Fréchet and the logic of the constitution of abstract spaces from concrete reality (Q1408667)

This paper examines Fréchet's philosophy of mathematics, both in its cognitive aspects and its pedagogical implications. Starting from Poincaré's conventionalism Fréchet insisted that conventions were not arbitrary but derived from experience. This includes mental experience. However, mathematical ideas that are ``truly new and important'' are suggested by natural problems. He believed that mathematics was constructed by ``successive schematising or inductive synthesis.'' This last has pedagogical implications, which are relevant in contemporary pedagogical debates. He [Fréchet] believed that ``the discovery method was also the most adequate way of communicating knowledge.'' This was conformal to the French method of instruction as contrasted with the German. Fréchet did not reject axiomatization (indeed had used it himself in famous work). However, axiomatization was not appropriate as a pedagogical device; a student should be led to discoveries, whence an axiom is formulated; to be ``so clear and simple that when the lesson is over the student can say to himself: 'how is it that I did not think like this before myself?' '' The student also must be made to understand that mathematics contributes only an approximate reality. The paper touches on the influence on Fréchet of Jean-Louis Destouches, for whom inductive synthesis was a preliminary part of any physical theory, to be followed by ``axiomatic utterance'' and deduction. Also considered are Fréchet's relations to (or avoidance of) the Kantian categories, and his debate with Enriques' views. The paper is unfortunately marred by a number of linguistic solecisms that occasionally cause a reader hesitation. But it is well worth reading.