One of the most interesting developments in the history of mathematics in the last decade has been the renewed appreciation of David Hilbert’s work on physics. No longer can the great algebraist, number theorist, and driver for new foundations of mathematics be seen shorn of his abiding interest in physics, upon which he lectured for 30 years and to which he made many original contributions. This splendidly edited and rich volume sets a remarkable store of Hilbert’s ideas before the public, and should do much to correct a narrow picture that grew up in the 1970s, 1908s, and 1990s.
Here, for example, will be found his work in 1915 on general theory of relativity, which has provoked an at times ill-informed debate about exactly what Hilbert and Einstein knew and when, together with a clear commentary setting out what can actually be said with confidence. But Hilbert worked on much else: statistical mechanics, the mathematical formulation of quantum mechanics, and foundational issues in physics in the aftermath of quantum theory and relativity. All these interests are represented here. We are given, for example, Hilbert’s lectures on physics of 1916–1917. Here Hilbert began with an axiomatic treatment of the kinematics of special relativity, with an eye to the dynamics of the electron (here treated as a rigid electron along the lines of Max Abraham’s theory, modified by Hilbert’s students Erich Hecke and Wilhelm Behrens, and contrasted with Gustav Mie’s approach). In the second part of the course Hilbert developed then generally covariant extension of these ideas. This involved him in describing two-, three- and four-dimensional differential geometry. Then he turned to matters of current research: deriving the field equations, the empirical status of Minkowskian space-time, the Schwarzschild metric and its singularities, and the integration of the theories of electro-magnetism and gravitation.
Another chapter gives us, inter alia, Hilbert’s Bucharest lectures of 1918 on the general theory of relativity. Another gives us Hilbert’s lectures of 1921 entitled Natur und mathematisches Erkennen (this is not the same object as the lectures Hilbert gave in 1919 with the same title, that were edited by David Rowe in 1992, as the editors might have made clear) and his Hamburg lectures of 1923. In these lectures Hilbert discussed epistemological questions, the scope of the laws of physics, causality, and the possibility of deducing the laws of chemistry from those of physics. The final two chapters cover radiation and quantum theory, and quantum theory itself. The second lot of lectures, given in 1926–1927 reflect the new quantum mechanics of Heisenberg, Born, Schrödinger, and Dirac. Is the editors quietly note (p. 505): “It is not by chance that in modern text books quantum mechanics is formulated in terms of abstract ‘Hilbert spaces’.”
The book and each of its six chapters have excellent introductions by the editors, Tilman Sauer and Ulrich Majer, and in a book of nearly 800 pages they have taken the sensible decision to leave Hilbert’s text in its original German. That will perhaps reduce its readership, but they have judged that the best (a translation) would have been the enemy of the good (a well edited book that actually exists) and for choosing the good, and much else, we should thank them.
Jeremy Gray is a 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 Plato’s Ghost: The Modernist Transformation of Mathematics.