Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal of Artificial General Intelligence

The Journal of the Artificial General Intelligence Society

3 Issues per year

Open Access
See all formats and pricing
More options …

On Mathematical Proving

Petros Stefaneas / Ioannis M. Vandoulakis
Published Online: 2015-12-30 | DOI: https://doi.org/10.1515/jagi-2015-0007


This paper outlines a logical representation of certain aspects of the process of mathematical proving that are important from the point of view of Artificial Intelligence. Our starting-point is the concept of proof-event or proving, introduced by Goguen, instead of the traditional concept of mathematical proof. The reason behind this choice is that in contrast to the traditional static concept of mathematical proof, proof-events are understood as processes, which enables their use in Artificial Intelligence in such contexts, in which problem-solving procedures and strategies are studied.

We represent proof-events as problem-centered spatio-temporal processes by means of the language of the calculus of events, which captures adequately certain temporal aspects of proof-events (i.e. that they have history and form sequences of proof-events evolving in time). Further, we suggest a “loose” semantics for the proof-events, by means of Kolmogorov’s calculus of problems. Finally, we expose the intented interpretations for our logical model from the fields of automated theorem-proving and Web-based collective proving.

Keywords: mathematical proof; proof-event; problem solving; agents; calculus of events; Kolmogorov’s calculus of problems; Polymath project; T. Gowers; J. Goguen; R. Kowalski; A.N. Kolmogorov


  • Aczel, Amir D. 1997. Fermat’s Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem. Delta (first published 1996).Google Scholar

  • Bashmakova, Izabella and Smirnova Galina. 2000. The Beginnings and Evolution of Algebra. Translated from the Russian by Abe Shenitzer with the editorial assistance of David A. Cox. The Mathematical Association of America.Google Scholar

  • Bazhanov, Valentin A. 2011. “Mathematical Proof as a Form of Appeal to a Scientific Community,” Russian Studies in Philosophy, vol. 50, no. 4 (Spring 2012), pp. 52–72.Google Scholar

  • Davidson, Donald. 1967b, “Causal Relations”, Journal of Philosophy, 64, pp. 691–703; reprinted in: Casati, R., and Varzi, A.C. (eds.), Events, Dartmouth, Aldershot, 1996, pp. 401–13, and in” Davidson, D., Essays on Actions and Events, Oxford: Clarendon Press, 1980, pp. 149–62.Google Scholar

  • Demidov, Sergei S. (Демидов С. С.). 1966. «К истории проблем Гильберта» [On the history of Hilbert’s problems], Историко-математические исследования, 17, pp. 91-122 (in Russian).Google Scholar

  • Demidov, Sergei S. (Демидов С. С.). 2001. ««Математические проблемы» Гильберта и математика XX века» [“Hilbert’s mathematical problems” and mathematics of the 20th century], Историко-математические исследования, 41 (6), pp. 84-99 (in Russian).Google Scholar

  • Fauconnier, Gilles and Mark Turner. 2008. The way we think: Conceptual blending and the mind’s hidden complexities. Basic Books.Google Scholar

  • Goguen, Joseph A. 1999a. “An introduction to algebraic semiotics, with applications to user interface design”. In Nehaniv, C. (ed.), Computation for Metaphors, Analogy and Agents. Springer, pp. 242-291.Google Scholar

  • Goguen, Joseph A. 1999b. “Social and Semiotic Analyses for Theorem Prover User Interface Design”, Formal Aspects of Computing 11, pp. 272–301. (Special Issue on User Interfaces for Theorem Provers.)Google Scholar

  • Goguen, Joseph A. 2001. “What is a proof”, http://cseweb.ucsd.edu/~goguen/papers/proof.html

  • Goguen, Joseph A. 2003. “Semiotic morphisms, representations, and blending for interface design”. Proceedings, AMAST Workshop on Algebraic Methods in Language Processing. AMAST Press. 1–15. Conference held in Verona, Italy, 25-27 August.Google Scholar

