Mind, Matter, and Mathematics

There are those who hold that there is nothing else but Mind and there are those who think there is nothing else but Matter. Those who hold these views often tend to be very certain that they are right and that the others are wrong. In this case right and wrong often become moral as well as logical judgements. Faced with such conflicting certainties it is tempting to call a plague on both houses. No matter! Never mind! 1 But it does matter and I, for one, do mind. So rather than sit on the fence between these cartesian dualities, it may be worth seeking a third mediating term. The same issue is often found in the case of Mathematics. The problematic dichotomies re-appear as soon as we ask questions about the nature of mathematical reality. There are those who hold that mathematics is constructed in the inner world of the mind and there are those who think that it is found in an outer world of matter. Those who hold these views often tend to be very certain that they are right and that the others are wrong. In this case right and wrong often become moral as well as logical judgements. Faced with such conflicting certainties it is tempting to call a plague on both houses. Neither constructivism! Nor empiricism! But these points of view are both in some way indispensable, they cannot be arbitrarily discarded and many people find it difficult to choose one at the expense of the other. So rather than sit on the fence between these cartesian dualities, it may be worth seeking a third mediating term. A mediating third is often found in an independent world of ideas. It is not surprising, then, that many perhaps most professional mathematicians see themselves as platonists (at any rate, some of them say, on weekdays). This point of view is rejected (often quite intemperately) by those mathematics educationists who are currently taken by some form of constructivism. It would also be rejected by those who take an empirical point of view, for example by those neuroscientists who have in recent years become very interested in how and where mathematical activity occurs in the brain. Their approach tends to link mind and matter in that they seek to find a physical description of mental activity in terms of electrical and chemical activity in neurons and synapses. My current interest in these issues and indeed the title I chose for this review article derives from a remarkable book. Conversations on mind, matter, and mathematics 2, which translates some conversations between Jean-Pierre Changeux, a neuroscientist author of an important book on the biology of the mind, and Alain Connes, a mathematician and Fields medallist. The background question is whether a machine could reproduce the activity of the brain. This would involve knowing more about the way the brain works. Since mathematical activity occurs in the brain, neuroscientists are interested in the nature of mathematical activity. Hence this interdisciplinary dialogue which ranges very widely and provides some exciting cut and thrust of argument, with at tim s the feel of a genuine conversation, a meeting of minds. I cannot here cover all the fascinating issues they raise. In general, the speakers keep to one r two ce tral themes. The opening third of the book addresses the n ture of mathematical objects, about which they take opposing sides. The central third discusses a neurobiological point of view. The final third consists of a chapter on "thinking machines", another chapter tha was dded on for he English edition in order to tie up some loose ends, and a final digressive epilogue almost entirely a onologue by Changeux on some ethical questions. There are throughout various other digressions into technical matters (which are sometimes too condensed and difficult to follow despite the provided glossaries of biol gical and ma hematical terms) but the conversation always returns to the way in which mathematics might be held to mediate between mind and matter. In this article, I give a brief account of some highly selected passages of a book which I commend to anyone also interested in its main themes, and I end with some further comments of my own.