Out there or in here?

Published in France in 1989, under the title *Matière à pensée*, to admiring reviews, Conversations on Mind, Matter, and Mathematics is the record of an intellectual encounter between two of France’s leading scientific figures. The book has been rendered into English with the conversational warts of the original – elisions, abrupt transitions and fragmentary remarks – removed; it contains some two dozen explanatory diagrams and is supplied with an afterthought written for this edition. The result is a smooth, easy-to-read representation of a protracted, interesting but ultimately frustrating exchange between Alain Connes, an eminent mathematician at the Collège de France and Fields medalist for his contribution to topology, and Jean-Pierre Changeux, a distinguished biologist known for his work on Darwinian models of neuronal development and the Director of the Neurobiology Laboratory at the Institut Pasteur.

The customary formula for a debate, a series of confrontational, seemingly irreconcilable alternatives, is followed here. Do numbers and other mathematical objects enjoy a timeless existence outside the human mind, or are they produced by human brains? Do we discover them and their properties, as many believe, or do we invent them? Does mathematics constitute a universal language that would be the natural basis for communicating with extra-terrestrials, or is it a local construction peculiar to the neuronal characteristics and evolutionary history of life on earth? Doe the physical universe obey mathematical laws, as the tradition from Galileo onwards assures us, or is the overwhelming presence of mathematics in physics the result of the scientific practice of insisting we make sense of the world using mathematics?

These questions are old chestnuts of debates about mathematics, and anyone reasonable familiar with such discussions will recognize them as different versions of the Platonism/constructivism stand-off. They are, of course, no less interesting for that, and Changeux and Connes go at them with a great deal of energy and a certain freshness that comes, perhaps, from no having been over-exposed to too many past debates. They chase them over various terrains, from the psychology of mathematical invention that goes back to the idea of Hadamard, Poincaré and Changeux that mathematical objects might be subject to quasi-Darwinian mechanisms of selection, through the “auscultation” of quantum mechanics, the multiple levels of organization of the brain, models for the working of long-term memory, Gödel’s theorem, Turing machines and the theory of the S-matrix in physics “as analogue to functionalism in psychology”, to whether the brain is a computer and whether it is necessary – in order to answer such a question – to be able to construct a “self-evaluating machine that can suffer”. The exigencies of conversation enforce directness, and this, together with the particular skills of the participants here, ensure that, in spite of the abstruse and specialized nature of much of the ground they cover, their exchange only rarely becomes off-puttingly technical or inaccessible.

Though it is clearly a two-hander, Changeux dominates the dialogue: he appears to have set the agenda, initiates most of the discussion, is the one who brings in what philosophical, historical and socio-cultural layers there are in to the proceedings, and writes the epilogue. On the whole, Connes’s role is that of (an ultra-intelligent, narrowly focused) responder; he is the one-who-knows, whom Changeux can challenge and interrogate. This imbalance seems natural: mathematics the older, more arcane, difficult and abstract discipline being put under the materialist microscope of scientific neurology; and it has distinct advantages for the non-mathematical reader, for whom Changeux is a tireless proxy, asking Connes to explain this or that aspect of mathematical thought or elaborate on how mathematicians imagine, feel and carry on the way they do (how do you think topology?), or trying as a fellow scientist to get Connes to suggest how mathematics might tackle the difficult neurological questions about the nature of logic and thought that exercise him (how do you topologize thinking?) The extent to which such readers (as well as more numerate ones) will be satisfied with the answers Connes gives – not to mention the form of Changeux’s questions – is another matter. For it soon becomes evident that Changeux and Connes are figures whose prior metaphysical attachments make it virtually impossible for them to engage in a common space.

The problem is that, like the majority of mathematicians. Connes is an unregenerate, won’t-budge-an-inch, Platonist: “I hold that…there exists, independently of the human mind, a raw and immutable mathematical reality”, is one of his opening statements which gets repeated and defended throughout the dialogue, while Changeux never departs from, no recognizes the limits of, a materialist-based mentalism – constructivism he calls it – according to which “mathematical objects exist materially in your brain”. The result is a pretty unilluminating replay of the constructivist/realist divide: mathematics is “out there”, waiting to be discovered (Connes), mathematics is “in here”, constantly being invented (Changeux), and so on.

Whenever Changeux points to anything external to mathematics that might have or might have had an impact on the subject, Connes counters by insisting on the separation between tools for the apprehension of the “archaic reality” of mathematics – which he admits are socially “tainted” – as opposed to the pure, immutable nature of the reality itself. As somebody (one of many) who has battled against the self-serving incoherence of mathematical Platonism. I can readily identify with Changeux’s repeated incredulity and ultimate frustration with Connes’ s metaphysics. How can you be a materialist, he more than once asks Connes, and maintain the sort of belief you have in some mysterious, transcendental realm of objects? How indeed. Nevertheless, as a mathematician, I can sympathize with Connes’s refusal to go along with what he sees as Changeux’s reductionism – which he likens to studying ink patterns to illuminate Shakespeare’s plays – whereby mathematical objects are identified with the neural activity that undoubtedly accompanies the thinking of them: Frege long ago made that kind of psychologim impossible to maintain.

All this is not to say that the discovery/invention of mathematical objects, the universality/restricted ness of mathematics as a language, or the puzzle of whether the physical universe is mathematical or is made by us to be so, are not vital questions of great contemporary relevance. Far from it; they are indeed important matières à pensée. What is clear, however, however, is that the terms in which they are asked have to go beyond those in evidence here, if anything more than an unprofitable impasse is to emerge. The simple and inescapable truth is that mathematics, before anything else (before its claims to “truth”, its status as play, its utility), is a business with signs. This means that, like any such, it is accompanied by and generates mental activity. So one can agree with Changuex and others that mathematics is a species of mind game that takes place in the head. But this is only part of the story. What appears as a private game is at the same time a public language. Mathematics is manipulative sign practice, an intersubjective act of imagination neither reducible to mentalism (purely neurological phenomena) nor to a metaphysically realist science (the study of transcendental, pre-human objects). In other words, mathematical signs are objective and subjective from the beginning. However, as is abundantly evident, mathematicians will not relinquish their Platonist understanding of this intersubjectivity as a pure, presemiotic objectivity easily; and certainly not unless an explanation is given to them as to why believing it is so natural and obviously correct (but illusory) an element of their practice. Whether it is possible so to convince them or whether the theology itself, the assurance of being in touch with immutable and eternal truth and beauty, is what they really want and what they go to mathematics for, is another story.