The book under review is a compendium of articles on various themes having to do with our post-Einsteinian world, collected into four parts titled, respectively, “At the limits of the classical worldview,” “Contexts of the relativity revolution,” “The emergence of the relativistic worldview,” and “A new worldview in the making.” Most of the articles have a solid scientific orientation (but not really a mathematical one), and there are some that would indeed try the patience of the average mathematical reader. It is hard, for example, to imagine any of us warming up to a discussion of how literature and the arts received Einstein’s relativity in the decades from the ’20s to the ’50s; and a discussion of ideology pertaining to Einstein’s 1925 visit to South America is not every mathematician’s cup of tea either.
But there are a number of very interesting discussions to be had from the other side of the scientific aisle, so to speak, e.g., an article on Hilbert’s work on the foundations of physics, its focus falling on the notorious business of reconciling causality with the axiom of general invariance. And then, to get down to scientific brass tacks, there are contributions having to do with constrained Hamiltonian dynamics, quantum field theory in curved spacetime, the (notorious) border region between relativity and quantum theory, and some material concerning quantum gravity: these are certainly themes of considerable scientific and even mathematical interest, of course.
On the other hand, as far as mathematics properly so-called goes, it is fair to say that, due to its interdisciplinarity, the book under review does not really offer anything solidly mathematical. This is not really a criticism; it is a parochial complaint, if anything. The audience the authors intend is that of attendees at “interdisciplinary conferences held since 1986 in locations alternating between the United States and Europe,” and the thrust of the enterprise is “dialogue fostered among historians, philosophers, and physicists, looking at the development of the foundations of general relativity from different perspectives.” Mathematicians are not part of the intended game.
Moreover, the authors say in their Introduction that their “focus is on the conditions and consequences of Einstein’s pathbreaking achievements that sealed the decline of the classical notions of space, time, radiation, and matter. Particular attention is paid to the implications of conceptual conflicts for scientific views of the world at large, thus providing the basis for a comparison of the demise of the mechanical worldview around 1900 with the challenges presented by cosmology around 2000.” This, while more oriented toward, e.g. the history of ideas or history of science, is of course legitimate scholarship and provides fascinating insights to any scientifically cultured person, mathematicians included. Be forewarned however: no theorems or proofs are to be found here. But, there is of course the article by Brading and Ryckman on Hilbert, for example, already alluded to above, i.e. “Hilbert’s Axiomatic Method and His ‘Foundations of Physics’” on pp. 175–199, which sports a lot of serious mathematics, so we are not all that far away from home after all.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.