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