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.
Notes
I am discussing mathematics as a written discipline, for we know little about the oral mathematics that preceded it.
Some developments in mathematics do not assume conservation, such as Cantor’s (1999) theory of transfinite set theory (ℵ0 = ℵ0 + ℵ0) and Boolean algebra (1 + 1 = 1).
These assumptions are an application of the Sapir-Whorf hypothesis, namely that our language “cuts up” the world into the way we conceptualize it. “We dissect nature along lines laid down by our native language.” Whorf (1956) p. 212.
Number mysticism survives to this day, for example in hotels having no floor labelled 13.
Learners have to overcome unavoidable “epistemological obstacles” (Bachelard, 1938) in learning of mathematics, for example, when numbers are expanded from the natural numbers (with a least number) to the integers (no least number).
Previously, I rejected the claim that mathematical knowledge is objective and superhuman and can be known with absolute certainty (Ernest, 1991, 1998). By redefining objectivity and certainty as culturally circumscribed by the limits of human knowing, I am happy to acknowledge the certainty of mathematical knowledge.
References
Bachelard, G. (1938). La formation de l’esprit scientifique. Paris: Libraire Philosophique J. Vrin.
Bell, E. T. (1953). Men of mathematics (Vol. 1). London: Pelican Books.
Bernal, M. (1987). Black Athena. London: Free Association Books.
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.
Bloor, D. (1991). Knowledge and social imagery. Chicago: University of Chicago Press.
Boyer, C. B. (1989). A history of mathematics. New York: Wiley.
Cantor, G. (1999). Contributions to the founding of the theory of transfinite numbers. New York: Dover Books.
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.
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.
Ernest, P. (1991). The philosophy of mathematics education. London: Routledge.
Ernest, P. (1998). Social constructivism as a philosophy of mathematics. Albany, New York: State University of New York Press.
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.
Ernest, P. (2013). The psychology of mathematics. Amazon: Kindle Books.
Fuller, S. (1988). Social epistemology. Bloomington, Indiana: Indiana University Press.
Gelman, R., & Galistel, C. R. R. (1978). The child’s understanding of number. Cambridge, Massachusetts: Harvard University Press.
Gentzen, G. (1936). Die widerspruchfreiheit der reinen zahlentheorie. Mathematische Annalen, 112, 493–565.
Gillies, D. A. (Ed.). (1992). Revolutions in mathematics. Oxford: Clarendon.
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.
Harding, S. (1986). The science question in feminism. Milton Keynes: Open University Press.
Hersh, R. (1997). What is mathematics, really? London: Jonathon Cape.
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.
Høyrup, J. (1994). In measure, number, and weight. New York: State University of New York Press.
Ingold, T. (2012). Toward an ecology of materials. Annual Review of Anthropology, 41, 427–442.
Kirk, G. S., & Raven, J. E. (Eds.). (1957). The pre-Socratic philosophers. Cambridge: Cambridge University Press.
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.
Kline, M. (1980). Mathematics: The loss of certainty. Oxford: Oxford University Press.
Knuth, D. E. (1985). Algorithmic thinking and mathematical thinking. American Mathematical Monthly, 92, 170–181.
Kuhn, T. S. (1970). The structure of scientific revolutions. Chicago: Chicago University Press.
Lakatos, I. (1962). Infinite regress and the foundations of mathematics. Aristotelian Society Proceedings (Supplementary Volume), 36, 155–184.
Lakatos, I. (1976). Proofs and refutations. Cambridge: Cambridge University Press.
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.
Lyotard, J. F. (1984). The postmodern condition. Manchester: Manchester University Press.
MacKenzie, D. (1993). Negotiating arithmetic, constructing proof: The sociology of mathematics and information technology. Social Studies of Science, 23, 37–65.
Muis, K. R. (2004). Personal epistemology and mathematics: a critical review and synthesis of research. Review of Educational Research, 74(3), 317–377.
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.
Piaget, J. (1952). The child’s conception of number. New York: Norton.
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.
Tymoczko, T. (Ed.). (1986). New directions in the philosophy of mathematics. Boston: Birkhäuser.
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)
Whorf, B. (1956). Language, thought, and reality. Boston: MIT Press.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ernest, P. The problem of certainty in mathematics. Educ Stud Math 92, 379–393 (2016). https://doi.org/10.1007/s10649-015-9651-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10649-015-9651-x