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Foundations of Mathematics and Physics One Century After Hilbert

Joseph Kouneiher, editor
Publication Date: 
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[Reviewed by
Michael Berg
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To be of a certain age, as the discreet saying used to have it, has benefits as well as liabilities. For me, as far as the latter are concerned, these days it’s about bad knees and a bad back (many years of judo), as well as a pate that is progressively more sensitive to sunburn, a creeping need for more sleep, well, you get the idea — “the thousand natural shocks that flesh is heir to,” as Hamlet would have it in a characteristically maudlin mood. But there’s the other side of the old coin: for instance, as a senior member of my department, I get to teach courses that allow me to delve a little into the history and background of the subject, and I get to bring in quite a variety of perspectives as I see fit. In recent years, in addition to History of Mathematics, I’ve taught a senior seminar in number theory, a graduate course in geometry, and reading courses in differential geometry and physics. One common theme in these courses is the imperative that the matter of “foundations” be given its due, and since I love this material, I go at it whole hog: there’s nothing like an audience held captive because of the threat of a bad grade.

The player on center stage in much of this is, of course, David Hilbert. For me, the first introduction to his watershed role, both in the foundations of mathematics and in the foundations physics, came about altogether painlessly in the context of a book I never tire of recommending, namely, Hilbert, by Constance Reid. Since Reid was the sister of the prominent mathematical logician Julia Robinson, it’s not surprising that the theme of the foundations of mathematics and physics should get a lot of airplay in her book. Prior to my devouring this biography, I had already read that classic storybook of mathematics and mathematicians, Men of Mathematics by the unsinkable E. T. Bell, with Riemann, Cantor, and Kronecker featured. So I had heard about the controversial 19th and early 20th century battles concerning the foundations of mathematics, but it was only with Hilbert that I started to see the bigger picture, and Hilbert’s centrality.

Early on, Hilbert defended Cantor against Kronecker and the latter’s fellow travelers. Surveying the evolving mathematical landscape, it was Hilbert who saw the need to look at the foundations of mathematics formally when the echoes of the paradoxes and antinomies of Russell, Burali-Forti, etc., refused to die down. It was also Hilbert who, in Paris in 1900, challenged the mathematical world to go at these themes and settle them — his biggest dreams were eventually thwarted by Gödel. Hilbert waged full-fledged war on the intuitionists, specifically their doyen L. E. J. Brouwer, launching what Einstein referred to as the frog-and-mouse battle among the mathematicians.

Speaking of Einstein, relativity and quantum mechanics are the two subjects which cleft modern physics from classical physics. Not only was Hilbert was intimately involved with both, but he competed with Einstein in the race for the general theory of relativity. Finally, his role in quantum mechanics is illustrated by, for example, the Privatdozenten and assistants who worked under him, such as Max Born and John von Neumann. Note that the non-classical statistics at the heart of the Copenhagen interpretation of quantum mechanics are due to the former, while the most mathematical version of quantum mechanics came from the pen of the latter. Indeed, it was von Neumann who transformed the subject into largely an application of spectral theory of unbounded operators on a Hilbert space (of quantum mechanical states). In any event, there is no doubt whatsoever that Hilbert took the foundations of physics very seriously — indeed his sixth Paris problem was explicitly concerned with providing a proper axiomatic foundation for physics.

Hilbert’s focus on foundations was both sweeping and pervasive; indeed, as Kouneiher indicates already in the Preface of the present book, Hilbert lectured on foundations of mathematics as well as physics from 1891 to 1933. Moreover, these lectures, long unpublished, have become available over the last couple of decades in a sextet of tomes launched by (who else?) Springer-Verlag, a bonanza for scholarship in the area of Hilbert’s role in the modern movements toward providing foundations for mathematics and physics. The book under review contains number of essays by prominent scholars addressing the matter: what has transpired in the wake of Hilbert’s work? Additionally, these essays look into the ecumenical interplay between physics and mathematics, another theme dear to Hilbert’s heart, given the famous ethos of Göttingen. It is a wonderful compendium indeed.

And the contributions contain the following cross section: Kouneiher and John Stachel, Hilbert and Einstein; Colin McLarty on Grothendieck’s unifying vision of geometry; Atiyah on the 6-sphere; Gromov on positive scalar curvature; Connes, Geometry and the quantum; Witten on string theory; Penrose on twistors; and Smolin on quantum gravity. Here’s a sample from McLarty (cf. p. 109):

Grothendieck was bitterly aware ‘it has been good manners in “high society” to look down on those who dare pronounce the word “topos” ‘ [from Recoltes et Semailles, p. 182]. This repugnance is linked with logical foundations because Grothendieck’s idea of topos poses problems for naïve set theory. So do the ideas of Abelian category, derived functor, and a forteriori their characterizations by universal properties, all of which are standard textbook material today. But the issue is associated with topos theory since Grothendieck wrote about it in Théorie des Topos et Cohomologie Étale des Schémas. Grothendieck cared to have a precise logical foundation. So he asked Pierre Samuel to write 30 pages on set theoretic universes, signed N. Bourbaki, as an appendix to [Vol. 1 of SGA 4]. Many geometers regret this.

Talk about a dense paragraph … One is moved to say, “Who knew?” Well, evidently Grothendieck did.

The bottom line is that this wonderful book is full of such insights, reflections, analyses, appraisals, and even anecdotes, and should be a smash hit: we should all seek to know more about these foundations, and we have a wonderful set of essays here, based off Hilbert’s original wonderful lectures, and a lot more.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

See the table of contents int he publisher's webpage.