credit of Beth and of his final position, which was charac
terised, as he says in his introduction, by "the constant effort to see every
point of view as a reasonable one" and by an appreciation of "the need
for a kind of doctrinal synthesis of the various trends of contemporary
thought". In 1950 I published a work on the operational mechanisms of
logic, which my publisher decided to call Traite de Logique: Beth criticised
it very severely in the journal Methodos. Father Bochenski, who had
requested this review, refused to publish my reply, which I then reduced
to a few lines, saying, in effect, that if two authors fail to understand each
other because their points of view are so divergent, the only way of
achieving some useful and objective result is for them to co-operate in the
preparation of a joint work, where the same data are investigated one by
one until a mutually satisfactory assimilation of their positions is reached.
It was along such lines that I wrote to Beth and invited him to participate
in various meetings, where we discussed the psychological formation and
development of certain elementary logical structures. It is a measure of
XI