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Ernst Zermelo: Collected Works II: Calculus of Variations, Applied Mathematics and Physics

Ernst Zermelo, edited by Hans-Dieter Ebbinghaus and Akihiro Kanimori
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Schriften der Mathematisch-naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften 23
[Reviewed by
Jeremy J. Gray
, on

This is the second and final volume of the collected papers of Ernst Zermelo. Today he is probably best remembered for his account of the axiom of choice and his axiomatisation of set theory, which was corrected and refined in the 1920s to become the Zermelo-Fraenkel axioms for set theory. His axiomatisation of 1909 and his subsequent papers on set theory (and related topics) form volume 1, published in 2010. The new volume, edited by Heinz-Dieter Ebbinghaus and Akihiro Kanamori with several assistants and published in 2013, includes his papers on analysis and mathematical physics, most of which were published before he turned to set theory. They are less known, but they are far from insignificant.

Hans Ebbinghaus opens volume 2 with a curriculum vitae of Zermelo, who began his career by writing a PhD on the calculus of variations under the supervision of H. A. Schwarz. He then wrote two papers criticising Ludwig Boltzmann’s statistical theory of heat, before applying to write his Habilitation thesis under Felix Klein in Göttingen. Once there he was supported by David Hilbert, and he moved into a study of set theory. But by 1914 his life was increasingly marred by ill health, and a mixture of achievements and setbacks followed until he became blind in the late 1940s and died in 1953, at age 81.

The dispute with Boltzmann was fuelled by Zermelo’s appreciation of Poincaré’s recurrence theorem. Jos Uffink gives a clear account of the conflicting approaches of Boltzmann and Planck to statistical mechanics, and then discusses how Zermelo argued that Poincaré’s theorem showed that the theories of mechanics and (Boltzmann’s) thermodynamics were incompatible. Boltzmann naturally disagreed, and tried to move the dispute into one about rare initial states of a gas, but Zermelo was not persuaded, and Uffink argues that in the end the two men were talking past each other. A likely verdict would be that more precise interpretations of what Boltzmann said are vulnerable to Zermelo’s criticisms, but that a deeper insight into statistical mechanics would vindicate Boltzmann’s intuitions.

Other papers by Zermelo in the early period of his career and again in the early 1930s deal with hydrodynamics, geodesics, the calculus of variations, and aerodynamics (inspired, it seems, by the flight of a Zeppelin).

The two handsome volumes of Zermelo’s Collected Works are well edited and richly informative. The individual papers are republished in the original with an English translation on facing pages. The result is a fine representation of the life and work of Ernst Zermelo.

Jeremy Gray is Professor of the History of Mathematics at the Open University, and an Honorary Professor at the University of Warwick, where he lectures on the history of mathematics. In 2009 he was awarded the Albert Leon Whiteman Memorial Prize by the American Mathematical Society for his work in the history of mathematics. His latest book is Hidden Harmony — Geometric Fantasies: The Rise of Complex Function Theory, written with Umberto Bottazzini.