  • Goguen, Joseph A. and Harell, D. Fox. 2004a. “Style as a choice of blending principles”. Shlomo Argamon, Shlomo Dubnov, and Julie Jupp (Eds), Style and Meaning in Language, Art Music and Design, AAAI Press, pp. 49-56.Google Scholar

  • Goguen, Joseph A. and Harell, D. Fox. 2004b. “Information Visualization and Semiotic Morphisms,” in Multidisciplinary Approaches to Visual Representations and Interpretations, Grant Malcolm (Ed.) Oxford: Elsevier, pp. 83-98.Google Scholar

  • Goguen, Joseph A. and Harell, D. Fox. 2010. “Style: A Computational and Conceptual Blending-Based Approach”, Shlomo Argamon, Kevin Burns, Shlomo Dubnov (Eds), The Structure of Style. Algorithmic Approaches to Understanding Manner and Meaning. Springer, pp. 291-316.Google Scholar

  • Hales, Alfred, and Robert Jewett. 1963. “Regularity and Positional Games.” Transactions of the American Mathematical Society 106, pp. 222–29.Google Scholar

  • Henkin, L. Monk, J.D. and Tarski, A. 1971. Cylindric Algebras, Part I, North-Holland.Google Scholar

  • Heyting, A. (1931) “The intuitionist foundations of mathematics,” reprinted in: P. Benacerraf and H. Putnam (eds) Philosophy of Mathematics: Selected Readings, 2nd ed, Cambridge: Cambridge University Press, 1983, pp. 52–61.Google Scholar

  • Heyting, A. 1955. Les Fondements des Mathématiques. Intuitionnisme. Théorie de la Démonstration. Paris : Gauthier-Villars. Title of the original: Mathematische Grundlagenforschung. Intuitionismus. Beweistheorie. Springer-Verlag, Berlin, 1934.Google Scholar

  • Heyting, A. 1956. Intuitionism: An Introduction, Amsterdam: North-Holland Publishing, 3rd revised edition, 1971.Google Scholar

  • Heyting, A. 1958. “Intuitionism in mathematics”, in: R. Klibansky (ed.), Philosophy in the Mid-Century. A survey (La Nuova Italia Editrice, Firenze) 101-115.Google Scholar

  • Huntington, E.V. 1933a. “New sets of independent postulates for the algebra of logic”, Transactions of American Mathematical Society 35, pp. 274-304.Google Scholar

  • Huntington, E.V. 1933b. “Boolean algebra: A correction”, Transactions of American Mathematical Society 35, pp. 557-558.Google Scholar

  • Kauffman, Louis H. 1990. “Robbins Algebra”. Proceedings of the Twentieth International Symposium on Multiple Valued Logic. IEE Computer Society Press, pp. 54-60.Google Scholar

  • Kauffman, Louis H. 2001. “The Robbins Problem: computer proofs and human proofs” Kybernetes - The International Journal of Systems and Cybernetics, Gordon Pask Remembered and Celebrated: Part I, Bernard Scott and Ranulph Glanville (eds), 30 (5/6), pp.726-752.Google Scholar

  • Kleene, Stephen Cole. 1945. “On the interpretation of intuitionistic number theory”, Journal of Symbolic Logic, 10, 109-124.Google Scholar

  • Klein, Jacob. 1968. Greek Mathematical Thought and the Origin of Algebra. Translated by Eva Brann from the German original Griechische Logistik und die Entstehung der Algebra. Reprinted by Dover, 1992.Google Scholar

  • Kleiner Israel. 2000. “From Fermat to Wiles: Fermat’s Last Theorem Becomes a Theorem”. Elemente der Mathematik 55, pp. 19-37.Google Scholar

  • Kolmogorov, Andrei N. 1932. „Zur Deutung der intuitionistischen Logik“, Mathematische Zeitschrift 35, pp. 58–65. English translation in V.M. Tikhomirov (ed.) Selected Works of A.N. Kolmogorov. Vol. I: Mathematics and Mechanics, 151-158. Kluwer, Dordrecht, 1991.Google Scholar

