Mathematics, The Computer Revolution and the Real World
James FRANKLIN · Philosophica · 1988
It is widely agreed that, the foundationalist program being exhausted, it is time for new direction~ in the philosophy of mathematics.Unfortunately, the plural in "directions" has been too rarely justified.We have seen a series of sociological case studies of the discovery of proofs in such areas as number theory, logic and the foundations of calculus.These have been not without interest, but are their choices of subject-matter perhaps too narrow?Sociology, like such sciences as psychology, history and anthropology, is a two-edged sword (or, perhaps more accurately, a boomerang).For these sciences are empirical (at least potentially), and subject to certain norms and standards of procedure and evidence.When Joseph de Maistre remarked, "You might as well say sheep are bor.ncarnivorous, and everywhere they eat grass", he exposed the fact that Rousseau was not an actual anthropologist, and that the Noble Savage was an inhabitant of the European mind's stock of paradigms, thought experiments, rational reconstructions, or myths, not of Tahiti.If philosophers of mathematics wish to embrace the social sciences, they must take account of the appropriate methodol~gy -and this means the current methodology, not that of speculative forerunners like Sir James George Frazer or Franz Anton Mesmer.If history is to be studied, it must be actual history, not reconstructions of it, rational or otherwise.If the psychology of mathematicians is to be used, a fair sample of mathematicians must be interviewed, and the investigator must be able to explain what experiments or surveys have been conducted, and at what level of significance were the results.A sociological or anthropological study must design and report its studies in the field in the standard way.In particular, with all such sciences, strict attention must be paid to whether any case studies used are "typical".Now "being typical" is not an absolute term, but a relation