I. Lakatos의 數理哲學의 敎育的 硏究

Abstract

The philosophical views about the problem of how mathematical knowledges grow have exerted the deepest influence upon mathematical education. Recently, according to the Zeitgeist regarding "teaching to think" or problem solving as a primary aim of school mathematics, the fallibilist philosophy of mathematics and the logic of mathematical discovery, which were developed by I. Lakatos under the influence of Hegel's dialectic, Popper's critical philosophy, and Polya's mathematical heuristics, have became a matter of deep concern to mathematics educators. The present study was undertakened to analyse the fallibilist position of the works of Lakatos and discuss the implications of it for the teaching of mathematics. Lakatos challenged to the mathematical formalism striking at the traditional heartland of infallibilsm, mathematical proofs, and characterized mathematics as a quasi-empirical science. According to his views, informal mathematics is so called a thought experimental science, which doesn't grow through a monotonous increase of the number of indubitably established theorem, but through improvement of conjectures by the logic of "proofs and refutations," and informal mathematical knowledges are no more than conjectures. Lakatos' philosophy of mathematics and logic of mathematical discovery suggest it as a major goal of mathematical education to develop the students' abilities and attitudes of critical and reasonable thinking and to learn how to do mathematics through genuine bona fide experience of mathematical rediscovery consisting of guessing, checking, proving, refuting, improving conjectures, and proof-generating concepts. Lakatos' philosophical position also rejects the traditional Euclidean deductive approach with the Parmenides-Platonic philosopy and the logic of inductive generalization, and suggests a way of humanizing mathematical education, realizing the idea of 'activism' in mathematical education, through critical fallibilist approach, that is, genetic-heuristic-Socratic- problematic situational- apprehensive approach. Lakatos' views challenge the formalist tradition of mathematics teaching, but this does't mean disregarding the logical construction of mathematics, rather require harmonizing the systematic deductive approach to ready-made mathematics and the heuristic approach to mathematics 'in statu nascendi.'