  • Kolmogorov, Andrei N. 1988. “Letters of A. N. Kolmogorov to A. Heyting”, Успехи математических наук (Uspekhi Matematicheskikh Nauk), 43 (6), pp. 75-77; English translation, Russian Mathematical Surveys, 43 (6), pp. 89-93.Google Scholar

  • Kowalski, Robert. 1992. “Database updates in the event calculus”, Journal of Logic Programming, 12, 121-146.Google Scholar

  • Kowalski, Robert and Sergot Marek. 1986 “A Logic-Based Calculus of Events”, New Generation Computing 4, pp. 67–95.Google Scholar

  • McCune, William. 1997. “Solution of the Robbins Problem”, Journal of Automated Reasoning 19(3), pp. 263-76.CrossrefGoogle Scholar

  • Mac Lane, Saunders. 1997. “Despite Physicists, Proof is Essential in Mathematics”, Synthese 111, pp. 147–154.Google Scholar

  • Martin-Löf, Per. 1984. Intuitionistic type theory. (Notes by Giovanni Sambin of a series of lectures given in Padua, June 1980). Napoli, Bibliopolis.Google Scholar

  • Medvedev, Yu. 1962. “Finite problems”, Soviet Mathematics Doklady, 3, 227-230.Google Scholar

  • Polymath. 2009. A New Proof of the Density Hales-Jewett Theorem. At http://arxiv.org/abs/0910.3926, arXiv:0910.3926v2 [math.CO], accessed April 2, 2012.

  • Polymath. 2010a. Deterministic Methods to Find Primes. At http://arxiv.org/abs/1009.3956, arXiv:1009.3956v3 [math.NT], accessed April 2, 2012.

  • Polymath. 2010b. Density Hales-Jewett and Moser Numbers. At http://arxiv.org/abs/1002.0374, arXiv:1002.0374v2 [math.CO], accessed April 2, 2012.

  • Rudin, Walter. 1953. Principles of Mathematical Analysis, New York: McGraw-Hill.Google Scholar

  • Miller, Rob and Shanahan Murray. 1999. “The event-calculus in classical logic — alternative axiomatizations”, Electronic Transactions on Artificial Intelligence 3(1), pp. 77-105.Google Scholar

  • Singh, Simon. 1998. Fermat’s Enigma. New York: Anchor Books.Google Scholar

  • Stefaneas, Petros and Vandoulakis, Ioannis. 2012. “The Web as a Tool for Proving” Metaphilosophy. Special Issue: Philoweb: Toward a Philosophy of the Web: Guest Editors: Harry Halpin and Alexandre Monnin. Volume 43, Issue 4, pp. 480–498. Reprinted in: Harry Halpin and Alexandre Monnin (Eds) Philosophical Engineering: Toward a Philosophy of the Web. Wiley-Blackwell, 2014, pp. 149-167.Google Scholar

  • Stefaneas, Petros and Vandoulakis, Ioannis. 2014. Proofs as spatio-temporal processes”, Pierre Edouard Bour, Gerhard Heinzmann, Wilfrid Hodges and Peter Schroeder-Heister (Eds) “Selected Contributed Papers from the 14th International Congress of Logic, Methodology and Philosophy of Science”, Philosophia Scientiæ, 18(3), pp. 111-125.Google Scholar

  • Stefaneas Petros, Vandoulakis Ioannis, Martínez Maricarmen and Foundalis Harry. 2012. “Web-Based Mathematical Problem-Solving with Codelets”, Computational Creativity, Concept Invention, and General Intelligence Tarek R. Besold, Kai-Uwe Kuehnberger, Marco Schorlemmer, Alan Smaill (Eds.), pp. 38–41.Google Scholar

  • Stefaneas Petros, Vandoulakis Ioannis, Martínez Maricarmen and Foundalis Harry. 2015. “Collective Discovery Events: Web-based Mathematical Problem-solving with Codelets”. In: Tarek R. Besold, Marco Schorlemmer, Alan Smaill (Eds) Computational Creativity Research: Towards Creative Machines. Atlantis Thinking Machines (Book 7) Atlantis/Springer, pp. 371-392.Google Scholar

