# The problem of certainty in mathematics

- 785 Downloads
- 3 Citations

## 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

- Bachelard, G. (1938).
*La formation de l’esprit scientifique*. Paris: Libraire Philosophique J. Vrin.Google Scholar - Bell, E. T. (1953).
*Men of mathematics*(Vol. 1). London: Pelican Books.Google Scholar - Bernal, M. (1987).
*Black Athena*. London: Free Association Books.Google Scholar - Bernays, P. (1935).
*Platonism in mathematics*. Retrieved from http//:www.phil.cmu.edu/projects/bernays/Pdf/platonism.pdf - 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 - Bloor, D. (1991).
*Knowledge and social imagery*. Chicago: University of Chicago Press.Google Scholar - Boyer, C. B. (1989).
*A history of mathematics*. New York: Wiley.Google Scholar - Cantor, G. (1999).
*Contributions to the founding of the theory of transfinite numbers*. New York: Dover Books.Google Scholar - 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 - 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 - Ernest, P. (1991).
*The philosophy of mathematics education*. London: Routledge.Google Scholar - Ernest, P. (1998).
*Social constructivism as a philosophy of mathematics*. Albany, New York: State University of New York Press.Google Scholar - 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 - 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 - Ernest, P. (2013).
*The psychology of mathematics*. Amazon: Kindle Books.Google Scholar - Fuller, S. (1988).
*Social epistemology*. Bloomington, Indiana: Indiana University Press.Google Scholar - Gelman, R., & Galistel, C. R. R. (1978).
*The child’s understanding of number*. Cambridge, Massachusetts: Harvard University Press.Google Scholar - Gentzen, G. (1936). Die widerspruchfreiheit der reinen zahlentheorie.
*Mathematische Annalen, 112*, 493–565.CrossRefGoogle Scholar - Gillies, D. A. (Ed.). (1992).
*Revolutions in mathematics*. Oxford: Clarendon.Google Scholar - 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 - Harding, S. (1986).
*The science question in feminism*. Milton Keynes: Open University Press.Google Scholar - Hersh, R. (1997).
*What is mathematics, really?*London: Jonathon Cape.Google Scholar - 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 - Høyrup, J. (1994).
*In measure, number, and weight*. New York: State University of New York Press.Google Scholar - Ingold, T. (2012). Toward an ecology of materials.
*Annual Review of Anthropology, 41*, 427–442.CrossRefGoogle Scholar - Kirk, G. S., & Raven, J. E. (Eds.). (1957).
*The pre-Socratic philosophers*. Cambridge: Cambridge University Press.Google Scholar - 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 - Kline, M. (1980).
*Mathematics: The loss of certainty*. Oxford: Oxford University Press.Google Scholar - Knuth, D. E. (1985). Algorithmic thinking and mathematical thinking.
*American Mathematical Monthly, 92*, 170–181.CrossRefGoogle Scholar - Kuhn, T. S. (1970).
*The structure of scientific revolutions*. Chicago: Chicago University Press.Google Scholar - Lakatos, I. (1962). Infinite regress and the foundations of mathematics.
*Aristotelian Society Proceedings (Supplementary Volume), 36*, 155–184. Google Scholar - Lakatos, I. (1976).
*Proofs and refutations*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - 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 - Lyotard, J. F. (1984).
*The postmodern condition*. Manchester: Manchester University Press.Google Scholar - MacKenzie, D. (1993). Negotiating arithmetic, constructing proof: The sociology of mathematics and information technology.
*Social Studies of Science, 23*, 37–65.CrossRefGoogle Scholar - 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 - 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 - Piaget, J. (1952).
*The child’s conception of number*. New York: Norton.Google Scholar - 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 - Tymoczko, T. (Ed.). (1986).
*New directions in the philosophy of mathematics*. Boston: Birkhäuser.Google Scholar - 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 - Whorf, B. (1956).
*Language, thought, and reality*. Boston: MIT Press.Google Scholar