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Hilbert's Programs and Beyond

Wilfried Sieg
Oxford University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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Playing fast and loose with historical minutiae, one can say that Hilbert’s famous approach to proof theory was precipitated by two conflicts. There were Hilbert’s early dealings with the constructivist school vis à vis the proof of what is now called the Hilbert basis theorem, and there was Hilbert’s championing of Cantor’s set theory in the face of attacks by, for example, Kronecker and Poincaré. Kronecker had also been an early opponent of Hilbert in connection with the aforementioned basis theorem. Later there was the famous, or, rather, notorious, fight with L. E. J. Brouwer whose intuitionism threatened to undermine the whole mathematical enterprise in various pernicious ways. Arguably Brouwer’s greatest threat aimed at the removal of the law of the excluded middle as a principal tenet of mathematical reasoning, and Hilbert’s oft-quoted retort to Brouwer was that to take that law from a mathematician was akin to forbidding a boxer the use of his fists. Beyond these internecine polemics, the feud between Hilbert and Brouwer resulted in Hilbert firing the entire staff of editors of the Mathematische Annalen, including himself as the principal editor, in order to get rid of Brouwer and curtail his influence. It was this maneuver that moved Einstein (an associate editor) to quip, “What is this frog-and-mouse battle between the mathematicians?” Furthermore, on a specifically scholarly front, Hilbert launched what is now called formalism, roughly the proposal to establish that mathematics was consistent using only finite methods of formal logic. This campaign was dramatically derailed by Gödel with his earth-shattering incompleteness and undecidability results. And all of these mathematical overtures on Hilbert’s part led toward the discipline of “proof theory.”

Indeed, even as Hilbert’s original dream failed to come to fruition, the program in proof theory that was launched with this campaign survived, if in an evolved form. For example, , Gerhard Gentzen, who was a student of Hilbert’s student Paul Bernays, did a good deal of later revolutionary work on proof theory (see: Thus, what Hilbert started has had quite an aftermath, and the book under review is concerned with all of it, as well as with both the prelude to Hilbert’s work and that work itself.

Hilbert’s Programs and Beyond is laid out in four parts: an introduction, including a very informative “Perspective on Hilbert’s Programs,” and three sections: “Mathematical roots,” “Analyses” (split into two parts, “Historical” and “Systematical”), and “Philosophical Horizons.” In the first section we encounter Dedekind in a major way, while in the historical part of the second section we meet Gentzen and Gödel; there is also a long and deep discussion of Hilbert’s and Bernays’s reaction and subsequent work in the face of what Gödel had wrought. The book’s last section is perhaps something of a dubious departure from strict mathematical logical practice in the sense that it includes an excursion into the realm where philosophy and psychology meet: witness the subsections titled “Aspects of mathematical experience” and “Searching for proofs (and uncovering capacities of the mathematical mind).” I guess there is plenty of precedent for this sort of thing, including the rationale Brouwer gave for his intuitionistic philosophy. Additionally, there are any number of books on the market these days dealing with the nature of the mathematical experience, and while this sort of thing is not every one’s cup of tea, it does excite interest in lots of places. Fair enough.

All this having been said, the readership for Hilbert’s Programs and Beyond is clearly rather narrow and specialized. Certainly mathematical logicians with a historical bent will eat it all up like candy. But others will, too. It is, or at least should be, the case that all of us have some awareness of the controversies of the early 20th century and the role they played in bringing about the shape of contemporary mathematics. Moreover, some of us are captivated, too, by the human element in all this. To revisit these themes and explore certain of their facets in great detail is a beneficial and pleasant experience.

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

In.1. A perspective on Hilbert's Programs
In.2. Milestones

I. Mathematical roots
I.3. Dedekind's analysis of number
I.4. Methods for real arithmetic
I.5. Hilbert's programs: 1917-1922

II. Analyses

II.1. Finitist proof theory: 1922-1934
II.2. After Königsberg
II.3. In the shadow of incompleteness
II.4. Gödel at Zilsel's
II.5. Hilbert and Bernays: 1939

II.6. Foundations for analysis and proof theory
II.7. Reductions of theories for analysis
II.8. Hilbert's program sixty years later
II.9. On reverse mathematics
II.10. Relative consistency and accessible domains

III. Philosophical horizons 
III.1. Aspects of mathematical experience
III.2. Beyond Hilbert's reach?
III.3. Searching for proofs