This book, written by a professor emeritus of mathematical physics, is a bold attempt to describe specific features of mathematical thinking. It presents and elaborates many methodological and philosophical problems connected with this frequently posed and until now unsolved problem. It aroused considerable attention immediately after its publication and critical reviews and discussions are still running. There was a criticism concerning the choice of the word brain instead of mind in the title but Ruelle’s Johann Bernoulli lecture presented in Groningen in April 1999 explains his position, namely the comparison of PC and brain activities. The scope of the book is “to present a view of mathematics and mathematicians that will interest those without training as well as the many who are mathematically literate”. An investigation of the activities of a human brain can be found in the Greek and Renaissance periods and the names and results of Leonardo da Vinci, Descartes, Newton and Galileo Galilei are mentioned and discussed in the book. However, Ruelle’s attention is primarily focused on personalities that directly influenced mathematics of the 20th century and, consequently, B. Riemann (in particular his conjecture concerning primes), F. Klein (Erlangen program), G. Cantor, D. Hilbert and K. Gödel are frequently recalled throughout the book, and beside the life works also the life stories of A. Turing and A. Grothendieck are described at length.
Ruelle also tries to answer many philosophical problems connected with mathematics and human thinking. However, a serious critique of his conclusions was heard just from philosophical circles and some of the reproaches are worth mentioning. The thesis that “the structure of human science is dependent on the special nature and organization of human brain” is questionable because of our limited knowledge of the brain’s activities. Ruelle’s conclusion that “mathematics is the unique endeavour where the use of a human language is, in principle, not necessary” is hardly correct. Also his link to previous investigators of the same theme, in particular to H. Weyl (Philosophy of Mathematics and Natural Science, 1949) and Saunders Mac Lane (Mathematics: Form and Function, 1986) is insufficient. Perhaps also Ruelle’s inclination to physics and the way of thinking in physics was an obstacle to attaining more general and correct conclusions. However, all referees praise the book for its ability to present the exceptionally creative ways of mathematical thinking as well as their impact on mathematicians’ lives and they recommend it as a daring attempt to elucidate the uniqueness of mathematical thinking in comparison to other scientific activities as well as the similarities in the behaviour of all scientists.