Intensionality and the Gdel Theorems

Philosophers of language have drawn on metamathematical results in varied ways. Extensionalist philosophers have been particularly impressed with two, not unrelated, facts: the existence, due to Frege/Tarski, of a certain sort of semantics, and the seeming absence of intensional contexts from mathematical discourse. The philosophical import of these facts is at best murky. Extensionalists will emphasize the success and clarity of the model theoretic semantics; others wiU emphasize the relative poverty of the mathematical idiom; still others will question the aptness of the standard extensional semantics for mathematics. In this paper I will investigate some implications of the G6del Second Incompleteness Theorem for these positions. I shall argue that the realm of mathematics, proof theory in particular, has been a breeding ground for intensionality and that satisfactory intensional semantic theories are implicit in certain rigorous technical accounts. One moral to be drawn is that intensionality does not, as a matter of course, involve incoherence. The central argument will conclude that an extensional semantics would attribute falsity to the G6del Second Incompleteness Theorem. Since we have good reason to believe the theorem true, the correct inference is to the insufficiency of extensional semantics. The intensional context lurking in the Second Theorem is that of indirect discourse. This indirect discourse is, contra Quine, not "at variance with the characteristic objectivity of science". In fact, rigorous generalizations of the Second Theorem provide the proper non-extensionalist treatment. Each of the matters just broached involves significant subtleties even the matter of justifying the truth of the Second Theorem. Moreover, the relevant technical work is not well reported in the philosophical literature 1 and has only recently been assimilated into logic texts. 2 In what follows, therefore, the methodological and metamathematical surveys are necessary preliminaries.