Computing with Mathematical Arguments



Thanks to developments in the last few decades in mathematical logic and computer science, it has now become possible to formalize non-trivial mathematical proofs in essentially complete detail. we discuss the philosophical problems and prospects for such formalization enterprises. We show how some perennial philosophical topics and problems in epistemology, philosophy of science, and philosophy of mathematics can be seen in the practice of formalizing mathematical proofs.


Epistemic justification mathematics formal proof inferentialism philosophy of mathematics 


  1. Boolos, G., 1984, “Don’t Eliminate Cut”, in: Journal of Philosophical Logic 13, 4, pp. 373–378.Google Scholar
  2. Cutland, N., 1980, Computability: An Introduction to Recursive Function Theory. Cambridge: Cambridge University Press.Google Scholar
  3. Davis, M., 1981, “Obvious Logical Inferences”, in: Proceedings of the 7th International Joint Conference on Artificial Intelligence (IJCAI), pp. 530–531.Google Scholar
  4. Grabowski, A., Korniłowicz, A., and Naumowicz, A., 2010, “Mizar in a Nutshell”, in: Journal of Formalized Reasoning, 3, 2, pp. 153–245.Google Scholar
  5. Hales, T. C., 2008, “Formal Proof”, in: Notices of the American Mathematical Society 55, 11, pp. 1370–1380.Google Scholar
  6. Lakatos, I., 1976, Proof s and Refutations: The Logic of Mathematical Discovery. Cambridge: Cambridge University Press.Google Scholar
  7. Orevkov, V P., 1993, Complexity of Proof s and their Transformations inAxiomatic Theories, Vol. 128 of Translations of Mathematical Monographs. Providence, RI: American Mathematical Society. Translated by Alexander Bochman from the original Russian manuscript, translation edited by David Louvish.Google Scholar
  8. Rudnicki, P., 1987, “Obvious Inferences”, in: Journal of Automated Reasoning 3, 4, pp. 383–393.Google Scholar
  9. Urban, J., and Sutcliffe, G., 2008, “Atp-based Cross-verification of Mizar Proof s: Method, Systems, and First Experiments”, in: Mathematics in Computer Science 2, 2, pp. 231–251.Google Scholar
  10. Wang, H., 1960, “Toward Mechanical Mathematics”, in: IBM Journal of Research and Development 4, 1, pp. 2–22.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Center for Artificial IntelligenceNew University of LisbonCaparicaPortugal
  2. 2.Center for Artificial Intelligence and Departamento de Matemática-FCTNew University of LisbonCaparicaPortugal

Personalised recommendations