Mathematical Logic: What Has It Done for the Philosophy of Mathematics?

The aim of this paper is not to provide any systematic reconstruction of Kreisel’s views but only 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’s views greatly influenced me in the Sixties andthe 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, I have come to the conclusion that Kreisel’s criticism has not 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 instead of stressing the need of putting the subject on new grounds replacing it by a more adequate approach. This seems absolutely necessary, not only to rectify misconceptions about the nature of mathematics, such as those spelled out in [Cr], for which mathematical logic has great responsibility, but also to meet the challenges that logic has to face as a result of the development of entirely new subjects such as computer science and artificial intelligence. To such challenges mathematical logic has been so far hopelessly unequal, contrary to McCarthy’s [Mc, p. 69] hope that the relationship between these new subjects and mathematical logic could be as fruitful as the one between analysis and physics. In this paper I will try to explain the reasons of my disagreement with Kreisel, as much as it can be done in the limited space allowed to me.