Formal and Natural Proof: A Phenomenological Approach

  • Merlin CarlEmail author
Part of the Synthese Library book series (SYLI, volume 407)


In this section, we apply the notions obtained above to a famous historical example of a false proof. Our goal is to demonstrate that this proof shows a sufficient degree of distinctiveness for a formalization in a Naproche-like system and hence that automatic checking could indeed have contributed in this case to the development of mathematics. This example further demonstrates that even incomplete distinctivication can be sufficient for automatic checking and that actual mistakes may occur already in the margin between the degree of distinctiveness necessary for formalization and complete distinctiveness.



We thank Marcos Cramer for the kind permission to use his Naproche version of Rav’s proof Sect. 14.4.1. We thank Dominik Klein and an anonymous referee for various valuable comments on former versions of this work that led to considerable improvements. We also thank Heike Carl for her thorough proofreading.


  1. Avigad, J., Donnelly, K., Gray, D., & Raff, P. (2006). A formally verified proof of the prime number theorem. ACM Transactions on Computational Logic, 9(1),Google Scholar
  2. Azzouni, J. (2005a). Tracking reason. Proof, consequence, and truth. New York: Oxford University Press.Google Scholar
  3. Azzouni, J. (2005b). Is there still a sense in which mathematics can have foundations? In G. Sica (Ed.), Essays on the foundations of mathematics and logic (pp. 9–47). Monza: Polimetrica.Google Scholar
  4. Azzouni, J. (2009). Why do informal proofs confirm to formal norms? Foundations of Science, 14, 9–26.CrossRefGoogle Scholar
  5. Bauer, G., & Nipkow, T. (2006). Flyspeck I: Tame graphs. Archive of Formal Proofs. Available online at
  6. Benzmüller, C. E., & Brown, C. E. (2007). The curious inference of Boolos in Mizar and OMEGA. Studies in Logic, Grammar and Rhetoric, 10(23), 299–386.Google Scholar
  7. Boolos, G. (1987). A curious inference. Journal of Philosophical Logic, 16(1), 1–12.CrossRefGoogle Scholar
  8. Carl, M. An Introduction to elementary number theory for humans and machines. Work in progress.Google Scholar
  9. Carl, M., & Koepke, P. (2010). Interpreting Naproche – An algorithmic approach to the derivation-indicator view. Paper for the International Symposium on Mathematical Practice and Cognition at the AISB 2010.Google Scholar
  10. Carl, M., Cramer, M., & Kühlwein, D. (2009). Chapter 1 of Landau in Naproche, the first chapter of our Landau translation. Available online:
  11. Cauchy, A. (1821). Cours d’analyse, p. 120.Google Scholar
  12. Cramer, M., Koepke, P., Kühlwein, D., & Schröder, B. (2009). The Naproche System, paper for the Calculemus 2009.Google Scholar
  13. Feferman, S. (1979). What does logic have to tell us about mathematical proofs? The Mathematical Intelligencer, 2(1), 20–24.CrossRefGoogle Scholar
  14. Fleck, A., Maennschen, Ph., & Perron, O. (1909). Vermeintliche Beweise des Fermatschen Satzes. Archiv der Mathematik und Physik, 14, 284–286.Google Scholar
  15. Hartimo, M., & Okada, M. (2016). Syntactic reduction in Husser’s early phenomenology of arithmetic. Synthese, 193(3), 937–969.CrossRefGoogle Scholar
  16. Husserl, E. (1900). Formale und transzendentale Logik. Versuch einer Kritik der logischen Vernunft. Tübingen: Niemeyer.Google Scholar
  17. Husserl, E. (1952). Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie. Drittes Buch. Haag: Matrinus Nijhoff.Google Scholar
  18. Husserl, E. (1999). Erfahrung und Urteil. Untersuchungen zur Genealogie der Logik. Hamburg: Felix Meiner.Google Scholar
