Princeton University Press
May 7, 1995
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 the 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 Alain 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, modelling, or actually reproducing it 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 relation 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). Alain Connes holds the Chair in Analysis and Geometry at the College de France. Winner of the 1982 Fields Medal, he is the author of Noncommunitative Geometry (Academic), among other works. Both Changeux and Connes are members of the French Academy of Sciences.