What is Hilbert’s 24th Problem?

Abstract

In 2000, a draft note of David Hilbert was found in his Nachlass concerning a 24th problem he had consider to include in the his famous problem list of the talk at the International Congress of Mathematicians in 1900 in Paris. This problem concerns simplicity of proofs. In this paper we review the (very few) traces of this problem which one can find in the work of Hilbert and his school, as well as modern research started on it after its publication. We stress, in particular, the mathematical nature of the problem.1

Mathematizing as a virtuous practice: different narratives and their consequences for mathematics education and society

“…Numerous further examples can be found inLakoff and Núñez (2000).11 Apparently, there is also literature on the Hilbert Thesis, see for instanceKahle and Oitavem (2018) orHipolito and Kahle (2019). See below.12 Formal systems become more important due to recent developments in formal mathematics.…”

Discussing Hilbert's 24th problem

“…mathematical objects which can be studied by mathematical tools. 6 When mathematical proofs are represented by formal proofs, we may get a faithful translation. But neither does this imply that mathematical proofs ought to be formal proofs-see [24]-nor could we even expect that such a translation tells us something about the method of proof.…”

“…We follow here the presentation of[6] where one can find a more detailed discussion of these threads 2. It his talk, Hilbert didn't read the full paper and presented only 10 of the 23 problems explicitly, see[7, p. 68] 3.…”