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Henri Poincaré. A scientific biography. (English) Zbl 1263.01002
Princeton, NJ: Princeton University Press (ISBN 978-0-691-15271-4/hbk; 978-1-400-84479-1/ebook). xiii, 592 p. (2013).
This book was written to commemorate the centenary of the death of the universal French mathematician and physicist Henri Poincaré (1854–1912). The latter is acknowledged as one of the world’s two leading mathematicians around 1900, the other being David Hilbert of Göttingen. The author, the English historian of mathematics, Jeremy Gray, is a leading authority on the mathematics of Poincaré, with a deep knowledge of Poincaré’s physics and philosophy as well.
In the first part of the 20th century, Poincaré’s contributions were at times obscured by the fame of Hilbert, because the French mathematician followed the “Hilbert challenge” (as another book by Gray from the year 2000 is entitled) with respect to pursuing important single problems (such as the Poincaré conjecture in topology) and was not so imbued with the axiomatic and set-theoretic spirit which dominated the work of German and American mathematicians and later the young generation of French mathematicians influenced by Bourbaki. In the past fifty years or so, developments in dynamical systems, stability theory, nonlinear differential equations and mathematical physics have led to a reevaluation of Poincaré’s work which, however, had always been present and influential in the research of individual mathematicians such as the American George David Birkhoff, and particularly in the Russian school of nonlinear mechanics and dynamical systems, which had been less visible in the West.
Gray introduces his book on Poincaré as “the first full-length study covering all the main areas of his contributions to mathematics, physics, and philosophy” (p. 1). He calls the book a “scientific biography …confined entirely to his public life” (p. 2), announcing a full biography of Poincaré to be published by “a team of scholars at the Archive Henri Poincaré at the University of Lorraine” in his birthplace Nancy. Gray acknowledges the work of these scholars and often avoids using detailed quotes by globally (although rarely with page numbers) referring to the publications and the correspondence of Poincaré, almost completely provided online by the Archives in Nancy.
In eleven chapters of a voluminous, but – in view of Poincaré’s versatility – still slim book of 592 pages, the author reserves about a quarter for Poincaré “The essayist”, while another 50 pages or so are devoted to “Poincaré’s career”, the latter being the most personal of the chapters. This leaves two thirds of the book for the discussion of the “hard facts” of Poincaré’s mathematical and physical work. Indeed, the book seems primarily written for mathematicians and physicists, although it contains in the Appendices some explanations of technical terms such as elliptic and abelian functions, and of Maxwell’s equations. However, given the sheer extent of Poincaré’s oeuvre, the high percentage of verbal argumentation in Poincaré’s own work, and due to the possibility of referring to online resources, Gray’s text is not overly technical but highly analytical-historical. The author, while clearly aware of the use of Poincaré’s results in modern mathematics, does not aim at following all lines of Poincaré’s heritage leading into present research. He deliberately avoids a detailed discussion of hot topics such as chaos theory, fractals, and the recent solution of Poincaré’s conjecture in topology, and he rather focusses as a historian on how fundamental notions and methods such as genus of a surface and the sweeping out method (balayage) in potential theory, flows on surfaces or limit cycles (not all of these notions are included in the subject index) occurred or originated in Poincaré’s work on topology, nonlinear differential equations and celestial mechanics. In the chapter on algebraic topology, Gray points to Poincaré’s important work on what he called order of connectivity of surfaces (soon to be replaced by genus) and on the relations to group theory, but does not discuss in detail implications for later developments of topology in the German school or in the work of Brouwer. Gray gives, however, much attention to the contemporary mathematical and philosophical discussion, for instance with respect to Poincaré’s relationship to Cantor’s then new theory of sets. Both Poincaré’s philosophical reservations and his early use of these new notions and methods are analyzed by Gray in a nuanced manner.
