Educational Studies in Mathematics

, Volume 92, Issue 3, pp 379–393

The problem of certainty in mathematics

Article

Abstract

Two questions about certainty in mathematics are asked. First, is mathematical knowledge known with certainty? Second, why is the belief in the certainty of mathematical knowledge so widespread and where does it come from? This question is little addressed in the literature. In explaining the reasons for these beliefs, both cultural-historical and individual psychological factors are identified. The cultural development of mathematics contributes four factors: (1) the invariance and conservation of number and the reliability of calculation; (2) the emergence of numbers as abstract entities with apparently independent existence; (3) the emergence of proof with its goal of convincing readers of certainty of mathematical results; (4) the engulfment of historical contradictions and uncertainties and their incorporation into the mathematical narrative of certainty. Individual learners of mathematics internalize ideas of invariance, reliability and certainty through their classroom experiences and exposure to such cultural factors. Lastly, with regard to the first question, it is concluded that mathematics can be known with a certainty circumscribed by the limits of human knowing.

Keywords

Certainty Objectivity Mathematical knowledge Beliefs Proof Social construction 

References

  1. Bachelard, G. (1938). La formation de l’esprit scientifique. Paris: Libraire Philosophique J. Vrin.Google Scholar
  2. Bell, E. T. (1953). Men of mathematics (Vol. 1). London: Pelican Books.Google Scholar
  3. Bernal, M. (1987). Black Athena. London: Free Association Books.Google Scholar
  4. Bernays, P. (1935). Platonism in mathematics. Retrieved from http//:www.phil.cmu.edu/projects/bernays/Pdf/platonism.pdf
  5. Bloor, D. (1984). A sociological theory of objectivity. In S. C. Brown (Ed.), Objectivity and cultural divergence (Royal Institute of Philosophy lecture series 17) (pp. 229–245). Cambridge: Cambridge University.Google Scholar
  6. Bloor, D. (1991). Knowledge and social imagery. Chicago: University of Chicago Press.Google Scholar
  7. Boyer, C. B. (1989). A history of mathematics. New York: Wiley.Google Scholar
  8. Cantor, G. (1999). Contributions to the founding of the theory of transfinite numbers. New York: Dover Books.Google Scholar
  9. Cohen, P. J. (1971). Comments on the foundations of set theory. In D. Scott (Ed.), Axiomatic set theory (pp. 9–15). Providence, Rhode Island: American Mathematical Society.CrossRefGoogle Scholar
  10. Dubinsky, E. (1994). A theory and practice of learning college mathematics. In A. Schoenfeld (Ed.), Mathematical thinking and problem solving (pp. 221–243). Hillsdale: Erlbaum.Google Scholar
  11. Ernest, P. (1991). The philosophy of mathematics education. London: Routledge.Google Scholar
  12. Ernest, P. (1998). Social constructivism as a philosophy of mathematics. Albany, New York: State University of New York Press.Google Scholar
  13. Ernest, P. (2007). The philosophy of mathematics, values, and Kerala mathematics. The philosophy of mathematics education journal, 20. Retrieved from http://people.exeter.ac.uk/PErnest/pome20/index.htm
  14. Ernest, P. (2008). Towards a semiotics of mathematical text (Parts 1, 2 & 3). For the Learning of Mathematics, 28(1), 2–8; 28(2), 39–47; & 28(3), 42–49.Google Scholar
  15. Ernest, P. (2013). The psychology of mathematics. Amazon: Kindle Books.Google Scholar
  16. Fuller, S. (1988). Social epistemology. Bloomington, Indiana: Indiana University Press.Google Scholar
  17. Gelman, R., & Galistel, C. R. R. (1978). The child’s understanding of number. Cambridge, Massachusetts: Harvard University Press.Google Scholar
  18. Gentzen, G. (1936). Die widerspruchfreiheit der reinen zahlentheorie. Mathematische Annalen, 112, 493–565.CrossRefGoogle Scholar
  19. Gillies, D. A. (Ed.). (1992). Revolutions in mathematics. Oxford: Clarendon.Google Scholar
  20. Gödel, K. (1931/1967). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. In J. van Heijenoort (Ed.), Frege to Gödel: A source book in mathematical logic (pp. 592–617). Cambridge, Massachusetts: Harvard University Press.Google Scholar
  21. Harding, S. (1986). The science question in feminism. Milton Keynes: Open University Press.Google Scholar
  22. Hersh, R. (1997). What is mathematics, really? London: Jonathon Cape.Google Scholar
  23. Høyrup, J. (1980/1987). Influences of institutionalized mathematics teaching on the development and organisation of mathematical thought in the pre-modern period. In J. Fauvel, & J. Gray, (Eds.), The history of mathematics: A reader (pp. 43–45). London: Macmillan.Google Scholar
  24. Høyrup, J. (1994). In measure, number, and weight. New York: State University of New York Press.Google Scholar
  25. Ingold, T. (2012). Toward an ecology of materials. Annual Review of Anthropology, 41, 427–442.CrossRefGoogle Scholar
  26. Kirk, G. S., & Raven, J. E. (Eds.). (1957). The pre-Socratic philosophers. Cambridge: Cambridge University Press.Google Scholar
  27. Kitcher, P., & Aspray, W. (1988). An opinionated introduction. In W. Aspray & P. Kitcher (Eds.), History and philosophy of modern mathematics (pp. 3–57). Minneapolis: University of Minnesota Press.Google Scholar
  28. Kline, M. (1980). Mathematics: The loss of certainty. Oxford: Oxford University Press.Google Scholar
  29. Knuth, D. E. (1985). Algorithmic thinking and mathematical thinking. American Mathematical Monthly, 92, 170–181.CrossRefGoogle Scholar
  30. Kuhn, T. S. (1970). The structure of scientific revolutions. Chicago: Chicago University Press.Google Scholar
  31. Lakatos, I. (1962). Infinite regress and the foundations of mathematics. Aristotelian Society Proceedings (Supplementary Volume), 36, 155–184. Google Scholar
  32. Lakatos, I. (1976). Proofs and refutations. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  33. Lakatos, I. (1978). A renaissance of empiricism in the recent philosophy of mathematics? In I. Lakatos, Philosophical papers (Vol. 2, pp. 24–42). Cambridge: Cambridge University Press.Google Scholar
  34. Lyotard, J. F. (1984). The postmodern condition. Manchester: Manchester University Press.Google Scholar
  35. MacKenzie, D. (1993). Negotiating arithmetic, constructing proof: The sociology of mathematics and information technology. Social Studies of Science, 23, 37–65.CrossRefGoogle Scholar
  36. Muis, K. R. (2004). Personal epistemology and mathematics: a critical review and synthesis of research. Review of Educational Research, 74(3), 317–377.CrossRefGoogle Scholar
  37. Paris, J., & Harrington, L. (1977). A mathematical incompleteness in Peano arithmetic. In J. Barwise (Ed.), Handbook of mathematical logic (pp. 1133–1142). Amsterdam: North Holland.CrossRefGoogle Scholar
  38. Piaget, J. (1952). The child’s conception of number. New York: Norton.Google Scholar
  39. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1–36.CrossRefGoogle Scholar
  40. Tymoczko, T. (Ed.). (1986). New directions in the philosophy of mathematics. Boston: Birkhäuser.Google Scholar
  41. Vico, G. (1988). On the most ancient wisdom of the Italians. (L. M. Palmer, Trans.). Ithaca, United States: Cornell University Press. (Original work published 1710)Google Scholar
  42. Whorf, B. (1956). Language, thought, and reality. Boston: MIT Press.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Graduate School of EducationExeter UniversityExeterUK

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