27.11.2024 | Philosophische Fakultät, Philosophisches Seminar

Hilbert's program and the status of ideal elements

The focus in this talk will be on the mathematical roots of Hilbert’s conservativity program, i.e., the attempt of showing the conservativity of ideal over real mathematics. It is well-known that Hilbert's foundational work from the 1920s and 1930s was strongly influenced by preceding developments in nineteenth-century mathematics. Specifically, his program was clearly inspired by the "method of ideal methods“ in mathematics (cf. Hilbert 1926, 1928). In the present talk, I will argue that Hilbert’s discussion of the usefulness and eliminability of “ideal constructs” in his proof-theoretic work was directly motivated by a particular understanding of ideal elements in nineteenth-century projective geometry. Moreover, I will show that a closer comparison with different accounts of ideal elements, as discussed by different synthetic geometers at the period in question, will allow us to reassess Hilbert’s reductive instrumentalism underling his proof-theoretic program.