Sets and Numbers

Abstract

Various positions in the philosophy of mathematics have been grouped together under the heading or Platonism.' The central tenets of these views are that mathematical statements are either true or false, that they are so by virtue of the properties of independently existing mathematical objects (not the makeup of the human mind, the idiosyncracies of human language, etc.), and that their possessing these truth values is independent of our ability (or inability) to determine which truth values they possess. (Dummett [4], p. 147, Putnam [ 19], pp. 69-70) Views of this sort allow a standard Tarskian semantics for mathematical discourse, square with the pre-philosophical views of many mathematicians, and simplify our account of applied mathematics. (Putnam [19], pp. 337-341) Still, mathematical realisms seem susceptible to popular challenges of two types. Those of the first type are basically epistemological: for example, given that mathematics is about objects of a curious sort, how are human beings able to interact with these objects in ways appropriate for referring to or knowing about them? (Benacerraf [1], Lear [13], Jubien [10]) Those of the second type, the ontological challenges, apply to mathematical realisms with logicist, or more accurately, set-theory-ist tendencies; for example, if numbers are sets, which sets are they?2 (Benacerraf [2], Kitcher [11]) My goal here is to indicate how one particular form of mathematical realism with set-theory-ist tendencies might begin to meet these challenges. This particular view, which I will call set theoretic realism, derives from the writings of G6del ([8] and [9]), though some deviations and extrapolations from the texts have been necessary. I hope that set theoretic realism will seem attractive, or at least interesting, in its own right, because I won't argue for it directly. To repeat, my limited objective is to meet some forms of the above-mentioned popular challenges. Even this is a fairly large task, so it is fortunate that my defense of set theoretic realism against some recent versions of the epistemological challenge is contained elsewhere ([14]).3 Here I will