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On the Foundations of Mathematics: MS 7 (c. 1903?)

By Charles S. Peirce Transcribed by Gary Furhman, April 2014

 The Robin Catalogue's mathematics section describes MS 7 as follows: 7. On the Foundations of Mathematics (Foundations) A. MS., n.p., [c.1903?], pp. 1-16, with 3 rejected pages; 17-19 of another draft. Mathematics as dealing essentially with signs. The MSS. below (Nos. 8-11) are drafts of this one, and all are concerned with the nature of signs. A large PDF file (2.6 megabytes) of images of MS 7 is at this location at the I.U.P.U.I. peirce-l archive. The images of MS 7 are also stored in a mammoth 26-megabyte PDF at the website of Grupo de Estudios Peirceanos; said PDF is linked at its page of manuscripts from 1887-1914. Manuscript page numbers have been inserted between backslashes by B.U. In each case, the first and larger number was assigned by Peirce, and the parenthesized second and smaller number was stamped on the side by a cataloguer. — B.U.

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On the Foundations of Mathematics

§1. Mathematics deals essentially with Signs. All that we know or think is so known or thought by signs, and our knowledge itself is a sign. The word and idea of a sign is familiar but it is indistinct. Let us endeavor to analyze it.

This attempt to begin an analysis of the nature of a sign may seem to be unnecessarily complicated, unnatural, and ill-fitting. To that I reply that every man has his own fashion of thinking; and if such is the reader's impression let him draw up a statement for himself. If it is sufficiently full and accurate, he will find that it differs from mine chiefly in its nomenclature and arrangement. [Not unlikely he might insist on distinctions which I avoid as irrelevant.] He will find that, in some shape, he is brought to recognize the same three radically different elements that I do. Namely, he must recognize, first, a mode of being in itself, corresponding to my quality; secondly, a mode of being constituted by opposition, corresponding to my object; and thirdly, a mode of being of which a branching line Y is an analogue, and which is of the general nature of a mean function corresponding to the sign.

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§2. Partly in hopes of reconciling the reader to my statement, and partly in order to bring out some other points that will be pertinent, I will review the matter in another order.

The reference of a sign to the quality which is its ground, reason, or meaning appears most prominently in a kind of sign of which any replica is fitted to be a sign by virtue of possessing in itself certain qualities which it would equally possess if the interpretant and the object did not exist at all. Of course, in such case, the sign could not be a sign; but as far as the sign itself went, it would be all that [it] would be with the object and interpretant. Such a sign whose significance lies in the qualities of its replicas in themselves is an icon, image, analogue, or copy. Its object is whatever that resembles it its interpretant takes it to be the sign of, and [it is a] sign of that object in proportion as \15(16)\ it resembles it. An icon cannot be a complete sign; but it is the only sign which directly brings the interpretant to close quarters with the meaning; and for that reason it is the kind of sign with which the mathematician works. For not only are geometrical figures icons, but even algebraical arrays of letters have relations analogous to those of the forms they represent, although these relations are not altogether iconically represented.

The reference of a sign to its object is brought into special prominence in a kind of sign whose fitness to be a sign is due to its being in a real reactive relation,—generally, a physical and dynamical relation,—with the object. Such a sign I term an index. As an example, take a weather-cock. This is a sign of the wind because the wind actively moves it. It faces in the very direction from which the wind blows. In so far as it \16(17)\ does that, it involves an icon. The wind forces it to be an icon. A photograph which is compelled by optical laws to be an icon of its object which is before the camera is another example. It is in this way that these indices convey information. They are propositions. That is they separately indicate their objects; the weather-cock because it turns with the wind and is known by its interpretant to do so; the photograph for a like reason. If the weathercock sticks and fails to turn, of if the camera lens is bad, the one or the other will be false. But if this is known to be the case, they sink at once to mere icons, at best. It is not essential to an index that it should thus involve an icon. Only, if it does not, it will convey no information. A cry of “Oh!” may be a direct reaction from a remarkable situation. But it will convey, perhaps, no further information. \17(18)\ The letters in a geometrical figure are good illustrations of pure indices not involving any icon, that is they do not force anything to be an icon of their object. The cry “Oh!” does to a slight degree; since it has the same startling quality as the situation that compells it. The index acts compulsively on the interpretant and puts it into a direct and real relation with the object, which is necessarily an individual event (or, more loosely, a thing) that is hic et nunc, single and definite.

A third kind of sign, which brings the reference to an interpretant into prominence, is one which is fit to be a sign, not at all because of any particular analogy with the quality it signifies, nor because it stands in any reactive relation with its object, but simply and solely because it will be interpreted to be a sign. I call such a sign a symbol. As an example of a symbol, Goethe's book on the Theory \18(19)\ of Colors will serve. This is made up of letters, words, sentences, paragraphs etc.; and the cause of its referring to colors and attributing to colors the quality it does is that so it is understood by anybody who reads it. It not only determines an interpretant, but it shows very explicitly the special determinant, (the acceptance of the theory) which it is intended to determine. By virtue of thus specially showing its intended interpretant (out of thousands of possible interpretants of it) it is an argument. An index may be, in one sense, an argument; but not in the sense here meant, that of an argumentation. It determines such interpretant as it may, without manifesting a special intention of determining a particular interpretant. It is a perfection of a symbol, if it does this; but it is not essential to a symbol that it should do so. Erase the conclusion of an argumentation and it becomes \19(20)\ a proposition (usually, a copulative proposition). Erase such a part of a proposition that if a proper name were inserted in the blank, or if several proper names were inserted in the several blanks, and it becomes a rhema, or term. Thus, the following are rhematic:
Guiteau assassinated ______
______ assassinated ______
Logicians generally would consider it quite wrong for me to call these terms; but I shall venture to do so.

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