Logic and Modern Philosophy of Mathematics

Meanwhile, contentment with Euclidian rigor was dissolving. From the 1860s non-Euclidean geometries had been accepted as legitimate theories, especially due to the insights of Bernhard Riemann. In addition, various mathematicians, including Peano, had noticed that Euclidian geometry itself needed several more axioms than Euclid had stated.

These developments made mathematicians still more aware of finding and expressing the assumptions involved in a theory. Another stimulus was the rise in importance of algebras (for example, among several, Boole's), each with its own basic laws. David Hilbert, the leading mathematician of his generation, studied geometries and algebras intensively, and was led around 1900 to seek for a mathematical means of studying axiom systems. He found it in "metamathematics" (his later name for it), in which a system was examined to establish its completeness, consistency, and independence.

One of Hilbert's axioms for geometry was a meta-assumption that the other axioms supplied all the objects required. This soon led the young American mathematician Oswald Veblen to consider two different sets of objects satisfying an axiom system; if the members of each could be put in one-one correspondence, then the system was "categorical." The study of (non-)categoricity enriched model theory considerably.

Set theory itself was influenced by these developments, for in 1908 Hilbert's follower Ernst Zermelo axiomatized it. His system included his own discovery, the "axiom of choice," a nonconstructive assumption that some colleagues found doubtful. A strong debate broke out; most participants agreed that the axiom was unavoidable. However, it was unacceptable for "constructivist" mathematicians, who admitted only procedures that built up mathematical objects in explicit stages: for them the axiom was outlawed, along with some standard proof methods such as by contradiction (to prove theorem T, assume not-T and get into a logical mess). The most prominent figure in this tradition was the Dutch mathematician L. E. J. Brouwer, who elaborated his alternative "intuitionistic" mathematics especially in the 1920s. At that time Hilbert's metamathematical program was in its definitive phase; but in 1927 Brouwer derided it as "formalism," a misleading name that Hilbert himself never used but that has regrettably become standard.