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.

David Hilbert (; German: [ˈdaːvɪt ˈhɪlbɐt]; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of all time. Hilbert discovered and developed a broad range of fundamental ideas including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, sp David Hilbert (; German: [ˈdaːvɪt ˈhɪlbɐt]; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of all time. Hilbert discovered and developed a broad range of fundamental ideas including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics (particularly proof theory). He adopted and defended Georg Cantor's set theory and transfinite numbers. In 1900, he presented a collection of problems that set a course for mathematical research of the 20th century. Hilbert and his students contributed to establishing rigor and developed important tools used in modern mathematical physics. He was a co-founder of proof theory and mathematical logic. Hilbert Bernays Project Hilbert's 23 Problems Address ICMM 2014 dedicated to the memory of D.Hilbert Works by David Hilbert at Project Gutenberg Works by or about David Hilbert at the Internet Archive Works by David Hilbert at LibriVox (public domain audiobooks) Hilbert's radio speech recorded in Königsberg 1930 (in German) Archived 14 February 2006 at the Wayback Machine, with English translation Archived 12 November 2020 at the Wayback Machine Wolfram MathWorld – Hilbert'Constant David Hilbert at the Mathematics Genealogy Project O'Connor, John J.; Robertson, Edmund F., "David Hilbert", MacTutor History of Mathematics Archive, University of St Andrews 'From Hilbert's Problems to the Future', lecture by Professor Robin Wilson, Gresham College, 27 February 2008 (available in text, audio and video formats). Newspaper clippings about David Hilbert in the 20th Century Press Archives of the ZBW

Abstract The study is focused on the relation between theology and mathematics in the situation of increasing secularization. My main concern in the second part of this paper is the early-twentieth-century foundational crisis of mathematics. The hypothesis that pure mathematics partially fulfilled the functions of theology at that time is tested on the views of the leading figures of the three main foundationalist programs: Russell, Hilbert and Brouwer.