Perception and Mathematical Intuition

Abstract

S ET theoretic realism is a view whose main tenets are that sets exist independently of human thought, and that set theory is the science of these entities.1 The foremost advocate of this position, the late Professor Gddel, has stressed an analogy between mathematics and physical science.2 According to G6del, higher set theory bears a relation to the rest of mathematical knowledge and to practical mathematical dealings of everyday life which is analogous to the relation borne by theoretical physics to physical science in general and to common sense knowledge of the world. Sense perception gives us knowledge of simple facts about physical objects, and a faculty of mathematical intuition gives us knowledge of sets, numbers,3 and of some of the simpler axioms concerning them. In both cases, theories involving unobservable entities or processes (that is, entities or processes beyond the range of sense perception or mathematical intuition) are formed in order to explain, predict, and systematize the elementary facts (of perception or intuition) and are judged by their success. A view of this sort has several attractive features: i) it allows a straightforward Tarskian semantics for set theoretic discourse, ii) it makes no mystery of how mathematical premises can combine with physical ones to yield testable consequences in physical science (that is, both sorts of premises are true in the same sense), iii) it squares with the prephilosophical views of most working