Quine’s “predilection” for finitism

Abstract

Rudolf Carnap’s notes of conversations he had in 1940–1941 with Alfred Tarski and W. V. Quine (a German transcription and English translation of which are now available in Frost-Arnold 2013) are packed with fascinating and sometimes puzzling clues about these logicians’ views. I shall focus here on just one of these clues: Quine’s commitment in the conversations to an austere sort of ontological finitism. Greg Frost-Arnold suggests that Quine’s interest in finitism in 1940–1941 was rooted in an epistemological foundationalism that he later recanted (FrostArnold 2013, 35–36). I shall try to show, on the contrary, that Quine’s interest in finitism in 1940–1941 was an early expression of an attitude toward ontological questions that was integral to his philosophy from the late 1930s on. Tarski is clearly the leader in the 1940–1941 conversations about finitism. He announces that he ‘‘truly understand[s] only a finite language S1; only individual variables, whose values are things; whose number is not claimed to be infinite (but perhaps also not the opposite).’’ On this view, a number is just one of a finite progression of concrete things. It follows that ‘‘many arithmetical sentences cannot be proved here, since we do not know how many numbers there are’’ (January 31, 1941; 156–157). Tarski rejects Carnap’s proposal that we view infinitary mathematics as analytic (March 6, 1940, 139–140) and says he ‘‘understands’’ the language of classical infinitary arithmetic—S2 for short—only insofar as he knows the formal rules (logical syntax) of S2; to ‘‘truly’’ understand S2, according to Tarski, a knowledge of its logical syntax is not enough (January 31, 1941; 157). Carnap’s notes record Quine as saying, in the weeks before Tarski introduces his finitism, that mathematics, like physics, involves ‘‘hypostases’’ that are underdetermined by experience. Quine also stresses that the paradoxes of set theory show that ‘‘familiar common sense results for finite classes’’ do not determine a general