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29 Supplement [on Continuity] [Peirce 1908f] This is the second of Peirce’s fragmentary attacks on the complex of problems he raises in selection 27. There is much more explicit attention to the part/whole relation in this version, including an elaborate taxonomy of the different kinds of parts, which Peirce never gets around to actually using in a definition of continuity. His general account of a part, as something which is necessarily present whenever its whole is, is perhaps more useful for the analysis of continuity than the detailed distinctions, which are based in Peirce’s categories; the distinctions are hard to evaluate because the text breaks off before they are put to their intended use. (It is clear from what comes later that homogeneous parts would have played a central role, if Peirce had gotten that far.) Peirce promises two definitions each of ‘imperfect continuum’ and ‘perfect continuum’ but only delivers one definition of the latter. Continuity is held to consist in a kind of regularity or homogeneity; this has a strongly logical flavor at first (“conformity to one Idea”) but eventually the more “topical ” idea of unbroken passage between contiguous parts becomes dominant. The role of time emerges more clearly here than in the later version: the “passage” from part to part is spelled out in temporal, indeed in quasi-mental , terms. Here the manuscript breaks off, perhaps because Peirce decided that it would be better to make the definition more thoroughly topical from the start. Or perhaps the published note gives us what would have been the continuation of this one; in that case this selection would be a first draft and not an alternative solution. As in selection 27, Peirce gets distracted just when it comes time to actually deliver his definition, in this instance by a charming but doubtfully relevant recollection from his boyhood. This fragment and selection 27, taken together, give a somewhat mixed picture. There are enough common elements, and the elements cohere well enough, for there to be a discernible shift in Peirce’s approach to the problem of continuity, and for some of the broad outlines of the new approach to be tolerably clear. At the same time, Peirce’s tendency in these texts to break off altogeter at the moment of truth is more discouraging. It remains to be seen whether these intriguing but unfinished hints can be assembled and augmented so as to point clearly in a promising new direction.1 Beings of Reason.book Page 221 Wednesday, June 2, 2010 6:06 PM 222 | Philosophy of Mathematics Supplement. 1908 May 24. In reading the proofs of this article, which was written nearly a year ago, I find myself in a condition to make, as it seems to me, a long stride toward clearing up the important question of whether Cantor and Dedekind, supported by an interpretation of Riemann’s celebrated memoir on the hypotheses of geometry,2 and followed, apparently , by the general body of mathematicians are right in holding that a collection of mathematical points each absolutely unextended and the collection being of the smallest but one of all infinite multitudes, and less than any of an endless series of multitudes each greater than those which precede it in the series, this collection being suitably arranged, constitutes a truly continuous line; or whether I am right in contending, solus, that though such a series of points no doubt has what is called continuity in the calculus and theory of functions, it has not the continuity of a line. While Cantor’s theory is that the continuity of space of all kinds, line[a]r, superfi[ci]al, and solid is constituted by there being this fewest of all multitude[s] of points, among those in which there are more units than in a simple endless series, so that it would be absolutely refuted the instant it was shown that there could be more numerous points upon a line, surface, or solid, my own theory nowise necessitates there being more, or upon there necessarily being any at all, although it does suppose that there is room for a good many (not necessarily, I think, even an infinite multitude). I proceed at once to define what I think it is that constitutes a true continuum, or continuous object. I begin by defining a part of any whole, in a sense of the [term] much wider [than] any in current use, though it is not obsolete in the vocabulary...

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