The author, a mathematician, recounts his journey to the “heart of his brain”, where he describes his understanding of mathematics. Grothendieck admitted that he had trouble understanding why people do not understand math: according to him, and David Bessis, it is because they have lost their confidence as children, so that their understanding is hampered by the fear of failing. .
It is true that the language of mathematics is off-putting: the definitions are arid before one distinguishes their scope! Mathematics is not innate, but we would all be able to grasp it if we had confidence in our abilities. If learning to ride a bike is more or less difficult, few people fail and the pleasure of riding is a reward. Similarly, it is a very powerful spiritual experience to know that there are many kinds of infinities, and understanding Cantor’s diagonal proof requires attention but little mathematical knowledge.
The author testifies to the possibility of making “emerge from an apparent nothingness of impalpable mists” an understanding or even a mathematical discovery. In his experience, the sensation of “seeing” a mathematical property is like the sensation of a red color in the head. He can visualize during daydreams guided by intuition four or five dimensional spaces. The author obviously does not reject the need for mathematical definitions: a definition, the birth certificate of a new word, is “the guide for the precise assembly of a new mental image”.
Reading David Bessis’ book is a beautiful journey in mathematical thought. It would be interesting to know if his personal experience is generalizable. A symposium on experiments in mathematical creation would be fascinating.