  • Tieszen, R. L. 1989. Mathematical Intuition: Phenomenology and Mathematical Knowledge, Dordrecht: Kluwer.Google Scholar

  • Tieszen, R. L. 1992. “What is a Proof?” In: Detlefsen, M. (ed.) Proof, Logic and Formalization, London: Routledge. pp. 57–76.Google Scholar

  • Tieszen, R. L. 2000. “Intuitionism, Meaning Theory and Cognition”, History and Philosophy of Logic 21, pp. 179–194.Google Scholar

  • Tieszen, R. L. 2005. Phenomenology, Logic, and the Philosophy of Mathematics. Cambridge: Cambridge University Press.Google Scholar

  • Vandoulakis, Ioannis. 1998. “Was Euclid’s Approach to Arithmetic Axiomatic?” Oriens–Occidens Cahiers du Centre d’histoire des Sciences et des philosophies arabes et Médiévales, 2 (1998), pp. 141-181.Google Scholar

  • Vandoulakis, Ioannis. 2009. “Styles of Greek arithmetic reasoning,” 数学史の研究 Study of the History of Mathematics RIMS 研究集会報告集 Kôkyûroku No 1625, pp. 12-22.Google Scholar

  • Vandoulakis, Ioannis. 2010. “A Genetic Interpretation of Neo-Pythagorean Arithmetic,” Oriens–Occidens Cahiers du Centre d’histoire des Sciences et des philosophies arabes et Médiévales, 7 (2010), pp. 113-154.Google Scholar

  • Vandoulakis, Ioannis. 2015. “On A.A. Markov’s attitude towards Brouwer’s intuitionism”, Pierre Edouard Bour, Gerhard Heinzmann, Wilfrid Hodges and Peter Schroeder-Heister (Eds) “Proceedings of the 14th Congress of Logic, Methodology and Philosophy of Science”, Philosophia Scientiæ, 19(1), pp. 143-158.Google Scholar

  • Vandoulakis, Ioannis & Stefaneas, Petros. 2013a. “Conceptions of proof in mathematics”, В.А. Бажанов А.Н. Кричевец, В.А. Шапошников (ред.) Доказательство. Очевидность, достоверность и убедительность в математике. [Proof. Evidence, reliability and convincingness]. Труды Московского семинара по философии математики, pp. 256-281.Google Scholar

  • Vandoulakis, Ioannis & Stefaneas, Petros. 2013b. “Proof-events in History of Mathematics”, Gaņita Bhāratī, 35 (1-4), 2013, 257-295.Google Scholar

  • Vandoulakis, Ioannis & Stefaneas, Petros. 2014. “On the semantics of proof-events”, Труды XII Международных Колмогоровских Чтений (Proceedings of the 12th International Kolmogorov Conference), 20-23 May 2014, Yaroslavl’, Russia, pp. 137-144.Google Scholar

  • Winker, Steven K. 1990. “Robbins Algebra: Conditions That Make a Near-Boolean Algebra Boolean”, Journal of Automated Reasoning 6(4), pp. 465-489.Google Scholar

  • Winker, Steven K. 1992. “Absorption and idempotency criteria for a problem in near-Boolean algebras”, Journal of Algebra 153 (2), pp. 414-423.CrossrefGoogle Scholar

About the article

Received: 2015-05-18

Accepted: 2015-11-19

Published Online: 2015-12-30

Published in Print: 2015-12-01

1 Both authors contributed equally to this paper.

Citation Information: Journal of Artificial General Intelligence, Volume 6, Issue 1, Pages 130–149, ISSN (Online) 1946-0163, DOI: https://doi.org/10.1515/jagi-2015-0007.

Export Citation

© 2015 Petros Stefaneas et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Ioannis M. Vandoulakis, S.V. Ivanova, and E.V. Nikulchev
SHS Web of Conferences, 2016, Volume 29, Page 01074

Comments (0)

Please log in or register to comment.
Log in