Eléments d'histoire des mathématiques

Description

PDF pp. 308ff. are on Lie groups and algebras

This fascinating volume assembles the historical notes from Bourbaki's various Éléments. The first edition was reviewed [Hermann, Paris, 1960; MR0113788], as was the Italian edition [Feltrinelli, Milan, 1963; MR0194304] and the second edition [Hermann, Paris, 1974; MR0349314]. The present volume is a reprint of the 1974 edition, with no apparent change except for the welcome addition of a table of contents. The bibliography is full, as is the index. However, page numbers beyond 300 in the index may need to be jogged upwards; Marczewski p. 304 to p. 306, Lazard 326 to 332, Frobenius 323 to 328, etc.
In virtue of its origin as appendices to separate texts on topics of current interest, these elements of history are just that: Former mathematics as it seems now to Bourbaki, and not as it seemed to its practioners then. In the terminology of historiography, it is "Whig history''. But for mathematicians, the various chapters are full of interesting connections and insights—Riemann's decisive influence on the origins of topology (analysis situs), the idea that various sets naturally have more than one topology, the fact that Leibniz had the notion of isomorphism, but that it was Emmy Noether who straightened up the terminology. On the way to these insights, there may be some minor annoyances. Thus the Gibbs-Wilson vector analysis (1900), which has dominated notation in Anglo-Saxon physics for 85 years, is dismissed as a vulgarization of the ideas of Hamilton and Grassmann—despite the fact that Gibbs understood the tensor product of vector spaces well before Bourbaki. The chapters, some short and some long, cover many topics, but not complex analysis. After all, Bourbaki started in revolt against the earlier French enthusiasm for this topic.
The long introductory chapter on foundations, logic, and sets is very short on logic (except for praises for the ineffective work of Leibniz) and long on the French. Thus the French semi- intuitionists, Borel, Baire, Hadamard, and Lebesgue get a full page on a par with the much more influential Principia mathematica of Whitehead and Russell. The idea that logic involves rules of inference is well hidden by some discussion of metamathematics. The section on the solution of the paradoxes emphasizes the present day favorite, solution by Zermelo's axioms, but not Russell's type theory, published in the same year, 1908. It is argued with considerable reason that set theory arose from the necessities of analysis, but the additional origins of set theory in philosophy (classes and properties) are downplayed. These are examples of Whig history. Despite such minor points, this book is splendid reading.

Reviewed by Saunders Mac Lane