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

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.

It is plain at the outset, first, that a sign is not any particular replica of it. If one casts one's eye down a printed page, every ‘the’ is the same word, and every e the same letter. The exact identity is not clear. Secondly, a sign may be complex; and the parts of a sign, though they are signs, may not possess all the essential characters of a more complete sign. Thirdly, a sign sufficiently complete must be capable of determining an interpretant sign, and must be capable of ultimately \2(3)\ producing real results. For a proposition of metaphysics which could never contribute to the determination of conduct would be meaningless jargon. On the other hand, the cards which, slipped into a Jacquard loom, cause appropriate figures to be woven, may very properly be called signs although there is no conscious interpretation of them. If not, it can only be because they are not interpreted by signs. In fact, in the present condition of philosophy, consciousness seems to be a mere quality of feeling which a formal science will do best to leave out of account. But a sign only functions as a sign when it is interpreted. It is therefore essential that it should be capable of determining an interpretant sign. Fourthly, a sign sufficiently complete must in some sense correspond to a real object. A sign cannot even be false unless, with some degree of definiteness, it specifies the real \3(4)\ object of which it is false. That the sign itself is not a definite real object has been pointed out under “firstly”. It is only represented. Now either it must be that it is one thing to really be and another to be represented, or else it must be that there is no such thing [a]s falsity. This involves no denial that every real thing may be a representation, or sign, but merely that, if so, there must be something more in reality than mere representation. Since a sufficiently complete sign may be false, and also since it is not any replica or collection of replicas, it is not real. But it refers to a real object. Consequently, a sign cannot have a sign as its sole object; though it may refer to an object through a sign; as if one should say, “Whatever the Pope, as such, may declare will be true,” or as a map may be a map of itself. But supposing the Pope not to declare anything, does that pro- \4(5)\ position refer to any real object? Yes, to the Pope. But, fifthly, even if there were no pope, still, like all other signs sufficiently complete, there is a single definite object to which it must refer; namely, to the ‘Truth,’ or the Absolute, or the entire Universe of real being. Sixthly, a sign may refer, in addition, and specially, to any number of parts of that universe. Seventhly, every interpretant of a sign need not refer to all the real objects to which the sign itself refers, but must, at least, refer to the Truth. Eighthly, an interpretant may refer to an object of its sign in an indefinite manner. Thus, given the sign, ‘Enoch was a man, and Enoch was translated,’ an interpretant of it would be ‘Some man was translated.’ Ninethly, a sign may refer to its interpretant in such a way that, in case the former sign is incomplete, the interpretant, being an interpretant of the completer sign, may refer to a sign to \5(6)\ which the first sign does not specially refer, but only generally refers. Thus, the sign ‘Any man there may be is mortal’ does not refer to any real man, unless it so happens that it is a part of a sign which otherwise refers to such a real thing. But if it be a part of a sign of which another part is ‘some man sings,’ the sign ‘some man is mortal’ becomes an interpretant of it. This may be more conveniently expressed by speaking of an ‘utterer’ and an ‘interpreter.’ Then the utterer says to the interpreter, “you are at liberty to understand me as referring to any man [of] whom you can get any indication, and of him, I say, ‘he is mortal.’” Tenthly, a sign sufficiently complete must signify some quality; and it is no more important to recognize that the real object to which a sign refers is not a mere sign than to recognize that the quality it signifies is not a mere \6(7)\ sign. Take the quality of the odor of attar. There is no difficulty in imagining a being whose entire consciousness should consist in this alone. But, it may be objected, if it were contrasted with nothing could it be recognized? I reply, no; and besides, such recognition is excluded by the circumstance that a recognition of the smell would not be the pure smell itself. It may be doubted by some persons, however, whether the feeling could exist alone. They are the persons whom it ought to be easiest for me to convince of my point. For they, at least, must admit that if such pure homogeneous quality of feeling were to exist alone, it would not be a sign. Everybody ought to admit it because it would be alone, and therefore would have no object different from itself. Besides, there would be no possible \7(8)\ replica of it, since each of two such things would be nonexistent for the