ABSTRACT
This essay is a study of foundationalism and rationalism in mathematics, and the relationship between one’s views on foundationalism and the appropriate or best, logic. I suggest that foundationalism or rationalism explicitly or implicitly dominated work in logic and foundations of mathematics until recently, most notably logicism and the Hilbert Program. But foundationalism has now fallen into disrepute. It might be recalled that it does suggest a rather straightforward criterion for evaluating proposed logics and foundations: namely, self-evidence. The issues here concern how much of the perspective is still plausible and how logic and foundations are to be understood in the prevailing antifoundationalist spirit. How are candidate logics to be evaluated now? One common orientation seems to be to regard logic and, perhaps, mathematics in general as an exception to the prevailing antifoundationalism, or anti-rationalism-a sort of last outpost as it were. Against this, I argue that we have learned to live with uncertainty in virtually every special subject, and we can live with uncertainty in logic and foundations of mathematics as well. In like manner, we can live without completeness in logic, and live well.
