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, Volume 51, Issue 5, pp 845–856 | Cite as

Aspects of freedom in mathematical proof

  • Gregor NickelEmail author
Original Article

Abstract

Mathematical argumentation is generally thought to be the paradigm of cogent reasoning. The concept of mathematical proof thus seems to be associated with necessity and enforcement, but not with freedom; however, in various ways a reference to freedom is also needed to understand the phenomenon of mathematical proof, for example, to distinguish it from a merely mechanical functioning. Moreover, without reference to these freedom-related aspects the value of incorporating proofs into the teaching of mathematics lacks an essential aspect of justification. The paper sheds light on some of these aspects and offers a picture of mathematical proof in tension between freedom and enforcement.

Keywords

Mathematical proof Freedom of choice Self determination Semiotics 

Notes

Acknowledgements

I wish to thank Michael Korey and the anonymous referees for many helpful remarks.

References

  1. Aristotle. (1960). Posterior analytics. Topica. In H. Tredennick (Ed.). Cambridge, MA: Harvard University PressGoogle Scholar
  2. Arnold, V. I. (1995). Will mathematics survive? The Mathematical Intelligencer, 17(3), 6–10.CrossRefGoogle Scholar
  3. Brown, J. R. (1999). Philosophy of mathematics. London: Routledge.Google Scholar
  4. Cantor, G. (1883). Über unendliche lineare Punktmannigfaltigkeiten. Mathematische Annalen, 21, 545–591. [New Edition in G. Cantor (1966), Abhandlungen mathematischen und philosophischen Inhalts. Hildesheim: Olms].CrossRefGoogle Scholar
  5. Carroll, L. (1895). What the Tortoise said to Achilles. Mind (new series), 4, 278–280.CrossRefGoogle Scholar
  6. Chudnoff, E. (2011). The nature of intuitive justification. Philosophical Studies, 153, 313–333.CrossRefGoogle Scholar
  7. Davis, P. J., & Hersh, R. (1981). The mathematical experience. Boston: Birkhäuser.Google Scholar
  8. Fischer, R. (2006). Materialisierung und Organisation. Wien: Profil.Google Scholar
  9. Giesinger, J. (2010). Die Vereinbarkeit von Willensfreiheit und Erziehung. Z Erziehungswiss., 13, 421–435.CrossRefGoogle Scholar
  10. Habermas, J. (1981). Theorie des kommunikativen Handelns (Vol. 1). Frankfurt am Main: Suhrkamp.Google Scholar
  11. Hadamard, J. (1945). The psychology of invention in the mathematical field. New York: Dover Publications.Google Scholar
  12. Hanna, G., Jahnke, H. N., & Pulte, H. (Eds.). (2010). Explanation and proof in mathematics: Philosophical and educational perspectives. Dordrecht: Springer.Google Scholar
  13. Hefendehl-Hebeker, L. (2018). Einwirkungen von Mathematik(unterricht) auf Individuen und ihre Auswirkungen in der Gesellschaft. In G. Nickel, M. Helmerich, R. Krömer, K. Lengnink, & M. Rathgeb (Eds.), Mathematik und Gesellschaft (pp. 173–182). Wiesbaden: Springer.CrossRefGoogle Scholar
  14. Heintz, B. (2000). Die Innenwelt der Mathematik. Wien: Springer.CrossRefGoogle Scholar
  15. Hilbert, D. (1988). Wissen und mathematisches Denken: Winter Semester 1922/23. Göttingen: Universität, Mathematisches Institut.Google Scholar
  16. Höffe, O. (2015). Kritik der Freiheit. München: Beck.CrossRefGoogle Scholar
  17. Kane, R. (2002). Introduction: The contours of contemporary free will debates. In R. Kane (Ed.), The Oxford handbook of free will. Oxford: Oxford University Press.Google Scholar
  18. Kant, I. (1781). Critik der reinen Vernunft. Riga: Hartknoch. [English edition: Kant, I. (1998). Critique of pure reason. (trans: Guyer, P. & Wood, A. W.). Cambridge: Cambridge University Press].Google Scholar
  19. Kant, I. (1783). Prolegomena. Riga: Hartknoch. [English edition: Kant, I. (2004). Theoretical Philosophy after 1781. (trans: Hatfield, G., Friedman, M., Allison, H., & Heath, P.). Cambridge: Cambridge University Press].Google Scholar
  20. Kanterian, E. (2012). Frege: A guide for the perplexed. London: Bloomsbury Academic.Google Scholar
  21. Kollosche, D. (2015). Gesellschaftliche Funktionen des Mathematikunterrichts. Wiesbaden: Springer.CrossRefGoogle Scholar
  22. Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  23. Leibniz, G. W. (1966). Hauptschriften zur Grundlegung der Philosophie (Vol. I). Hamburg: Meiner.Google Scholar
  24. Löwe, B., Müller, T., & Müller-Hill, E. (2010). Mathematical knowledge as a case study in empirical philosophy of mathematics. In B. Van Kerkhove, J. P. Van Bendegem, & J. De Vuyst (Eds.), Philosophical perspectives on mathematical practice (pp. 185–203). London: College Publications.Google Scholar