Two famous prize competitions stand out, which founded Poincaré’s fame, although he was only mentioned honorably for the first and won the second despite a significant mistake. The 1880 Paris Academy competition on linear differential equations led to Poincaré’s discussion of a class of complex functions which he named after the German Lazarus Fuchs. This attribution met with resistance by Felix Klein in Germany who was a competitor of Karl Weierstrass’ student Fuchs. In the following years Klein entered into a tense competition with Poincaré on the uniformization of Riemannian surfaces and automorphic functions, a battle he lost and which led to a premature end of his career as a research mathematician. It was Poincaré’s important and original, psychologically startling discovery in 1880 that the generalized periodicity of “Fuchsian” functions could be best understood by looking at connections with non-Euclidean geometry, which, according to Gray, “was to make Poincaré’s name among mathematicians” (p. 217). It is in his discussion of the prize competition of 1880 that Gray makes extensive use both of his book [Linear differential equations and group theory from Riemann to Poincaré. Boston-Basel-Stuttgart: Birkhäuser (1986; Zbl 0596.01018)] and of his later discovery and edition (together with S. Walter in 1997) of unpublished supplements to the prize essay, which, among other things, showed that Poincaré as late as 1880 was not aware of Riemann’s mapping theorem. The other famous prize competition which Poincaré entered and finally won was announced in 1885 by the Swedish King Oscar II, a former student of mathematics. The formulation of the competition, which apparently was due to Gösta Mittag-Leffler, the founder of Acta Mathematica (1884) and admirer of Poincaré’s, was directly related to several of Poincaré’s results. Poincaré’s prize winning essay dealt with the restricted three-body problem and contained among other things his famous recurrence theorem. Gray discusses in detail both the original (unpublished) prize essay of 1889 and the published extensive 270-pages version in [Acta Math. XIII, 1–270 (1890; JFM 22.0907.01)], which contains corrections concerning asymptotic surfaces, with the mistake basically having been found by Poincaré himself.
Gray’s book contains further substantial chapters on “Cosmogony” (rotating fluid masses) and on what he calls “Inventions in pure mathematics” (number theory, Lie theory, algebraic geometry). With the latter subtitle Gray apparently alludes indirectly to the inextricable links of much of Poincaré’s work with mathematical physics, which are discussed in detail in the chapters “Physics” (mainly electricity and relativity), “Theory of functions and mathematical physics” (mainly potential theory) and “Poincaré as a professional physicist” (mainly thermodynamics and probability). These chapters contain many fascinating topics and results from recent research done by historians of physics about the relationship (or rather non-relationship) between Poincaré and Einstein, including a discussion why Poincaré finally did not win a Nobel Prize in physics in spite of support by influential physicists and mathematicians.
The above discussion leads of course to Poincaré’s role as a philosopher of science, among other things to his conventionalist position which was very influential at the time and not least propagated by many of Poincaré’s widely published semi-popular presentations on “Science and method” and similar titles. Gray devotes to these questions a short, separate and concluding chapter where he partly draws on a recent book on Poincaré’s philosophy by E. Zahar [Poincaré’s philosophy: from conventionalism to phenomenology. Chicago, IL: Open Court (2001; Zbl 1097.00005)]. However, Gray reflects about Poincaré’s opinions about the relation of space and time and similar topics throughout the entire book. This is very much the case in the very first and most extensive chapter of all, “The essayist”, where the reader is so impressed by the author’s erudition and immersion into Poincaré’s spirit that he sometimes has problems distinguishing between Poincaré’s and Gray’s opinions. More direct quotes from Poincaré’s essays would have helped here and would also have given an impression of the mathematician’s attractive literary style.
An impressive, voluminous and multifaceted book like this, which has very few typographical errors, may nevertheless contain some points to be polished. Gray calls Poincaré’s article “Hazard” (1907) “lightweight”, an article which, however, according to B. Bru [“Souvenirs de Bologne”, J. SFdS 144, No. 1–2, 135–226 (2003)] is an important link to the second and revised edition of Poincaré’s book on probability (1912). Given that George D. Birkhoff as Poincaré’s important American heir is discussed in various places, it would have been helpful to stress that the so-called Birkhoff-Witt Theorem (1937) on the structure of Lie algebra is related to work by Birkhoff’s son Garrett. Gray, using historical research by T. Ton-That and T.-D. Tran [Rev. Hist. Math. 5, No. 2, 249–284 (1999; Zbl 0958.01012)], shows the theorem to have been proven by Poincaré himself already.
Jeremy Gray’s book on Poincaré’s mathematics, physics, and philosophy is an important contribution to the literature and a huge step towards a full biography of this pioneer of modern science.

01-02 Research exposition (monographs, survey articles) pertaining to history and biography
01A55 History of mathematics in the 19th century
01A60 History of mathematics in the 20th century
01A70 Biographies, obituaries, personalia, bibliographies
Biographic References:
Poincaré, Henri