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.
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Notes
- 1.
Long (1998) p.269
- 2.
Motterlini (1999) pp.375-382; translated by Ninon Leader. While he was preparing this speech, Lakatos organised a protest against a party official’s doctoral thesis that was critical of the late professor of pedagogy, Sándor Karácsony (1891-1952). After midnight and much acrimonious discussion, the panel of examiners rejected the thesis. We should not suppose that this was a simple defence of scholarly independence from politics. As a piece of Stalinist party work, this thesis was, by September 1956, behind the political times. The occasion of its formal public defence presented an opportunity for Lakatos and others to rehearse the political revolution to come later that autumn.
- 3.
- 4.
Motterlini (1999) p.380.
- 5.
This was already part of his thought in 1947, when he published a review of Karoly Jeges’ I Learn Physics. While he approved of Jeges’ aim of introducing physics to non-specialists, Lakatos complained that Jeges introduced concepts “in a scholastic manner, without making them real in terms of experiments” or giving “the historical dialectics of theories” (quoted from Long p. 266). In a short piece published in 1963, he complains that, “…science and mathematics teaching is disfigured by the customary authoritarian presentation. Thus presented, knowledge appears in the form of infallible systems hinging on conceptual frameworks not subject to discussion. The problem-situational background is never stated…” (1978b p.254).
- 6.
“Without refutations one cannot sustain suspicion” (Lakatos 1976 p.49).
- 7.
“…no scientific theory, no theorem can conclude anything finally…” (Motterlini (1999) p.379). Lakatos ended this speech thus, “At the last Party Congress in China, Teng Xiao Ping talked about guaranteeing the right to dissent and remarked that if, perchance, truth happened to be on the side of a minority, this right would facilitate the recognition of truth.” (Op. Cit. p.382).
- 8.
Except Marxism, of course. In his review of Jeges, Lakatos remarked, “It is incorrect to give the impression that physics is an eternal science” (quoted from Long p. 266). Marxist dialectical progressivism is distinct from liberal empirical fallibilism, but they both insist that today’s orthodoxies may be rationally superseded tomorrow. Hence, opposition to Stalinism’s fixed official truths provided a context in which Lakatos could move between them and eventually combine them.
Dialectics in Hegel and Marx is about progress through conflict, which in science means criticism. However, true communism is supposed to mark the resolution of all dialectical oppositions, therefore communist science has no need of criticism; nevertheless, it advances. This explains Lakatos’ otherwise perplexing remark that “dialectic tries to account for change without using criticism: truths are ‘in continual development’ but always ‘completely incontestable’” (1976 p.55n). He has in mind here Soviet or Stalinist ‘dialectic’, rather than the Hegelian-Marxist dialectics that he elsewhere mentions with approval. György Litván called Lakatos a “natural-born Trotskyite,” better fitted to Trotsky’s ‘permanent revolution’ than to Stalinist stasis (Long p.275). Long elaborates on the tension between Lakatos’ Trotskyite tendency and his need for order and clarity.
- 9.
Motterlini (1999) pp.379-380.
- 10.
Lakatos (1978b) p.249.
- 11.
- 12.
Though not always. See the case of organic farmer Mark Purdey’s work on organo-phosphate insecticides and bovine spongiform encephalopathy.
- 13.
For the spontaneous appearance of counterexamples, see (1976) pp.10, 11, 13, 15, 16, 19, 21 & 22.
- 14.
“What a dramatic series of volte-faces! Critical Alpha has turned into a dogmatist, dogmatist Delta into a refutationist, and now inductivist Beta into a deductivist!” (1976) p.75.
- 15.
Lakatos (1976) pp.10-11.
- 16.
Op. cit. pp.33ff.
- 17.
Op. cit. pp.57ff.
- 18.
Op. cit. pp.88ff.
- 19.
“The class is a rather advanced one. To Cauchy, Poinsot, and to many other excellent mathematicians of the nineteenth century these questions did not occur.” (Op. Cit. p.8n3). See also p.52n3.
- 20.
Op. cit. p.117.
- 21.
Op. cit. pp.70ff.
- 22.
Op. cit. p.11.
- 23.
Bandy and Long 2000 p.89.
- 24.
Lakatos (1976) pp.142-154.
- 25.
Op cit. p. 142.
- 26.
Op cit. p. 154.
- 27.
Op cit. p. 142n2.
- 28.
See Jha (2006) pp.258-260.
- 29.
Polya 1954 volume II (Patterns of Plausible Inference) p.147.
- 30.
Op. cit. p.148.
- 31.
Op. cit. p.149-152. Of rational reconstructions, Polya says, “…the best stories are not true. They must contain, however, some essential elements of the truth… The following is a somewhat ‘rationalised’ presentation of the steps that led me to the proof…” p.148.
- 32.
Polya (2004) pp.20-22.
- 33.
