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Pour l’honneur de l’esprit humain. Les mathématiques aujourd’hui. (French) Zbl 0703.00035

Histoire et Philosophie des Sciences. Paris: Hachette. 298 p. FF 178.00 (1987).
This solid book, written by a master-mathematician of our times, comprises a vast amount of historical and actual problems of mathematics. A lot of technical appendices illustrates notions such as Galois group, Fourier series, cardinals etc.
Chapter III discusses predominantly classical Euclidean geometry, chapter IV number theoretic problems of 17th until 19th centuries.
The greater part of the book deals with the changes, caused by the rise of mathematical structures in the 19th century (ch. V, 113-210) and with “Problems and pseudoproblems of the ‘foundations”’ (ch. VI, 211-264). A short survey of modern mathematics (164-176) and a biographical appendix (267-288) are given.
The first two chapters deal with the work of the mathematician and with the nature of mathematical problems, especially in relation to science. In his introduction (9-13) the author states a widespread ignorance of the public with regard to the profession of the mathematician. The twofold aim of this book is to investigate the reasons for this ignorance and to contribute to the latter’s dissipation.
The author considers “intellectual curiosity” as the main stimulus of mathematical work (39). Only 30-40 % of mathematics would stem from the needs of applications. The natural consequence would be a relative isolation of mathematics. No mathematics that is taught at schools was done after 1800. (What about set theory?). Almost no scientist (except some theoretical physicists) would use mathematics, which was created after 1860 (9). A mathematician would need a sheet of paper and a good library only (21). These strong thesises are weakened by the author himself hinting on various applications of probability theory (196) and on the use of computers in the proof of the four-colour-theorem.
Contrary to the author’s opinion the book seems to go to much into the details for nonmathematicians. The cited book of S. Lang’s “The beauty of doing mathematics” (New York 1985; Zbl 0574.00015) seems unequalled in this respect.
It is a stimulating book for students of mathematics and its history. For this audience the bibliography should have been more specific however. E.g. the author deals with the notion of mathematical strength discussing Euler’s polyhedron formula (249) without mentioning I. Lakatos classical book.

MSC:

00A30 Philosophy of mathematics
01A80 Sociology (and profession) of mathematics
01A45 History of mathematics in the 17th century
01A50 History of mathematics in the 18th century
01A55 History of mathematics in the 19th century
01A60 History of mathematics in the 20th century

Citations:

Zbl 0574.00015
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