We discuss the development of metamathematics in the Hilbert school, and Hilbert's proof-theoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how these considerations help frame our understanding of metamathematics and proof theory today.

Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s

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Abstract. I address the historical emergence of the mathematical infinite, and how we are to take the infinite in and out of mathematics. The thesis is that the mathematical infinite in mathematics is a matter of method. The infinite, of course, is a large topic. At the outset, one can historically discern two overlapping clusters of concepts: (1) wholeness, completeness, universality, absoluteness. (2) endlessness, boundlessness, indivisibility, continuousness. The first, the metaphysical infinite, I shall set aside. It is the second, the mathematical infinite, that I will address. Furthermore, I will address mathematical infinite by considering its historical emergence in set theory and how we are to take it in and out of mathematics. Insofar as physics and, more broadly, science deals with the mathematical infinite through mathematical language and techniques, my remarks should be subsuming and consequent. The main underlying point is that how the mathematical infinite is approached, assimilated, and applied in mathematics is not a matter of “ontological commitment”, of coming to terms with whatever that might mean, but rather of epistemological articulation, of coming to terms through knowledge. The mathematical infinite in mathematics is a matter of method. How we deal with the specific individual issues involving the infinite turns on the narrative we present about how it fits into methodological mathematical frameworks established and being established. The first section discusses the mathematical infinite in historical context, and the second, set theory and the emergence of the mathematical infinite. The third section discusses the infinite in and out of mathematics, and how it is to be taken. §1. The Infinite in Mathematics What role does the infinite play in modern mathematics? In modern mathematics, infinite sets abound both in the workings of proofs and as subject matter in statements, and so do universal statements, often of ∀ ∃ “for all there exists” form, which are indicative of direct engagement with the infinite. In many ways the role of the infinite is importantly “second-order ” in the sense that Frege regarded number generally, in that the concepts of modern mathematics are understood as having infinite instances over a broad range. 1 But