Polya (1954) volume II (Patterns of Plausible Inference) pp.144-145.
- 34.
Polya (2004) p.6.
- 35.
Op. Cit. p.v (from the preface to the first printing).
- 36.
“Teaching to solve problems is education of the will” Op. Cit. p.94.
- 37.
“We shall attain complete certainty when we shall have obtained the complete solution, but before obtaining certainty we must often be satisfied with a more or less plausible guess.” Op. Cit. p.113.
- 38.
“The mathematical experience of the student is incomplete if he never had an opportunity to solve a problem invented by himself.” Op. Cit. p.68. “…he should endeavour to make his first important discovery: he should discover his likes and his dislikes, his taste, his own line.” Op. Cit. p.206.
- 39.
“[The future mathematician] should look out for the right model to imitate. He should observe a stimulating teacher.” Op. Cit. p.206.
- 40.
Perhaps Lakatos had a similar figure in mind when he insisted that control of the curriculum at the LSE should remain exclusively with the professors. See also ‘The Traditional Mathematics Professor’ (Op.Cit.p.208)
- 41.
Aside from early attempts to prove the Descartes-Euler formula, Lakatos gives the example of Cauchy’s 1821 proof that the limit of any convergent series of continuous functions is continuous (Lakatos 1976 appendix 1). However, even in this case, Lakatos indicates that Cauchy and others knew straight away that something was not right (Op. Cit.p.131). Thus, even in his best example, Lakatos could not show us the dramatic refutation of an apparently secure theorem.
- 42.
Op. Cit. p.93.
- 43.
Though the heuristic patterns in Proofs and Refutations are instructive. Lakatos addressed the corresponding problem in the philosophy of science with his Methodology of Scientific Research Programmes (1978a). Some authors have tried to carry this model (or parts of it, with modifications) from natural science into mathematics (see Hallett 1979, Koetsier 1991, Corfield 2003); for criticism of MSRP see Larvor (1998) esp. chapters four and six; for criticism of Methodologies of Mathematical Research Programmes, see Op. Cit. and Larvor (1997). For a consideration of Kuhnian approaches to the question of progress in mathematics, see Gillies (1992). For semantic shifts in mathematics, see Derrida (1978) or Grosholz (2007).
- 44.
Polya gives some examples, including a proof that all girls have the same colour eyes.
References
Bandy, A., & Long, J. (2000). Dress rehearsal for a revolution? The Hungarian Quarterly, XLI, 157.
Corfield, D. (2003). Towards a philosophy of real mathematics. Cambridge/New York: Cambridge University Press.
Derrida, J. (1978). Edmund Husserl’s origin of geometry: an introduction. New York: Nicolas Hays (J. P. Leavey, Trans.).
Feyerabend, P., & Lakatos, I. (1999). For and against method. Chicago: University of Chicago Press. Motterlini (ed).
Gillies, D. (1992). Revolutions in mathematics. Oxford/New York: Clarendon Press/Oxford University Press.
Grosholz, E., & Breger, H. (2000). The growth of mathematical knowledge. Boston: Kluwer.
Grosholz, E. R. (2007). Representation and productive ambiguity in mathematics and the sciences. New York: Oxford University Press.
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.
Jha, S. R. (2006). Hungarian studies in Lakatos’ philosophies of mathematics and science - Editor’s Introduction. Perspectives on Science, 14(3), 257-262.
Koetsier, T. (1991). Lakatos’ philosophy of mathematics: a historical approach (Studies in the History and Philosophy of Mathematics Vol.3). Amsterdam: North Holland.
Lakatos, I. (1947). Eötvös Kollegium - Gyorffy Kollegium. Valóság, 3.
Lakatos, I. (1976). Proofs and refutations. In J. Worrall, & E. Zahar (Eds.) Cambridge: Cambridge University Press.
Lakatos, I. (1978a). The methodology of scientific research programmes (Philosophical papers, Vol. 1). In J. Worrall, & E. Curry (Eds.) Cambridge: Cambridge University Press.
Lakatos, I. (1978b). Mathematics, science and epistemology (Philosophical papers, Vol. 2) . In J. Worrall, & E. Curry (Eds.) Cambridge: Cambridge University Press.
Larvor, B. (1997). Lakatos as historian of mathematics. Philosophia Mathematica, 5(1), 42-64.
Larvor, B. (1998). Lakatos, an introduction. London: Routledge.
Long, J. (1998). Lakatos in Hungary. Philosophy of the Social Sciences, 28, 244-311.
Polya, G. (1954). Mathematics and plausible reasoning. Princeton, NJ: Princeton University Press.
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).
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Larvor, B. (2010). Authoritarian Versus Authoritative Teaching: Polya and Lakatos. In: Hanna, G., Jahnke, H., Pulte, H. (eds) Explanation and Proof in Mathematics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-0576-5_6
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