The metalinguistic perspective in mathematics∗

In this paper I consider three technical arguments in logic which have been held by various authors at diverse times to have consequences for the philosophy of mathematics. The first is Frege’s permutation argument, which Davidson (1979) has used to show that proper names have a determinate reference only in a relative sense. The second is the Löwenheim-Skolem theorem, which (like its close relative, the theorem on the existence of nonstandard models of arithmetic) apparently shows that the true sentences of a reasonably sophisticated first-order theory are not of themselves sufficient to fix the concepts of that theory. The third is the paradox of the set of all sets, which has been held by Lear (1977) and others to threaten a realist conception of set theory. It is the last of these arguments which interests me most and which was my starting point in writing this article. What led me to consider the other two was the realization that all three arguments have a common feature: they can be understood as being conducted in a metalanguage which does not coincide with the object language. The question I intend to address here is whether the metalanguage is to be understood merely as being external to the specific (mathematical) object language about which the argument attempts to draw philosophical conclusions, or whether it rather has to be understood, if the argument’s philosophical force is to be maintained, as external to all language. If the latter, there may be good reason to suppose the argument to be incoherent. Frege’s permutation argument is, of the three I shall be considering, the most mathematically straightforward and the most general in respect of the languages to which it may be applied. Suppose first that we have available to us an interpretation of some object-language. Suppose second that f is a non-trivial permutation of the domain of the interpretation. Define a new interpretation as follows: keep the domain unchanged; where the first scheme makes a name refer to an element a of the domain, let the second scheme make it refer to f(a); and where the first scheme makes an n-ary