other; nor could there be any third who should compare them. So, then, the whole question of whether such a quality is a sign or not resolves itself into the question of whether there could be such a tinge upon the consciousness of a being, supposing the being could be conscious (for I shall show presently that the fact that he would be asleep is only in my favor). In order to decide this question, it will be sufficient to look at any object parti-colored in bright red and bright blue and to ask oneself a question or two. Would there be any possibility of conveying the idea of that red to a person who had no feeling nearer to it than that blue? Plainly not, the quality of the red is in the red itself. The proximity of the blue heightens the shock up[on] the seer[']s organism, \8(9)\ emphasizes it, renders it vivid, perhaps slightly confuses the feeling. But the red quality is altogether positive and would remain if the blue were not there. If every other idea were removed, there would be no shock, and there would be sleep. But the quality of that sleep would be red, in this sense, that if it were taken away frequently and brought back so as to wake the being up, the tinge of his consciousness would be of that quality. A quality, in itself, has no being at all, it is true. It must be embodied in something that exists. But the quality is as it is positively and in itself. That is not true of a sign, which exists only by bringing an interpretant to refer to an object. A quality, then, is not a sign. Eleventhly, we may assume that this is as true of what is, with excusable inaccuracy, called a composite quality as of a simple one. \9(10)\ In itself, one quality is as simple as another. A person who should be acquainted with none but the spectral colors would get no idea of white by being told that it was the mixture of them all. One might as well tell him to make a mixture of water, patriotism, and the square root of minus one. Find a man who has had no idea of patriotism; and if you tell him that it is the love of one's country, if he knows what love is, and what a man's country, in its social sense, is, he can make the experiment of connecting ideas in his imagination, and noting the quality of feeling which arises upon this composition. Tell him this in the evening, and he will repeat the experiment several times during the night, and in the morning he will have a fair idea of what patriotism means. He will \10(11)\ have performed an experiment analogous to that of mixing colored lights in order to get an idea of white. If a treasure is buried in the midst of a plain, and there are four signal poles, the place of the treasure can be defined my means of ranges, so that a person who can take ranges and set up new poles can find the treasure. In like manner the name of any color may be defined in terms of four color disks so that a person with a color-wheel can experimentally produce the color and thereafter be able to use the name. Every definition to be understood must be treated as a precept for experimentation. The imagination is an apparatus for such experimentation that often answers the purpose, although it often proves insufficient. No point on the plain where the treasure is hid is more simple than other. Colors may be defined by various systems of coördinates, and \11(12)\ we do not know that one color is in itself simpler than another. It is only in a limited class of cases that we can define a quality as simply a mixture of two qualities. In most cases, it is necessary to introduce other relations. But even when that is the case, if a quality is defined as being at once a and b, there will always be another way of defining it as that which is at once c and d. Now all that is either a or c will have a certain quality p, common and peculiar to that class; the class of possible objects that are b or c will be similarly related to a quality, r; and the class of possible objects that are either b or d will be similarly related to a quality, s. Then that quality which was defined as, at once, a and b, can be more analytically defined as that which is at once p, q, r, and s; \12(13)\ and so on ad infinitum. We may not be able to make out these qualities; but there is reason to believe that any describable class of possible objects has some quality common and peculiar to it. It is certain that a pure quality, in its mode of being as a pure quality, does not cease to be because it is not embodied in anything. Every situation in life appears to have its peculiar flavor. This flavor is what it is positively and in itself. For the experiment by which it may be reproduced an adequate prescription may be given; but the definition will not itself have that flavor. To say that a flavor, or pure quality, is composed of two others, is simply to say that on experimentally mixing these others in a particular way, that first flavor will be reproduced. Every sufficiently complete sign determines a sign to the effect that on a certain occasion, that is, in a certain object a certain flavor or quality may be \13(14)\ observed.

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.