  19. Husserl, E. (2009). Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie. Hamburg: Meiner.Google Scholar
  20. Kamp, H., & Reyle, U. (2008). From discourse to logic: Introduction to model-theoretic semantics of natural language, formal logic and discourse representation theory. Netherlands: Springer.Google Scholar
  21. Koepke, P., & Schlöder, J. (2012). The Gödel completeness theorem for uncountable languages. Journal of Formalized Mathematics, 30(3), 199–203.Google Scholar
  22. Kreisel, G. (1965). Informal rigor and completeness proofs. In I. Lakatos (Ed.), Problems in the Philosophy of Mathematics. Proceedings of the International Collquium in the Philosophy of Science (Vol. 1).Google Scholar
  23. Lakatos, I. (1976). Proof and refutation. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  24. Landau, E. (2004). Grundlagen der Analysis. Heldermann, N.Google Scholar
  25. Lohmar, D. (1989). Phänomenologie der Mathematik: Elemente einer phänomenologischen Aufklärung der mathematischen Erkenntnis nach Husserl. Kluwer Academic Publishers.CrossRefGoogle Scholar
  26. Martin-Löf, P. (1980). Intuitionistic type theory (Notes of Giovanni Sambin on a series of lectures given in Padova). Available online:
  27. Martin-Löf, P. (1987). Truth of a proposition, evidence of a judgement, validity of a proof. Synthese, 73, 407–420.CrossRefGoogle Scholar
  28. MathOverflow-Discussion: Examples of common false beliefs in mathematics. (2010a).
  29. MathOverflow-Discussion: Widely accepted mathematical results that were later shown wrong. (2010b).
  30. Matiyasevich, Y. (1993). Hilbert’s 10th problem (MIT Press Series in the Foundations of Computing. Foreword by Martin Davis and Hilary Putnam). Cambridge: MIT Press.Google Scholar
  31. Monty Hall Problem. (1990). Wikipedia-article available at
  32. Naproche Web Interface. Available at
  33. Pak, K. (2011). Brouwer fixed point theorem in the general case. Journal Formalized Mathematics, 19(3), 151–153.CrossRefGoogle Scholar
  34. Rautenberg, W. (2006). A concise introduction to mathematical logic. New York: Springer.Google Scholar
  35. Rav, Y. (2007). A critique of a formalist-mechanist version of the justification of arguments in mathematicians’ proof practices. Philosophia Mathematica (III), 15, 291–320.CrossRefGoogle Scholar
  36. Rickey, V. F. Cauchy’s famous wrong proof. Available online:
  37. Rota, G.-C. (1997). The phenomenology of mathematical proof. Synthese, 111(2), 183–196.CrossRefGoogle Scholar
  38. Singh, S. (2000). Fermats letzter Satz. Die abenteuerliche Geschichte eines mathematischen Rätsels. dtv.Google Scholar
  39. Sundholm, G. (1993). Questions of proof. Manuscrito, 16, 47–70.Google Scholar
  40. Tieszen, R. (2005). Phenomenology, logic, and the philosophy of mathematics. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  41. Tieszen, R. (2011). After Gödel. Platonism and rationalism in mathematics and logic. Oxford: Oxford University Press.CrossRefGoogle Scholar
  42. Tragesser, R. (1992). Three insufficiently attended to aspects of most mathematical proofs: Phenomenological studies. In M. Detlefsen (Ed.), Proof, logic and formalization (pp. 71–87). London: Routledge.Google Scholar
  43. Turing, A. (1937). On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, Ser. 2, 42, 230–265.CrossRefGoogle Scholar
  44. van Atten, M. (2010). Construction and constitution in mathematics. The New Yearbook for Phenomenology and Phenomenological Philosophy, 10, 43–90.Google Scholar
  45. van Atten, M., & Kennedy, J. (2003). On the philosophical development of Kurt Gödel. The Bulletin of Symbolic Logic, 9(4), 425–476.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Europa-Universität FlensburgFlensburgGermany

Personalised recommendations