  25. McEvoy, M. (2008). The epistemological status of computer-assisted proofs. Philosophia Mathematica, 16(3), 374–387.CrossRefGoogle Scholar
  26. Nagel, F. (1984). Nicolaus Cusanus und die Entstehung der exakten Wissenschaften. Aschendorf: Münster.Google Scholar
  27. Nicholas, C. (1982). De Venatione Sapientiae. Opera Omnia (Vol. XII). Hamburg: Meiner. [English edition: Nicholas de Cusa (1998). De Venatione Sapientiae. (On the Pursuit of wisdom). (trans: J. Hopkins). Minneapolis, Minnesota: The Arthur J. Banning Press].Google Scholar
  28. Nicholas, C. (1988). De Beryllo. Opera Omnia (Vol. XII/1). Hamburg: Meiner. [English edition: Nicholas de Cusa (1998). De Beryllo. (On [intellectual] eyeglasses). (trans: J. Hopkins). Minneapolis, Minnesota: The Arthur J. Banning Press].Google Scholar
  29. Nickel, G. (2005). Zur Möglichkeit von theologischer Mathematik und mathematischer Theologie. In I. Bocken & H. Schwaetzer (Eds.), Spiegel und Porträt (pp. 9–28). Maastricht: Shaker.Google Scholar
  30. Nickel, G. (2006). Zwingende Beweise—zur subversiven Despotie der Mathematik. In J. Dietrich & U. Müller-Koch (Eds.), Ethik und Ästhetik der Gewalt (pp. 261–282). Paderborn: Mentis.Google Scholar
  31. Nickel, G. (2007). Mathematik und Mathematisierung der Wissenschaften. In J. Berendes (Ed.), Autonomie durch Verantwortung (pp. 319–346). Paderborn: Mentis.Google Scholar
  32. Nickel, G. (2010). Proof—Some Notes on a phenomenon between freedom and enforcement. In B. Löwe & T. Müller (Eds.), Philosophy of mathematics: Sociological aspects and mathematical practice (pp. 281–291). London: College Publications.Google Scholar
  33. Nickel, G. (2011). Mathematik—die (un)heimliche Macht des Unverstandenen. In M. Helmerich, K. Lengnink, G. Nickel, & M. Rathgeb (Eds.), Mathematik verstehen. Philosophische und didaktische Perspektiven (pp. 47–58). Wiesbaden: Vieweg + Teubner.Google Scholar
  34. Nickel, G. (2016). “Schlüsseltechnologie” oder Medium zur freien Entfaltung des Geistes—Bildende Beiträge der Mathematik. In U. Nembach (Ed.), Informationes Theologiae Europae. Internationales Jahrbuch für Theologie. Bd. 20 (pp. 141–162). Frankfurt am Main: Peter Lang.Google Scholar
  35. Nickel, G. (2017). Vorausgesetzt, ein Beweis überzeugt—Aspekte mathematischen Denkens. In U. Lüke & G. Souvignier (Eds.), Wie objektiv ist Wissenschaft? (pp. 124–139). Darmstadt: Wissenschaftliche Buchgesellschaft.Google Scholar
  36. Nida-Rümelin, J. (2005). Über menschliche Freiheit. Stuttgart: Reclam.Google Scholar
  37. Rathgeb, M. (2016). Zenon, Carroll, Winter. In G. Nickel & R. Krömer (Eds.), Siegener Beiträge zur Geschichte und Philosophie der Mathematik 7 (pp. 69–98). Siegen: UniverSi.Google Scholar
  38. Rav, Y. (2007). A critique of a formalist-mechanist version of the justification of arguments in mathematicians’ proof practices. Philosophia Mathematica, 15(3), 291–320.CrossRefGoogle Scholar
  39. Rotman, B. (1988). Towards a semiotics of mathematics. Semiotica, 72(1/2), 1–35. [Page numbers refer to the reprint in R. Hersh (Ed.) (2006), 18 unconventional essays on the nature of mathematics. (pp. 97–127). New York: Springer].CrossRefGoogle Scholar
  40. Rotman, B. (2000). Mathematics as sign. Writing, imagining, counting. Stanford: Stanford University Press.Google Scholar
  41. Skovsmose, O., & Penteao, M. G. (2015). Mathematics education and democracy. An open landscape of tensions, uncertainties, and challenges. In D. English & D. Kirshner (Eds.), Handbook of international research in mathematics education (pp. 359–372). New York: Routledge.Google Scholar
  42. Spaemann, R. (1972). Freiheit. In J. Ritter (Ed.), Historisches Wörterbuch der Philosophie. Wissenschaftliche Buchgesellschaft: Darmstadt.Google Scholar
  43. Trudeau, R. (2001). The non-Euclidean revolution. Basel: Birkhäuser.CrossRefGoogle Scholar
  44. Vohns, A. (2018). Rechnen oder Rechnen lassen? Mathematik(unterricht) als Bürgerrecht und Bürgerpflicht. In G. Nickel, M. Helmerich, R. Krömer, K. Lengnink, & M. Rathgeb (Eds.), Mathematik und Gesellschaft (pp. 203–220). Wiesbaden: Springer.CrossRefGoogle Scholar
  45. Wille, M. (2007). Die Mathematik und das synthetische Apriori. Paderborn: Mentis.Google Scholar
  46. Wille, M. (2008). Beweis und Reflexion. Paderborn: Mentis.Google Scholar
  47. Wille, M. (2012). Transzendentaler Antirealismus. Berlin: De Gruyter.Google Scholar

Copyright information

© FIZ Karlsruhe 2019

Authors and Affiliations

  1. 1.Universität SiegenSiegenGermany

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