During the first half of this century classical mathematics, as constructed within the foundational framework provided by ZFC was challenged primarily, but not effectively, by intuitionist doctrines. (We take ZFC here to be representative of those first order theories based on the iterative conception of set which could equally well have served as a foundation for the mathematics of the period.)
In the second half a wider range of influences from philosophy, mathematics and computer science have increased pressure on the iterative conception of sets as the foundation for modern mathematics. The most signficant of these pressures comes from Computer Science. Many academic computer scientists with theoretical inclinations are drawn to the power of abstraction provided by category theoretic methods, while category theoreticians themselves strain against the limitations on abstraction imposed by classical set theory. At the same time arguments for constructive mathematics have been plausibly aimed at computer scientists. Arguments that these are more relevant and practical in computing than classical methods are accepted enthusiastically and without scrutiny.
Academics will persue research provided only that they find the material interesting and that some else finds it fundable. Funding agencies in turn will often operate by peer review aiming primarily to fund the most able researchers whose opinions about the merits of various avenues of research are definitive.
In this context a philosophical interest in the reasons why particular lines are followed may be difficult to satisfy.
We will consider criticisms of classical foundations arising from the following general sources:
R.© Roger Bishop Jones; Created: 1995/3/5; Modified: 1995/3/5