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5 The Essence of Reasoning [Peirce 1893a] The ideality of mathematics is the center of attention in this brief but very rich excerpt from the Grand Logic. Peirce compares arithmetic , a branch of pure mathematics, with the applied fields of logic and geometry. Logic is “intermediate” between the other two: unlike arithmetic, it is concerned with “questions of fact,” but unlike geometry, it “knows nothing about the truths of nature.” Arithmetic and geometry are more sharply separated. Space, the subject matter of geometry, is “a matter of real experience ,” and geometrical concepts like straightness and length involve vision and “the sense of muscular action.” Moreover, some geometrical assertions at least are conceivably open to experimental refutation. With arithmetic, on the other hand, this is quite inconceivable. An apparent counterexample to some basic number theoretic truths is seen to fail because it does not “conform to the idea of number.” Here Peirce takes his stand with Frege, over against Mill and the empiricist tradition, in denying that the truths of arithmetic are empirical generalizations about the behavior of physical objects (Frege 1884, 12–13). The truths of mathematics are truths about ideas merely. They are all but certain. Only blundering can introduce error into mathematics. Questions of logic are questions of fact. Can the premise be true and the conclusion be false at the same time? But the logician, as such, knows nothing about the truth of nature. He only hopes that a few assumptions he makes may be near enough correct to answer his purpose in some measure. These assumptions are, for instance, that things are sufficiently steady for something to be true, and what contradicts it false, that nothing is true and false at once, etc. The assertion that mathematics is purely ideal requires some explanation . Thomson and Tait (Natural Philosophy §438) wisely remark that it is “utterly impossible to submit to mathematical reasoning the exact conditions of any physical question.”1 A practical problem arises, and the physicist Beings of Reason.book Page 37 Wednesday, June 2, 2010 6:06 PM 38 | Philosophy of Mathematics endeavors to find a soluble mathematical problem that resembles the practical one as closely as it may. This involves a logical analysis of the problem, a putting of it into equations. The mathematics begins when the equations, or other purely ideal conditions, are given. “Applied mathematics” is simply the study of an idea which has been constructed so as to be more or less like nature. Geometry is an example of such applied mathematics; although the mathematician often makes use of space imagination to form icons of relations which have no particular connection with space. This is done, for example, in the theory of equations and throughout the theory of functions. And besides such special applications of geometrical ideas, all a mathematician ’s diagrams are visually imagined, and involve space. But space is a matter of real experience; and when it is said that a straight line is the shortest distance between two points, this cannot be resolved into a merely formal phrase, like 2 and 3 are 5. A straight line is a line that viewed endwise appears as a point; while length involves the sense of muscular action. Thus the connection of two experiences is asserted in the proposition that the straight line is the shortest. But 2 and 3 are 5 is true of an idea only, and of real things so far as that idea is applicable to them. It is nothing but a form, and asserts no relation between outward experiences. If to a candle, a book, and a shadow,—three objects, is joined a book, one, the result is 5, because there will be two shadows; and if 5 more candles be brought, the total will be only 8, because the shadows are destroyed. But nobody would take such facts as violations of arithmetic; for the propositions of arithmetic are not understood as applicable to matters of fact, except so far as the facts happen to conform to the idea of number. But it is quite possible that if we could measure the angles of a triangle with sufficient accuracy, we should find they did not sum up to 180o, but either exceeded it or fell short. There is no difficulty in conceiving this; although, owing to numerous associations of ideas, it is necessary to devote some weeks to a careful study of the matter before it becomes perfectly clear. Accordingly, geometrical propositions and arithmetical propositions stand...

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