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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 maintained since,
or do we construct them -- as the Dutch mathematician L. E. J. Brouwer influentially
proposed in the first half of this century, prompting Wittgenstein to return to
doing philosophy and occasioning G. H. Hardy's famous defense of mathematical
Platonism in A Mathematician's Apology? Was the nineteenth century German
mathematician Leopold Kronecker right in asserting that "God made the integers;
all else is the work of man," or are the integers themselves the free creation
of the human mind, as Einstein came to believe in his later years? Does mathematics
constitute a universal language that in principle permits human beings to communicate
with extraterrestrial civilizations, or is it merely an earthly language that
owes its accidental existence to the peculiar evolution of neuronal networks in
the brain of Homo sapiens? Does the external world obey mathematical laws,
or does it seem to conform to them simply because physicists have increasingly
been able to make mathematical sense of physical phenomena? Jean-Pierre Changeux,
an internationally renowned neurobiologist, and Alan Connes, one of the most eminent
living mathematicians, find themselves deeply divided by these questions.
In a wide-ranging series of conversations, Changeux and Connes
discuss the development of the human brain as a function of natural selection
and variation, debate the character of human intelligence (and the obstacles that
stand in the way of simulating, modeling, or actually reproducing by mechanical
means), dispute the reasons for the "unreasonable effectiveness" of
mathematics in explaining the physical world, and differ over the sources of mathematical
creativity. In an epilogue they go on to inquire into the relations of mathematics
and science to ethics, asking whether a code of human morality consistent with
what is known about the structure and function of the human brain can be devised,
and whether the "enlargement of human sympathies" hoped for by Darwin,
Kropotkin, and others may be given a natural basis. This vivid record of profound
disagreement, and, at the same time, passionate search for mutual understanding,
follows in the modern tradition of Poincare, Turing, Hadamard, and von Neumann
in probing the limits of human rationality and intellectual possibility. Why order
should exist in the world at all -- and why it should be comprehensible by human
beings -- is the question that lies at the heart of these remarkable dialogues.
Jean-Pierre Changeux is Director of the Molecular Neurobiology
Laboratory at the Institut Pasteur in Paris and holds the Chair in Cellular Communications
at the College de France. Among his works translated into English are Neuronal
Man: The Biology of Mind (Pantheon/Oxford). Alan Connes holds the Chair in
Analysis and Geometry at the College de France. Winner of the 1982 Fields Medal,
he is the author of Noncommutative Geometry (Academic), among other works.
Both Changeux and Connes are members of the French Academy of Sciences. |
