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15 Recreations in Reasoning [Peirce 1897(?)b] Along with selection 10, this mathematico-philosophical treatment of the natural numbers is one of the strongest pieces of evidence in support of the claim that Peirce held to a kind of mathematical structuralism .1 It is also, as discussed in the introduction (xxxix), an important source of information about Peirce’s views on the metaphysics of mathematical structure. The selection culminates in a Dedekindian axiomatization of the natural numbers, and the derivation of some fundamental properties—including the Fermatian Principle, or mathematical induction—from the basic axioms.2 But Peirce’s first approach to numbers here is semiotic. A number is, in the first place, a “meaningless vocable” used in counting collections. Numerals are familiar examples of numbers in this primary sense, but so are some nonsense syllables from children’s games. Such numbers, recited in a standard order, are used to run through a collection; the one that exhausts the collection then functions as an adjective expressive of that objective attribute of the collection which Peirce calls its “multitude, or collectional quantity.” When Peirce says that numbers are “meaningless” he is in effect classifying them semiotically as indices; in Mill’s terminology, they are signs with denotation only and no connotation. He effects the classification here against the background of the associationist account of the inner and outer worlds he used also in selection 7. Here, as there, the associationist language masks his categories and his semiotics (though Thirdness comes through much more clearly here than there). But in this context it also provides a vocabulary in which he can couch his account of abstract number. The function of indices is to point things out, and if numbers are indices it is fair to ask what it is they point to. Peirce neither poses nor answers that question directly, but a natural answer is that each number points to a position in the order of counting. We can think of the theory of abstract number which Peirce develops in the second half of the selection as the theory of these positions, viewed in abstraction from the numbers (vocables) that indicate them. Peirce does not make systematic use of hypostatic abstraction in explaining how the concept of number, or the conception of numbers as objects, arises from the practice of counting. But abstraction is arguably implicit in the transition from meaningless vocables to adjectives, and in Beings of Reason.book Page 113 Wednesday, June 2, 2010 6:06 PM 114 | Philosophy of Mathematics Peirce’s classification of “multitude . . . [as] an attribute . . . of collections” [emphasis added]. This is one of the many avenues for philosophical exploration opened up by this richly suggestive text. When a number is mentioned, I grant that the idea of a succession, or transitive relation, is conveyed to the mind; and in so far the number is not a meaningless vocable. But then, so is this same idea suggested by the children ’s gibberish “Eeny, meeny, mony, mi.” Yet all the world calls these meaningless words, and rightly so. Some persons would even deny to them the title of “words,” thinking, perhaps, that every word proper means something. That, however, is going too far. For not only “this” and “that,” but all proper names, including such words as “yard” and “metre” (which are strictly the names of individual prototype standards), and even “I” and “you,” together with various other words, are equally devoid of what Stuart Mill calls “connotation.” Mr. Charles Leland informs us that “eeny, meeny,” etc. are gipsy numerals.3 They are certainly employed in counting nearly as the cardinal numbers are employed. The only essential difference is, that the children count on to the end of the series of vocables round and round the ring of objects counted; while the process of counting a collection is brought to an end exclusively by the exhaustion of the collection , to which thereafter the last numeral word used is applied as an adjective . This adjective thus expresses nothing more than the relation of the collection to the series of vocables. Still, there is a real fact of great importance about the collection itself which is at once deducible from that relation, namely, that the collection cannot be in a one-to-one correspondence with any collection to which is applicable an adjective derived from a subsequent vocable but only to a part of it; nor can any collection to which is applicable an adjective derived from a preceding collection4...

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