In lieu of an abstract, here is a brief excerpt of the content:

4 The Simplest Mathematics [Peirce 1902]1 The last of our selections on the nature of mathematics is the fullest and latest; it comes from a logic text (the Minute Logic) and its date of 1902 makes it roughly contemporaneous with the Harvard Lectures on pragmatism and the Lowell lectures excerpted in selections 9, 12, and 13. Peirce varies and enlarges upon themes from the preceding selections: the relationship between mathematics and logic, the competing definitions of mathematics , the virtual infallibility of mathematical reasoning. The reader will be well equipped, at this point, to understand his masterful treatment of these topics here. Two points about logic do deserve special mention. Like all of the selections so far (except for the first one) this discussion of mathematics is ancillary to a logical treatise; in particular it is preparatory, Peirce tells us at the outset, to an exposition of ‘‘certain extremely simple branches of mathematics which . . . [are of] utility in logic.’’ Though it does not get spelled out in this excerpt, Peirce is picking up here on the remark in the previous selection that the logical ‘‘algebra of Boole is nothing but the algebra of . . . the simplest conceivable system of quantity’’; this is the ‘‘simplest mathematics ’’ that gives the selection its title. So Peirce is very serious, and very specific , here about the mathematical underpinnings of logic. At the same time—and this is the second point—he urges towards the end of the selection that the mathematical part of logic (‘‘Formal Logic,’’ as he calls it here) is not the only or even the most important part of the subject. What impresses him now is not the mathematical, but rather the ethical, underpinnings of logic. This is because logic is concerned with the criticism of a certain kind of conduct (namely, reasoning) and thus involves a kind of ethical evaluation. The reader may want to compare these remarks on logic and ethics with the scheme of the normative sciences which is laid out at some length in the fifth Harvard lecture on pragmatism (Peirce 1903j): there logic is the third of a trio of normative sciences—aesthetics, ethics, and logic— which are preceded in the larger hierarchy of the sciences only by mathematics and phenomenology. In the present selection he leaves aesthetics out of the picture, and singles out mathematics as the most abstract science by noting that it alone (except for ethics itself) has no need of ethics! Beings of Reason.book Page 23 Wednesday, June 2, 2010 6:06 PM 24 | Philosophy of Mathematics In addition to these variations on familiar themes, Peirce introduces two other leading ideas of his philosophy of mathematics, both of which have been discussed in the introduction: hypostatic abstraction (xxxviii) and the corollarial/theorematic distinction (xxx). Here he introduces the distinction by contrasting the mainly corollarial reasoning of the philosopher, which is concerned with words and definitions, with the mathematician’s theorematic manipulation of diagrams. The contrast collapses when we notice that words function, in corollarial/philosophical reasoning, as schemata. (What Peirce says here about philosophical reasoning is worth comparing with his remarks on the same subject in selection 3.) Peirce says that an abstraction has ‘‘a mode of being that merely consists in the truth of propositions of which the corresponding concrete term is the predicate.’’ This sounds like nominalism, but in the introductory discussion of ‘‘dormitive virtue’’ he insists that there ‘‘really is in opium something which explains its always putting people to sleep,’’ and in an important footnote he asserts that ‘‘even a percept is an abstraction,’’ which makes it ‘‘difficult to maintain that all abstractions are fictions.’’ Abstractions are, then, realities in some sense, though he does not explain very fully here in just what sense they are real: he has a lot more to say about this in selections 9, 12, and 13. In this chapter, I propose to consider certain extremely simple branches of mathematics which, owing to their utility in logic, have to be treated in considerable detail, although to the mathematician they are hardly worth consideration. In Chapter IV, I shall take up those branches of mathematics upon which the interest of mathematicians is centred, but shall do no more than make a rapid examination of their logical procedure. In Chapter V, I shall treat formal logic by the aid of mathematics. There can really be little logical matter in these chapters; but they seem to me to be quite indispensable preliminaries to the study of...

Share