MATHEMATICAL LOGIC: WHAT HAS IT DONE FOR THE PHILOSOPHY OF MATHEMATICS?

Abstract

to discuss some claims concerning the relationship between mathematical logic and the philosophy of mathematics that repeatedly occur in his writings. Although I do not know to what extent they are representative of his present position, they correspond to widespread views of the logical community and so seem worth discussing anyhow. Such claims will be used as reference to make some remarks about the present state of relations between mathematical logic and the philosophy of mathematics. Kreisel’sviewsgreatlyinfluencedmeintheSixtiesandthe Seventies. His critical remarks on the foundational programs taught me that one could and should have an approach to the subject of mathematical logic less dogmatic, corporative and even thoughtless than the one the logical community sometimes used to have. This is even more true today when the professionalization of mathematical logic generates a flood of results but few new ideas and the lack of ideas leads to the sheer byzantinism of most current production in mathematical logic. In the past few years,however,IhavecometotheconclusionthatKreisel’scriticism hasnot been radical enough: his main worry seems to have been to preserve as much as possible- to save the savable- of the tradition of mathematical logic. His critical remarks have focused on the defects of the foundational schools, thus drawing attention away from the intrinsic defects of mathematical logic itself