Hilbert, David (1862–1943)

Hilbert is commonly regarded as one of the two leading mathematicians of the early 20th century, the other being Henri Poincaré (1854–1912). Hilbert made important contributions to a remarkable variety of subjects, including the theory of invariants, algebraic number theory (where he also wrote a widely acclaimed general report, the so-called Zahlbericht (1897)), algebraic geometry, the theory of integral equations and Hilbert spaces, mathematical physics and the foundations of mathematics.

Hilbert’s ideas in the foundations of mathematics were original and influential. The philosophically most significant were those concerning the development of his so-called formalist program. These included most importantly a conception of mathematical proof which allowed for the use of non-contentual, and more specifically symbolic methods of reasoning. By this is meant reasoning which makes no use of interpretations (or contents) of the expressions used in the reasoning. In accordance with this, Hilbert conceived a new plan for consistency proofs, specifically, an alternative to model construction as a means of proving the consistency of arithmetic (specifically, the arithmetic of the real numbers, or analysis). This was Hilbert’s so-called proof theory (Beweistheorie), a plan he had at least vestigially in mind in his early foundational work, and which he and his students developed more fully in later work.

In the summer of 1900 Hilbert addressed the international congress of mathematicians in Paris. In that address, he presented a list of problems for 20th-century mathematics. The second problem on this list called for a proof of consistency for arithmetic (specifically, the arithmetic of the real numbers). More specifically, he called for a direct proof of the consistency of arithmetic. By a direct proof of consistency, Hilbert mainly meant a proof which does not merely establish the consistency of arithmetic relative to that of some other axiomatic theory. Eventually, however, it came to mean something more – namely, proof which proceeds directly from a description of the basic acts of symbol manipulation which, in Hilbert’s view, constituted (at least in part) the fundamental operations of mathematical reasoning. In Hilbert’s view, this proof-theoretic approach to the consistency problem for arithmetic was ultimately the only satisfactory approach to it.