On Mathematics, Music and Autism

A discussion of research into the psychology of mathematicians, especially in relation to autism, and possible links with the psychology of musicians. Every science has its own culture and that of mathematics is quite distinctive. As Henri Poincare said ‘mathematics is the activity in which the human mind seems to take least from the outside world, in which it acts or seems to act only of itself and on itself, so that in studying the procedure of geometric thought we may hope to reach what is most essential in the mind of man.’ When Bertrand Russell asserted that ‘mathematics rightly viewed possesses not only truth but supreme beauty — a beauty cold and austere, like that of good sculpture ... supremely pure, and capable of a stern perfection such as only the greatest art can’ he was expressing an extreme view which applies, if at all, only to certain kinds of mathematics. Although Russell was writing a long time ago and expressing his personal feeling for the discipline there are modern mathematicians who would agree with him that mathematics has a special quality, different from that of any other kind of science. In fact many do not regard it as a science at all, rather as one of the arts, and research in the discipline as art for art’s sake. For the pure mathematician, the interest seems to lie in the mathematics itself, rather than its application to the real world. Poincare wrote about ‘the feeling of mathematical beauty, of the harmony of numbers and forms, of geometric elegance. This is a true aesthetic feeling which all real mathematicians know.’ The contemporary French mathematician Alain Connes explained that ‘exploring the geography of mathematics, little by little the mathematician perceives the contours and structure of an incredibly rich world. Gradually he develops a sensitivity to the notion of simplicity that opens up access to new, wholly unsuspected regions of the mathematical landscape.’ Psychologists, especially cognitive psychologists, have long been interested in mathematicians, and. mathematicians have long been interested in cognitive psychology. The Leipzig neurologist Paul Mobius (grandson of the mathematician, August Ferdinand Mobius), an enthusiast for the pseudoscience of phrenology, wrote about mathematicians in his book Die Anlage zur Mathematik. in the late nineteenth century. Just a century ago, Henri Poincare gave a famous lecture on the ‘Psychology of Mathematical Invention’ at a conference of psychologists in Paris, while about the same time Felix Klein ran a seminar on the subject at Gottingen. Hadamard’s well-known monograph The Psychology of Invention in the Mathematical Field [1] refers extensively to Poincare’s lecture during which reported extensively on his personal experiences. Poincare explained that he relied greatly on his unconscious mind. Hadamard’s investigations confirmed that many creative mathematicians also rely on intuition although perhaps not to the same extent. Cognitive scientists tell us that most thought is unconscious. They distinguish between verbal thinkers and visual thinkers. My impression is that many mathematicians think in pictures, for example geometers and mathematical physicists, but many others are more verbal thinkers, for example mathematical logicians, but no research has been done on this, as far as I know.