Conversations on Mind, Matter, and Mathematics

by Changeux, Jean-Pierre; Connes, Alain; Debevoise, M. B.

• ISBN13: 9780691004051
• ISBN10: 0691004056
• Format: Paperback
• Copyright: 1998-12-07
• Publisher: Princeton Univ Pr

Summary

Do numbers and the other objects of mathematics enjoy a timeless existence independent of human minds, or are they the products of cerebral invention? Do we discover them, as Plato supposed and many others have believed since, or do we construct them? Does mathematics constitute a universal language that in principle would permit human beings to communicate with extraterrestrial civilizations elsewhere in the universe, or is it merely an earthly language that owes its accidental existence to the peculiar evolution of neuronal networks in our brains? Does the physical world actually obey mathematical laws, or does it seem to conform to them simply because physicists have increasingly been able to make mathematical sense of it? Jean-Pierre Changeux, an internationally renowned neurobiologist, and Alain Connes, one of the most eminent living mathematicians, find themselves deeply divided by these questions.The problematic status of mathematical objects leads Changeux and Connes to the organization and function of the brain, the ways in which its embryonic and post-natal development influences the unfolding of mathematical reasoning and other kinds of thinking, and whether human intelligence can be simulated, modeled,--or actually reproduced-- by mechanical means. The two men go on to pose ethical questions, inquiring into the natural foundations of morality and the possibility that it may have a neural basis underlying its social manifestations. This vivid record of profound disagreement and, at the same time, sincere search for mutual understanding, follows in the tradition of Poincareacute;, Hadamard, and von Neumann in probing the limits of human experience and intellectual possibility. Why order should exist in the world at all, and why it should be comprehensible to human beings, is the question that lies at the heart of these remarkable dialogues.

Table of Contents

TRANSLATOR'S NOTE	vii	(4)
PREFACE TO THE ORIGINAL FRENCH EDITION	xi
1. MATHEMATICS AND THE BRAIN	3	(22)
Introductory Remarks	3	(3)
The Hierarchy of the Sciences in Question	6	(5)
Invention or Discovery?	11	(7)
Mathematics Has a History	18	(4)
Is Mathematics Merely a Language?	22	(3)
2. PLATO AS MATERIALIST?	25	(16)
The Intellectual Ascesis of the Materialist	25	(5)
The Psychoanalysis of Mathematics	30	(2)
Is a Mathematical Object a Cultural Representation Like Any Other?	32	(3)
The Darwinism of Mathematical Objects	35	(3)
Beliefs in Mathematics	38	(3)
3. NATURE MADE TO ORDER	41	(33)
Constructivist Mathematics	41	(6)
The "Unreasonable Effectiveness" of Mathematics	47	(4)
Einstein and Mathematics	51	(6)
The Usefulness of Mathematical Models in Biology	57	(7)
The Auscultation of Quantum Mechanics	64	(10)
4. THE NEURONAL MATHEMATICIAN	74	(48)
Illumination	74	(8)
The Brain and Its Multiple Levels of Organization	82	(8)
The Cellular Level	90	(7)
From Elementary Circuits to Mental Objects	97	(8)
The Neuropsychology of Mathematics	105	(2)
Transition among Levels by Variation-Selection	107	(9)
Mental Darwinism and Mathematical Creation	116	(6)
5. DARWIN AMONG THE MATHEMATICIANS	122	(31)
The Utility of the Darwinian Schema	122	(5)
Coding Stable Forms	127	(11)
The Organization of Long-Term Memory	138	(4)
Reasoning by Analogy	142	(1)
Linking Representations within Frameworks of Thought	143	(3)
The Natural Selection of Mathematical Objects	146	(7)
6. THINKING MACHINES	153	(26)
Are Intelligent Machines Possible?	153	(1)
Godel's Theorem	154	(8)
Turing's Thinking Machine	162	(3)
The Theory of the S-Matrix in Physics as Analogue to Functionalism in Psychology	165	(3)
Is the Human Brain a Computer?	168	(5)
A Self-Evaluating Machine that Can Suffer	173	(6)
7. THE REAL AND THE RATIONAL	179	(31)
The Nature of Mathematical Objects Revisited	180	(12)
The Construction of Mathematics by the Child	192	(4)
Order in the World	196	(14)
EPILOGUE: ETHICAL QUESTIONS	210	(27)
In Search of the Natural Bases of Ethics	210	(6)
Social Life and the Frontal Lobe	216	(3)
Prosocial Behavior in the Child and Its Cultural Imprint	219	(3)
The Functions of Morality	222	(2)
In Defense of a Revisable, Rational, and Natural Morality	224	(2)
The "Enlargement of Sympathy" and the Function of Aesthetics	226	(7)
Ethics and Mathematics	233	(4)
GLOSSARY OF MATHEMATICAL TERMS	237	(10)
GLOSSARY OF NEUROBIOLOGICAL TERMS	247	(6)
INDEX	253