What is Hilbert’s 24th Problem?

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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

[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.

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