What Finitism could not be.

SUMMARY: In his paper "Finitism" (1981), W.W. Tait maintains that the chief difficulty for everyone who wishes to understand Hilbert's conception of finitist mathematics is this: to specify the sense of the provability of general statements about the natural numbers without presupposing infinite totalities. Tait further argues that all finitist reasoning is essentially primitive recursive. In this paper, we attempt to show that his thesis "The finitist functions are precisely the primitive recursive functions" is disputable and that another, likewise defended by him, is untenable. The second thesis is that the finitist theorems are precisely the universal closures of the equations that can be proved in PRA.

KEY WORDS: finitist functions, primitive recursive functions, infinite totalities, finitist proof of the universal closure of an equation

1. Tait's Interpretation of Finitism

In his influential essay "Finitism" (1981), W.W. Tait sets himself the task (a) of explicating the notion of finitism by explaining a sense in which one can prove general statements about the natural numbers without assuming infinite totalities, and (b) of arguing for the thesis that all modes of finitist reasoning are essentially primitive recursive. Tait maintains that the significance of finitism is due to the fact that it is a minimal kind of reasoning presupposed by any nontrivial reasoning about the concept of number. In this sense, finitism is fundamental to mathematics, although Hilbert's attempt to found mathematics on finitism miscarries definitively, according to Tait (cf. Tait 1981, pp. 526, 540, 546).

In what follows, we try to show that one thesis advocated by Tait is questionable and another, likewise defended by him, indefensible. The thesis (Tait 1981, p. 533), "The finitist functions are precisely the primitive recursive functions" we call Tait's First Thesis. The thesis (Tait 1981, p. 537) "The finitist theorems are precisely the universal closures of the equations that can be proved in Primitive Recursive Arithmetic" (PRA) will be referred to as Tait's Second Thesis. (1)

At the outset of his paper, Tait attempts to pinpoint the main difficulty for everyone who wants to understand Hilbert's conception of finitist mathematics. He believes that this difficulty is embodied by the question as to how to specify the sense of the provability of general statements about the natural numbers without presupposing some infinite totality. Somewhat surprisingly, Tait does not distinguish explicitly between finitist mathematics and finitist metamathematics along the lines of Hilbert. When we talk of Hilbert's finitism, we ordinarily mean his finitist proof theory or metamathematics rather than what he takes to be the finitist portion of a formalized mathematical theory. As an example of a general statement, Tait chooses "[for all] xy(x + y = y + x)". This is a [[PI].sup.0.sub.1]-sentence (2) which appears to be a constituent of the language of formalized number theory rather than a sentence belonging to the (formalized or informal) language of metamathematics. We think that at this point Tait ought to have quoted a [[PI].sup.0.sub.1]-sentence whose formulability in the formalized language of metamathematics really counts for the proof theorist; "[Con.sub.pa]" would be an appropriate example. (3)

In any case, it seems to us that it would have been in Tait's own interest to set up the requirement that nontrivial finitist metamathematics be able to prove [[PI].sup.0.sub.1]-sentences (or [[PI].sup.0.sub.0]-formulae) without assuming some infinite totality. When he comes to describe a certain finitist framework for establishing some theses concerning finitist mathematics and PRA, he does so in terms of conditions which he himself regards as appropriate without being reasonably faithful to Hilbert. We thus face the question: Does Tait intend to present Hilbert's answer to his initial question concerning the finitist provability of general statements, or does he wish to give his own answer by endowing the term "finitism" with a sense which may be distinct from that one Hilbert attaches to it? To repeat: if we adopt Hilbert's finitist standpoint and if we wish to show [Con.sub.[tau]] for certain mathematical, axiomatizable theories T (with representation [tau]), we must clearly distinguish between finitist mathematics and (finitist) metamathematics. It is obvious that Tait, unlike Hilbert, considers metamathematics to be a formal mathematical theory. Thus, we ought to regard with reserve Tait's contention (1981, pp. 525f., 540, 546) that the only real divergence of his characterization of finitism from Hilbert's account concerns Hilbert's epistemological distinction between numbers and transfinite objects such as functions and sets in terms of representability in intuition. (4) For the present, we simply want to record that the only argument that Tait adduces in favour of the possibility of formulating finitistically [[PI].sup.0.sub.1]-sentences is this: it should be possible to establish the consistency of certain mathematical theories by finitist metamathematical means, and, thus, for Tait it should at least be possible to formulate consistency assertions [Con.sub.[tau]] metamathematically. (5)