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
[Ala14] Jesse Alama. The simplest axiom system for hyperbolic geometry revisited again. Studia Logica 102(3):609–615 2014.
[Ara17] Andrew Arana. On the alleged simplicity of impure proof. In Roman Kossak and Philip Ording editors Simplicity: Ideals of Practice in Mathematics and the Arts pages 205–226. Springer 2017.
[AZ09] M. Aigner and G.M. Ziegler. Proofs from THE BOOK. Springer 4th edition 2009.
[Ber67] Paul Bernays. Hilbert David. In Paul Edwards editor The Encyclopedia of Philosophy pages 496–505. Macmillan 1967.
[GG00] Ivor Grattan-Guinness. A sideways look at Hilbert’s twenty-three problems of 1900. Notices of the AMS 47(7):752–757 2000.
[Hil97] David Hilbert. Theorie der algebraischen Invarianten nebst Anwendungen auf Geometrie. 1897. lecture notes from Summer 1897 prepared by Sophus Marxsen Library of the Mathematical Institute of the University of Göttingen; English translation Theory of Algebraic Invariants R.C. Laubenbacher and B. Sturmfels (eds.) [using a different copy from the Mathematics Library of Cornell University] Cambridge University Press Cambridge 1993.
[Hil01a] DavidHilbert. Mathematical problems. Bulletin of the American Mathematical Society 8 1901.
[Hil01b] David Hilbert. Mathematische Probleme. Archiv für Mathematik und Physik 3. Reihe 1:44–63 213–237 1901. Reprinted in [Hil35 p. 290–329].
[Hil05] David Hilbert. Über die Grundlagen der Logik und der Arithmetik. In Adolf Krazer editor Verhandlungen des Dritten Internationalen Mathematiker-Kongresses in Heidelberg vom 8. bis 13. August 1904 pages 174–185. Leipzig 1905.
[Hil10] David Hilbert. The Foundations of Geometry. Open Court 1910.
[Hil18] David Hilbert. Axiomatisches Denken. Mathematische Annalen 78(3/4):405–415 1918. English translation: [Hil70].
[Hil35] David Hilbert. Gesammelte Abhandlungen Band III. Springer 1935. Second edition 1970.
[Hil70] David Hilbert. Axiomatic thinking. Philosophia Mathematica 1970.
[Hug06a] Dominic Hughes. Proofs without syntax. Annals of Mathematics 143(3):1065–1076 2006.
[Hug06b] Dominic Hughes. Towards Hilbert’s 24 problem: Combinatorial proof invariants: (preliminary version). Electr. Notes Theor. Comput. Sci. 165:37–63 2006.
[KO17] Roman Kossak and Philip Ording editors. Simplicity: Ideals of Practice in Mathematics and the Arts. Springer 2017.
[Lan17] Marc Lange. Because Without Cause. Oxford Studies in Philosophy of Science. Oxford University Press 2017.
[Loo68] ElishaScott Loomis. The Pythagorean Proposition. Ann Arbor Michigan 1968. Reprint of the 2nd edition from 1940. First published in 1927.
[Mac34a] Saunders MacLane. Abbreviated proofs in logic calculus. Bulletin of the American Mathematical Society 40(1):37–38 1934. Abstract.
[Mac34b] Saunders MacLane. Abgekürzte Beweise im Logikkalkul. PhD thesis Georg August-Universität zu Göttingen 1934.
[Man01] Paolo Mancosu. Mathematical explanation: Problems and prospects. Topoi 20:97–117 2001.
[Man08] Paolo Mancosu. Mathematical explanation: why it matters. In Paolo Mancosu editor The Philosophy of Mathematical Practice pages 134–149. Oxford University Press 2008.
[NvP14] S. Negri and J. von Plato. Proof Analysis — A Contribution to Hilbert’s Last Problem. Cambridge University Press 2014.
[Pam88] Victor Pambuccian. Simplicity. Notre Dame Journal of Formal Logic 29(3):396–411 1988.
[Pam11] Victor Pambuccian. The simplest axiom system for plane hyperbolic geometry revisited. Studia Logica 97(3):347–349 2011.
[Str05] Lutz Straßburger. What is a logic and what is a proof? In Jean-Yves Beziau editor Logica Universalis pages 135–145. Birkhäuser 2005.
[Str06] Lutz Straßburger. Proof nets and the identity of proofs. CoRR abs/cs/0610123 2006.
[Thi03] R. Thiele. Hilbert’s twenty-fourth problem. American Mathematical Monthly 110(1):1–24 January 2003.
[TW02] R. Thiele and L. Wos. Hilbert’s twenty-fourth problem. Journal of Automated Reasoning 29(1):67–89 2002.
[WP03] Larry Wos and GailW. Pieper. Automated Reasoning and the Discovery of Missing and Elegant Proofs. Rinton Press 2003.