Strict finitism

Conclusion Dummett's objections to the coherence of the strict finitist philosophy of mathematics are thus, at the present time at least, ill-taken. We have so far no definitive treatment of Sorites paradoxes; so no conclusive ground for dismissing Dummett's response — the response of simply writing off a large class of familiar, confidently handled expressions as semantically incoherent. I believe that cannot be the right response, if only because it threatens to open an unacceptable gulf between the insight into his own understanding available to a philosophically reflective speaker and the conclusions available to one confined to observing the former's linguistic practice; for an observer of our linguistic practice could never justifiably arrive at the conclusion that red, child, etc., are governed by inconsistent rules. But the Sorites is not the subject of this paper. The points I hope to have made plausible are: that a generalized intuitionist position cannot be so much as formulated and that even a most local intuitionism, argued for the special case of arithmetic, is hard pressed effectively to stabilize and defend itself; that strict finitism remains the natural outcome of the anti-realism which Dummett has propounded by way of support for the intuitionist philosophy of mathematics; that it is powerfully buttressed by the ideas of the latter Wittgenstein on rule-following; and that there is no extant compelling reason to suppose that its involvement with predicates of surveyability calls its coherence into question. The correct philosophical assessment of strict finitism, and its proper mathematical exegesis, remain absolutely open, almost virgin issues. This is not a situation which philosophers of mathematics should tolerate very much longer.The term was introduced by Kreisel in 6] to denote what he took to be an aspect of Wittgenstein's later philosophy of mathematics; and taken over by Kielkopf (Strict Finitism, Mouton 1970) — misunderstanding, as it seems to me, both Kreisel and Wittgenstein — as a label for Wittgenstein's later philosophy of maths. in its entirety. It is not a happy label for the ideas I am concerned with, since it is only from non-strict finitist points of view that the strict finitist can be straightforwardly seen as stressing the finitude of human capacities, countenancing only finite sets, etc. (See subsections 5 and 6 below). But we need a labeel; and Dummett in 3] has already followed Kreisel's lead. Anyway, a rose by any other name,...