What Finitism Could Not Be *

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.