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)