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By Greg Frost-Arnold
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Extra resources for Carnap, Tarski, and Quine at Harvard: Conversations on Logic, Mathematics, and Science
22 Tarski himself cites Chwistek (090–16–09) and Kotarbi´ nski (090– 16–28) for some of the ideas he presents, so I will first briefly outline the claims of these two Polish philosophers that are most relevant to the finitist-nominalist project. Next, I present possible indirect lines of influence that Russell’s ideas may have had on the formation of the 1941 FN project. Finally, I briefly sample contemporaneous skeptical complaints about infinity from Wittgenstein and Neurath. 1 The Poles: Chwistek, Kotarbi nski, Le´sniewski The finitist-nominalist project is originally Tarski’s proposal; thus, it is natural to look to the philosophical ideas he was exposed to during his intellectual development in Poland to find his inspiration for the FN conditions.
Therefore, you do not have as ultimate constituents of your world, these queer things that you are inclined to call numbers. (Russell 1918/1956, 270) The fact that the leading philosophical luminary of Carnap, Tarski, and Quine’s early careers called classes ‘fictions’ and declared numbers, the things successfully manipulated by six-year-old children, to be ‘queer things’ could play some role in inclining Tarski, Quine, and others to consider the refusal to allow numbers into the universe of discourse prima facie plausible or reasonable.
I only ‘understand’ any other language in the way I ‘understand’ classical mathematics, namely, as a calculus; I know what I can derive from what (rather, I have derived; ‘derivability’ in general is already problematic). With any higher ‘Platonic’ statements in a discussion, I interpret them to myself as statements that a fixed sentence is derivable (or derived) from certain other sentences. ) (090–16–28) The contrast between fully ‘intelligible language’ and ‘uninterpreted calculus’ also appears, albeit more briefly, elsewhere in the discussion notes (090–16– 25, –04, –05).