Foundations of Science

, Volume 16, Issue 4, pp 337–351 | Cite as

Objects and Processes in Mathematical Practice

  • Uwe V. Riss


In this paper it is argued that the fundamental difference of the formal and the informal position in the philosophy of mathematics results from the collision of an object and a process centric perspective towards mathematics. This collision can be overcome by means of dialectical analysis, which shows that both perspectives essentially depend on each other. This is illustrated by the example of mathematical proof and its formal and informal nature. A short overview of the employed materialist dialectical approach is given that rationalises mathematical development as a process of model production. It aims at placing more emphasis on the application aspects of mathematical results. Moreover, it is shown how such production realises subjective capacities as well as objective conditions, where the latter are mediated by mathematical formalism. The approach is further sustained by Polanyi’s theory of problem solving and Stegmaier’s philosophy of orientation. In particular, the tool and application perspective illuminates which role computer-based proofs can play in mathematics.


Argumentation Mathematical knowledge Mathematical practice Formal proof Informal proof Dialectic 


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  1. Appel K., Haken W. (1977) Every planar map is four colorable. Illinois Journal of Mathematics 21(3): 429–567Google Scholar
  2. Arkoudas K., Bringsjord S. (2007) Computers, justification, and mathematical knowledge. Minds and Machines 17(2): 185–202. doi: 10.1007/s11023-007-9063-5 CrossRefGoogle Scholar
  3. Bartlett J. M. (1964) I.-frege: On the scientific justification of a concept-script [1882]. Mind LXXIII(290): 155–160. doi: 10.1093/mind/LXXIII.290.155 CrossRefGoogle Scholar
  4. Bittner, T., & Smith, B. (2001). Granular partitions and vagueness. In: FOIS ’01: Proceedings of the international conference on formal ontology in information systems (pp. 309–320). ACM, New York, NY, USA. doi: 10.1145/505168.505197
  5. Colburn T., Shute G. (2007) Abstraction in computer science. Minds and Machines 17(2): 169–184. doi: 10.1007/s11538-006-9061-4 CrossRefGoogle Scholar
  6. Corcoran J. (1973) Gaps between logical theory and mathematical practice. In: Bunge M. (Ed.) The methodological Um’ty of science. D. Riedel, Dordrecht, pp 23–50CrossRefGoogle Scholar
  7. Davis P. J., Hersh R. (1998) The ideal mathematician. In: Tymoczko T. (Ed.) New directions in the philosophy of mathematics, rev sub edn. Princeton University Press, Princeton, pp 177–184Google Scholar
  8. Dove I. J. (2009) Towards a theory of mathematical argument. Foundations of Science 14(1–2): 137–152. doi: 10.1007/s10699-008-9156-5 CrossRefGoogle Scholar
  9. Ernest P. (1997) The legacy of lakatos: Reconceptualising the philosophy of mathematics. Philosophia Mathematica 5(2): 116–134. doi: 10.1093/philmat/5.2.116 Google Scholar
  10. Hersh, R. (Ed.) (2005). 18 Unconventional essays on the nature of mathematics (1st ed.). Springer,
  11. Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge University Press,
  12. Larvor B. P. (2001) What is dialectical philosophy of mathematics. Philosophia Mathematica 9(2): 212–229Google Scholar
  13. Máté A. (2006) Árpád szabó and imre lakatos, or the relation between history and philosophy of mathematics. Perspectives on Science 14(3): 282–301. doi: 10.1162/posc.2006.14.3.282 CrossRefGoogle Scholar
  14. Pietschmann, H. (1996). Phänomenologie der Naturwissenschaft: Wissenschaftstheoretische und philosophische Probleme der Physik. Springer,
  15. Polanyi M. (1957) Problem solving. The British Journal for the Philosophy of Science 8(30): 89–103CrossRefGoogle Scholar
  16. Polanyi, M. (1962). Personal knowledge: Towards a post-critical philosophy. University Of Chicago Press,
  17. Polanyi, M. (1966). The tacit dimension (1st edn.). Doubleday, Garden City, NY,
  18. Rav Y. (1999) Why do we prove theorems?. Philosophia Mathematica 7(1): 5–41. doi: 10.1093/philmat/7.1.5 Google Scholar
  19. 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. doi: 10.1093/philmat/nkm023 CrossRefGoogle Scholar
  20. Ropolyi L. (2002) Lakatos and lukács. In: Kampis G., Stöltzner M., Kvasz L. (eds) Appraising Lakatos: Mathematics, methodology and the man (Vienna Circle Institute Library) (1st ed., pp 303–338). Berlin, Heidelberg: SpringerGoogle Scholar
  21. Rotman, B. (2006). Towards a semiotics of mathematics. In: R. Hersh (Ed.), 18 Unconventional essays on the nature of mathematics (Chap. 16, pp. 97–127). New York: Springer. doi: 10.1007/0-387-29831-2_16.
  22. Ruben, P. (1978) Dialektik und Arbeit der Philosophie. Studien zur Dialektik, Pahl-Rugenstein,
  23. Ruben, P. (1979). Philosophie und Mathematik. B. G. Teubner, Leipzig,
  24. Stegmaier, W. (2008). Philosophie der Orientierung (1st ed.). Walter de Gruyter,
  25. Thomas R. S. D. (1991) Meanings in ordinary language and in mathematics. Philosophia Mathematica s2-6(1): 3–38. doi: 10.1093/philmat/s2-6.1.3 CrossRefGoogle Scholar
  26. Thurston W. P. (1994) On proof and progress in mathematics. Bulletin of the American Mathematical Society 30(2): 161–178. doi: 10.1090/S0273-0979-1994-00502-6 CrossRefGoogle Scholar
  27. Tymoczko T. (1986) Making room for mathematicians in the philosophy of mathematics. The Mathematical Intelligencer 8(3): 44–50. doi: 10.1007/BF03025789 CrossRefGoogle Scholar
  28. Tymoczko, T. (Ed.) (1998). New directions in the philosophy of mathematics, rev sub edn. Princeton, NJ: Princeton University Press,

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© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.SAP Research KarlsruheSAP AGKarlsruheGermany

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