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Synthese

, Volume 196, Issue 11, pp 4623–4656 | Cite as

Settings and misunderstandings in mathematics

  • Brice HalimiEmail author
Article
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Abstract

This paper pursues two goals. Its first goal is to clear up the “identity problem” faced by the structuralist interpretation of mathematics. Its second goal, through the consideration of examples coming in particular from the theory of permutations, is to examine cases of misunderstandings in mathematics fit to cast some light on mathematical understanding in general. The common thread shared by these two goals is the notion of setting. The study of a mathematical object almost always goes together with the choice of a particular setting, and the understanding of the workings of mathematical settings is an essential component of mathematical knowledge. It is claimed that the recognition of mathematical settings, as features distinct from both mathematical structures and the systems which instantiate those structures, allows one to classify most of understandable misunderstandings in mathematics, and also to solve the identity problem.

Keywords

Ante rem structuralism Identity problem Theory of permutations Mathematical settings Labels Parameters Mathematical cognition Misunderstandings in mathematics 
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Notes

Acknowledgements

I wish to thank Denis-Charles Cisinski, Étienne Fieux, Gerhard Heinzmann and Marco Panza, as well as anonymous referees, for very valuable suggestions and comments.

References

  1. Avigad, J. (2006). Mathematical method and proof. Synthese, 153(1), 105–159.CrossRefGoogle Scholar
  2. Baez, J. (2009). Torsors made easy. http://math.ucr.edu/home/baez/torsors.html.
  3. Benacerraf, P. (1973). Mathematical truth. The Journal of Philosophy, 70(19), 661–679.CrossRefGoogle Scholar
  4. Bermúdez, J. L. (2007). Indistinguishable elements and mathematical structuralism. Analysis, 67, 112–116.CrossRefGoogle Scholar
  5. Brandom, R. (1996). The significance of complex numbers for Frege’s philosophy of mathematics. Proceedings of the Aristotelian Society, 96, 293–315.CrossRefGoogle Scholar
  6. Coxeter, H., & Moser, W. (1984). Generators and relations for discrete groups (4th ed.). Berlin: Springer.Google Scholar
  7. Coxeter, H. S. M. (1989). Introduction to geometry (2nd ed.). New York: Wiley.Google Scholar
  8. Dochtermann, A. (2009). Hom complexes and homotopy theory in the category of graphs. European Journal of Combinatorics, 30(2), 490–509.CrossRefGoogle Scholar
  9. Keränen, J. (2001). The identity problem for realist structuralism. Philosophia Mathematica, 9(3), 308–330.CrossRefGoogle Scholar
  10. Keränen, J. (2006). The identity problem for realist structuralism II. A reply to Shapiro. In F. MacBride (Ed.), Identity and modality. New essays in metaphysics (chap. 6, pp. 146–163). Oxford: Oxford University Press.Google Scholar
  11. Knobloch, E. (2001). Determinants and elimination in Leibniz. Revue d’histoire des sciences, 54(2), 143–164.CrossRefGoogle Scholar
  12. Leitgeb, H., & Ladyman, J. (2008). Criteria of identity and structuralist ontology. Philosophia Mathematica, 16(3), 388–396.CrossRefGoogle Scholar
  13. Lengnink, K., & Schlimm, D. (2010). Learning and understanding numeral systems: Semantic aspects of number representations from an educational perspective. In B. Löwe, & T. Müller (Eds.), PhiMSAMP. Philosophy of mathematics: Sociological aspects and mathematical practice (pp. 235–264). London: College Publications.Google Scholar
  14. Linnebo, Ø. (2008). Structuralism and the notion of dependence. The Philosophical Quarterly, 58(230), 59–79.Google Scholar
  15. Macbeth, D. (2012). Proof and understanding in mathematical practice. Philosophia Scientiæ, 16(1), 29–54.CrossRefGoogle Scholar
  16. Magnus, W., Karrass, A., & Solitar, D. (1976). Combinatorial group theory. Presentations of groups in terms of generators and relations (2nd revised ed.). New York: Dover Publications.Google Scholar
  17. Peirce, C. S. (1998). On reasoning in general. In The Peirce Edition Project (Ed.), The essential Peirce. Selected philosophical writings (1893–1913) (Vol. 2, pp. 11–26, MS 595 (1895)), chapter 3. Indiana University Press: Bloomington and Indianapolis.Google Scholar
  18. Pettigrew, R. (2008). Platonism and Aristotelianism in mathematics. Philosophia Mathematica, 16(3), 310–332.CrossRefGoogle Scholar
  19. Resnik, M. D. (1997). Mathematics as a science of patterns. Oxford: Oxford University Press.Google Scholar
  20. Rizza, D. (2010). Discernibility by symmetries. Studia Logica, 96(2), 175–192.CrossRefGoogle Scholar
  21. Roberts, C. (2003). Uniqueness in definite noun phrases. Linguistics and Philosophy, 26, 287–350.CrossRefGoogle Scholar
  22. Shapiro, S. (1997). Philosophy of mathematics: Structure and ontology. New York: Oxford University Press.Google Scholar
  23. Shapiro, S. (2006a). The governance of identity. In F. MacBride (Ed.), Identity and modality. New essays in metaphysics (chapter 7, pp. 164–173). Oxford: Oxford University Press.Google Scholar
  24. Shapiro, S. (2006b). Structure and identity. In F. MacBride (Ed.), Identity and modality. New essays in metaphysics (chapter 5, pp. 109–145). Oxford: Oxford University Press.Google Scholar
  25. Shapiro, S. (2008a). Identity, indiscernibility and ante rem structuralism: The tale of \(i\) and \(-i\). Philosophia Mathematica, 16(3), 285–309.CrossRefGoogle Scholar
  26. Shapiro, S. (2008b). An ‘\(i\)’ for an \(i\). Reference and indiscernibility. In International workshop on mathematical understanding (Université Paris Diderot, Paris, June 9–13, 2008), talk given on June 13, 2008.Google Scholar
  27. Shapiro, S. (2012). An “\(i\)” for an \(i\): singular terms, uniqueness, and reference. Review of Symbolic Logic, 5(3), 380–415.CrossRefGoogle Scholar
  28. Whitehead, A. N. (1911). An introduction to mathematics. London: Williams & Northgate.Google Scholar

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Département de PhilosophieUniversité Paris NanterreNanterreFrance

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