Explanation and Proof in Mathematics pp 71-83 | Cite as

# Authoritarian Versus Authoritative Teaching: Polya and Lakatos

## Abstract

Lakatos argued that a proof, when presented in the usual “Euclidian” style, may leave the choice of theorem, definitions and proof-idea mysterious. To remove these mysteries, he recommended a “heuristic” style of presentation. This distinction was already present in the work of Polya. Moreover, Polya was directly concerned with teaching and consequently paid attention to the emotional and existential experience of the student. However, Polya lacked Lakatos’s account of proof analysis and was not a fallibilist. Therefore, the question of whether Lakatos advanced pedagogy from where Polya left it reduces to two questions: (1) does proof analysis have a place in the classroom? and (2) does fallibilism have a place in the classroom? In this paper, I argue that the answers are (1) Yes and (2) No.

## Keywords

Bovine Spongiform Encephalopathy Algebraic Version Plausible Inference Intellectual Honesty Semantic Shift## References

- Bandy, A., & Long, J. (2000). Dress rehearsal for a revolution?
*The Hungarian Quarterly, XLI*, 157.Google Scholar - Corfield, D. (2003). Towards a philosophy of real mathematics. Cambridge/New York: Cambridge University Press.CrossRefGoogle Scholar
- Derrida, J. (1978).
*Edmund Husserl’s origin of geometry: an introduction*. New York: Nicolas Hays (J. P. Leavey, Trans.).Google Scholar - Feyerabend, P., & Lakatos, I. (1999).
*For and against method*. Chicago: University of Chicago Press. Motterlini (ed).Google Scholar - Gillies, D. (1992).
*Revolutions in mathematics*. Oxford/New York: Clarendon Press/Oxford University Press.Google Scholar - Grosholz, E., & Breger, H. (2000).
*The growth of mathematical knowledge*. Boston: Kluwer.Google Scholar - Grosholz, E. R. (2007).
*Representation and productive ambiguity in mathematics and the sciences*. New York: Oxford University Press.Google Scholar - Hallett, M. (1979). Towards a theory of mathematical research programmes (in two parts).
*British Journal for the Philosophy of Science*, 30, 1-25, 135-159.Google Scholar - Jha, S. R. (2006). Hungarian studies in Lakatos’ philosophies of mathematics and science - Editor’s Introduction.
*Perspectives on Science, 14*(3), 257-262.CrossRefGoogle Scholar - Koetsier, T. (1991).
*Lakatos’ philosophy of mathematics: a historical approach*(Studies in the History and Philosophy of Mathematics Vol.3). Amsterdam: North Holland.Google Scholar - Lakatos, I. (1947). Eötvös Kollegium - Gyorffy Kollegium.
*Valóság*, 3.Google Scholar - Lakatos, I. (1976).
*Proofs and refutations*. In J. Worrall, & E. Zahar (Eds.) Cambridge: Cambridge University Press.Google Scholar - Lakatos, I. (1978a).
*The methodology of scientific research programmes (Philosophical papers, Vol. 1)*. In J. Worrall, & E. Curry (Eds.) Cambridge: Cambridge University Press.Google Scholar - Lakatos, I. (1978b).
*Mathematics, science and epistemology (Philosophical papers, Vol. 2)*. In J. Worrall, & E. Curry (Eds.) Cambridge: Cambridge University Press.Google Scholar - Larvor, B. (1997). Lakatos as historian of mathematics.
*Philosophia Mathematica, 5*(1), 42-64.Google Scholar - Larvor, B. (1998). Lakatos, an introduction. London: Routledge.Google Scholar
- Long, J. (1998). Lakatos in Hungary. Philosophy of the Social Science
*s, 28*, 244-311.CrossRefGoogle Scholar - Polya, G. (1954).
*Mathematics and plausible reasoning*. Princeton, NJ: Princeton University Press.Google Scholar - Polya, G. (2004).
*How to solve it: a new aspect of mathematical method*. Princeton Science Library Edition with a new foreword by John Conway. (Original publication 1945).Google Scholar