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Paradoxes in Paradise

Martin Davis

The Search for Certainty: A Philosophical Account of Foundations of Mathematics. Marcus Giaquinto. xii + 286 pp. Oxford University Press, 2002. $45.

David Hilbert (1862–1943) was arguably the leading mathematician of his time. In struggles over how mathematics was to accommodate new understandings of the infinite, the Dutch mathematician L. E. J. Brouwer was his most fervent opponent. When Hilbert's favorite student, Hermann Weyl, went over to the enemy, saying "Brouwer, that is the revolution," Hilbert was incensed. In a passionate address delivered in 1922, he proclaimed:

Weyl and Brouwer . . . seek to provide a foundation for mathematics by pitching overboard whatever discomforts them and declaring an embargo. . . . But this would mean dismembering and mutilating our science, and, should we follow such reformers, we would run the risk of losing a large part of our most valued treasures. Weyl and Brouwer outlaw the general notion of irrational number, of function, even of number-theoretic function, Cantor's [ordinal] numbers of higher number classes, etc. The theorem that among infinitely many natural numbers there is always a least, and even the logical law of the excluded middle, e.g., in the assertion that either there are only finitely many prime numbers or there are infinitely many: these are examples of forbidden theorems and modes of inference. I believe that impotent as Kronecker was to abolish irrational numbers . . ., no less impotent will their efforts prove today. No! Brouwer's [program] is not as Weyl thinks, the revolution, but only a repetition of an attempted putsch with old methods, that in its day was undertaken with greater verve yet failed utterly. Especially today, when the state power is thoroughly armed and fortified by the work of Frege, Dedekind, and Cantor, these efforts are foredoomed to failure.

A decade later Hilbert's own program for the foundations of mathematics lay in tatters, destroyed in an investigation by the young logician Kurt Gödel, which had initially been undertaken in an effort to contribute to that very program. Today, passions have cooled, and working mathematicians show little interest in foundational matters. The infinitary set theoretic methods that occasioned such controversy are casually absorbed in passing by the beginning graduate student and used unhesitatingly.

Like a military historian surveying the battlefield long after all the bodies have been cleared away, Marcus Giaquinto coolly revisits the controversies in The Search for Certainty. It all began with a necessary effort by mathematicians to put their house in order. Although the methods of the calculus had proved highly successful, their underlying logic had been in grave need of clarification. Calculus worked with numbers and functions, but no coherent theory of the so-called "real" numbers had been developed, and the notion of function had been stretched in the direction of a frightening arbitrariness. Richard Dedekind's elegant characterization of the real numbers in terms of "cuts" had a distinctly set-theoretic flavor, and Georg Cantor's new push into the "actual" infinite expanded the boundaries of the subject matter of mathematics into what Hilbert later characterized as "Cantor's paradise." But Cantor's infinite was plagued by paradox. The kernel of at least one of these paradoxes could manifest itself in what seemed like everyday reasoning, as Bertrand Russell showed in his famous paradox: If we consider the set S of all those sets that are not members of themselves, then S is a member of itself if and only if it is not a member of itself. This was particularly bad news for Gottlob Frege, whose ambitious logical system for the foundations of mathematics proved vulnerable to Russell's paradox and was thus seen to be inconsistent.

The highly influential but rather baroque three-volume Principia Mathematica was Bertrand Russell's effort (with coauthor Alfred North Whitehead) to revive Frege's wrecked program. Paradoxes such as Russell's were to be avoided by slicing up the universe of mathematical discourse into discrete successive "types," with set membership permitted only between adjacent types.

Meanwhile, Ernst Zermelo, who never accepted the stringent syntactic demands of formal logical systems, proposed a set of axioms for set theory. By the 1920s, it was realized that Zermelo's axioms provided the basis of a formal system rivaling that of Principia. In an important paper appearing in 1930, Zermelo proposed what came to be called the iterative notion of set, in which a hierarchy of sets is built from some initial collection of things by iterating indefinitely the operation of forming the set of all subsets of a given set. He observed that his axioms could be construed as being about just this notion. A few years later, in an address on the foundations of mathematics, Kurt Gödel emphasized that rather than being seen as a rival to Principia, when viewed from the perspective of the iterative notion of set Zermelo's system could be seen as the result of eliminating unnecessary complications and artificial restrictions from the Whitehead-Russell system. By the 1940s and '50s, set-theoretic methods had become a crucial part of the mathematician's toolbox.

Back in the 1920s, when passions were aflame, Hilbert developed an ingenious strategy by which he intended to overcome his opponents. He would establish the legitimacy of methods that Brouwer and Weyl considered dubious by encapsulating those methods in formal systems whose consistency would then be proved using only methods of which they approved. In a revolutionary paper in 1931, the young Gödel demonstrated not only that consistency could not be proved using only these restrictive methods, but also that the same negative conclusion held even if the entire panoply of methods encapsulated in the systems in question was brought to bear. After Gödel, the foundations of mathematics were seen as inevitably open-ended, with more and more propositions becoming provable as ever more powerful methods were employed. Gödel liked to emphasize that these more powerful methods could be thought of as being essentially a matter of venturing sufficiently far out in the iterative hierarchy of sets.

Giaquinto has provided a careful and judicious discussion and analysis of these matters, supplying needed technical background for readers who are not mathematicians. Although foundational questions have ceased to be of much importance to most mathematicians, controversies among specialists continue. Readers of this book will be well prepared to follow the current literature on foundations of mathematics.

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