The knowing mathematician

Mathematics is at the beginning of a new foundational crisis. Twenty years ago there was a firm consensus that mathematics is set theory and that set theory is Zermelo-Fraenkel set theory (ZF). That consensus is breaking down. It is breaking down for two quite different reasons. One of these is a turning away from the excesses of the tendency toward abstraction in post-war mathematics. Many mathematicians feel that the power of the method of abstraction and generalization has, for the time being, exhausted itself. We have done about as much as can be done now by these means, and it is time to return once more to hard work on particular examples. (For this point of view see Mac Lane [16, pp. 37–38].) Another reason for this turning back from abstraction is the economic fact that society is less prepared now than it was twenty years ago to support abstract intellectual activity pursued for its own sake. Those who support research are asking more searching questions than formerly about the utility of the results that can reasonably be expected from projects proposed. Mathematicians, moreover, are increasingly obliged to seek employment not in departments emphasizing pure mathematics, but in departments of computer science or statistics, or even in industry. Thus there is a heightened interest in applied and applicable mathematics and an increased tendency to reject as abstract nonsense what our teachers considered an intellectually satisfying level of generality. The set-theoretic account of the foundations of mathematics, however, is inextricably linked with just this tendency to abstraction for its own sake. Mathematics, on that account, is about abstract structures which, at best, may happen to be isomorphic to structures found in the physical world, but which are themselves most definitely not in the physical world. Thus as mathematicians turn away from pure abstraction, they also become increasingly dissatisfied with the doctrine that mathematics is set theory and nothing else.