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14 The Logic of Quantity [Peirce 1893c] Kant and Mill, the principal objects of Frege’s critical scrutiny in The Foundations of Arithmetic, were of great importance for Peirce as well, and in this selection he locates his epistemology for mathematics relative to theirs.1 He takes Kant to task for holding that analytic truths can be ascertained by “a simple mental stare,” and proceeds to rewrite Kant along the future-oriented lines of his own pragmatism: to find out what is involved in a concept, we must see what we can evolve from it by way of experimentation on diagrams. As in the somewhat later selection 7, Peirce uses a modified associationist framework to unify the inner experimentation of the mathematician and the outer experimentation of the natural scientist. He acknowledges that “the difference between the inward and outward worlds is very, very great, with a remarkable absence of intermediate phenomena”; but ultimately the difference is “merely one of how much.” This reconstruction of the analytic/synthetic distinction enables Peirce to put mathematics, in opposition to Kant, on the analytic side.2 His way of doing this looks, at first, like a kind of logicism, but he holds in the end to his usual view of mathematics as “prelogical.” So he is not saying that mathematics (in particular, arithmetic) rests on logic, but rather that (successful) mathematical reasoning does unfold what is “involved” in its hypotheses: this is just a quasi-Kantian way of saying that mathematics is “the science which draws necessary conclusions.” Yet mathematics is also an experimental science, and Peirce agrees with Mill that “experience is the only source of any kind of knowledge.” At the same time he denies that mathematics is experiential in Mill's sense. He concedes that putative counterexamples to simple arithmetical propositions are conceivable; indeed, he holds that they “often happen” but are not genuine counterexamples because the “arithmetical propositions are not understood in an experiential sense.” Their justification in inner experience renders them immune to the kind of refutations Mill envisions. But that justification does nonetheless involve experience broadly speaking, and so Mill is wrong to say that logically necessary propositions are, by virtue of their necessity alone, merely verbal. Peirce refuses to call them a priori because this suggests that their discovery is a matter of “[applying] plain rules to plain cases.” He prefers to call them innate, “because that may be innate which is Beings of Reason.book Page 107 Wednesday, June 2, 2010 6:06 PM 108 | Philosophy of Mathematics very abstruse, and which we can only find out with extreme difficulty.” In the final analysis, then, Peirce holds that arithmetic is analytic yet experimental , and neither a priori nor a posteriori, but innate. It may be doubted whether he really has found a “third way” between Kant and Mill. No doubt Frege would have felt that Peirce took too much from them both. In any case this selection bears close comparison with Frege’s more famous critique.§231. Kant, in the Introduction to his Critic of the Pure Reason,3 started an extremely important question about the logic of mathematics. He begins by drawing a famous distinction, as follows: In judgments wherein the relation of a subject to a predicate is thought . . . this relation may be of two kinds. Either the predicate, B, belongs to the subject, A, as something covertly contained in A as a concept ; or B is external to A, though connected with it. In the former case, I term the judgment analytical; in the latter synthetical. Analytical judgments , then, are those in which the connection of the predicate with the subject is thought to consist in identity, while those in which this connection is thought without identity, are to be called synthetical judgments. The former may also be called explicative, the latter ampliative judgments, since those by their predicates add nothing to the concept of the subject, which is only divided by analysis into partial concepts that were already thought in it, though confusedly; while these add to the concept of the subject a predicate not thought in it at all, and not to be extracted from it by any analysis. For instance, if I say all Bodies are extended, this is an analytical judgment. For I need not go out of the conception I attach to the word body, to find extension joined to it; it is enough to analyze my meaning, i.e. merely to become aware of the various...

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