Abstract
In the last decade of his life C.S. Peirce began to formulate a purely geometrical theory of continuity to supersede the collection-theoretic theory he began to elaborate around the middle of the 1890s. I argue that Peirce never succeeded in fully formulating the later theory, and that while that there are powerful motivations to adopt that theory within Peirce’s system, it has little to recommend it from an external perspective.
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Notes
\(^{1}\)There is at least one very late text on continuity (Peirce nd, p. 6) that does not clearly take the topical approach. Its existence was first brought to my attention by (Havenel (2008), p. 123). My initial reaction was in line with Havenel’s characterization of the text as “much more algebraical than topological.” Given the apparent finality of Peirce’s pronouncements in the texts to be discussed in this paper, the purely number- and order-theoretic treatment of this late fragment would indeed be, as Havenel rightly terms it, “surprising.” It would moreover very strongly support my thesis that Peirce could never work out the topical theory in detail. I have decided not to enter it into evidence just because it is so fragmentary, and so far from delineating a concept that anyone (let alone Peirce) would recognize as continuity.
By far the most comprehensive review of Peirce’s work on topology is Havenel (2010), which very usefully locates Peirce’s topological thinking within the discipline as it existed at the time, and as it has developed since. Chapter IX of Murphey (1961) continues to be a valuable resource. Both treatments drive home the differences between topology as Peirce understood it, and the subject as we study it today; Havenel stresses that “what Peirce calls topology corresponds roughly to algebraic topology, with the addition of some aspects of differential topology. If Peirce is concerned with point-set topology when he characterizes the continuum as being supermultitudinous, point-set topology is not topology for Peirce” (Havenel 2010, p. 283).
“In a theater, it happened that a fire started offstage. The clown came out to tell the audience. They thought it was a joke and applauded. He told them again and they became still more hilarious. This is the way, I suppose, that the world will be destroyed—amid the universal hilarity of wits and wags who think it is all a joke” (Kierkegaard 1843, p. 30). Cited by M. O’C. Drury, as quoted in (Rhees (1984a), p. xi.).
An especially dramatic example of this syndrome is the relative obscurity into which Myhill’s devastating objection (Murphey 1961, p. 261) has fallen, despite its fatal force for Peirce’s most thoroughly developed definition of continuity, and depite the crystal-clear exposition it receives in a classic survey of Peirce’s philosophy. This objection should be as well known to students of Peirce’s continuum as Russell’s Paradox is to students of logicism. Yet it is impossible not to stumble, at any gathering of Peirceans, over any number of devotees of what they take to be Peirce’s continuum, most of whom have presumably read Murphey’s book, and from whom Myhill’s name elicits no glimmers of recognition.
All other things being equal, a clear definition is a powerful thickening agent, whose absence virtually guarantees thinness. To state and explicate such a definition is to go most, if not all the way, to a theory in the thicker sense. This is why I read Potter and Shields, for example, as telling a developmental story about Peirce’s theories of continuity, even though their professed focus is on his definitions.
He gives a subtly fallacious proof of this—see (Peirce (2010), pp. 257–258n2), and the works cited there—in (Peirce (1897), pp. 191–192), which is very close in content, and I think also in date, to the Cambridge Conferences: that argument also purports to show that the multitudes of these collections are all the infinite multitudes there are. The fallacy in the argument is exposed by Myhill’s objection, referred to in footnote 4.
This is a remark about the logic of the theory, not about Peirce’s intentions, which even in the heyday of the Super-M theory would have made geometry the conceptual center of gravity.
Jérôme Havenel has distinguished in correspondence (email message to the author, 11 September 2011) between what he calls “external continuity” (which “deals with properties shared by objects belonging to the same class, which is itself defined according to a homeomorphism”) and what he calls “internal continuity” (which “deals with the mode of immediate connection of the parts of a continuum”). My view of the place of topology in Peirce’s theories of continuity can be summarized, in terms of Havenel’s distinction, as follows: topical ideas function in the Super-M theory as an organizing framework for the analysis of external continuity; it was not until Peirce became dissatisfied with the whole collection-theoretic approach that he attempted to craft a definition of his continuum in terms of internal continuity alone.
In editing (Peirce (2010), p. 251), I relied on Robin (1967), which gives 1905 as the date of this manuscript. When this paper was all but complete, Ahti-Veikko Pietarinen observed, and André De Tienne confirmed, that it is now thought to have been written around March of 1908. The later date substantially telescopes the gestation period of the topical theory, and makes the extrusion of collection-theoretic ideas look even more abrupt, and less thoroughly worked out.
Peirce is in complete earnest when he says that the Super-M theory is implied by “the common conception of a line.” In (Peirce (1897), p. 199), he writes that it is part of “the intuitional idea of a line with which the synthetic geometer really works” that the points on the line form a supermultitudinous, and therefore fused, collection. Similarly, he argues in (Peirce (1896), pp. 162–164) that the Super-M continuity of phenomenal time follows from “our natural common-sense belief ... that the flow of time is directly perceived” (p. 162). The skeptical value of these arguments is not diminished by Peirce’s inability to put the finishing touches on the Super-M theory: even if the arguments do not show that we must, to be true to common sense, take geometrical and temporal continua to be Super-M continua, they do effectively undermine the suggestion that common sense forces any other kind of continuity upon us. In particular, they can help to counteract the belief that Cantor/Dedekind continuity is something we just see in the geometrical line, as opposed to something we learn (with some difficulty) to see there.
The text in question is the definition of ‘multitude’ (Peirce and Fine 1902) contributed by Peirce and Fine to Baldwin’s Dictionary of Philosophy and Psychology. In the section of the third volume of the Collected Papers where that definition appears, Hartshorne and Weiss (p. 338) give 1911 as the publication date for the Dictionary. A revised edition did come out then, but all of the entries selected by Hartshorne and Weiss for inclusion in that part of Volume 3 were in the first edition, and hence had appeared in print by 1902.
I call this Peirce’s primary objective for two reasons. It is what most immediately concerns him in the late texts I am now discussing, which take Cantor and Dedekind to task precisely over reflexivity. Reflexivity is also, I contend, philosophically primary because it does the lion’s share of the philosophical work when Peirce applies his continuum, for example, in his metaphysics of generality. In singling out reflexivity as a primary concern I am of course not suggesting that it is his only concern. The desire to do justice to continuity as a purely geometrical concept is another driving force behind Peirce’s pursuit of his continuum, at least as longstanding and arguably as primary as reflexivity. It is not clear to me that there is all that much difference, as far as most philosophical applications go, between a continuum whose definition is based in the theory of collections, and one defined in terms of topical geometry. A purely geometrical definition may be more in line with our pretheoretical understanding of continuity, and hence would be more philosophically satisfactory if we wanted to keep our mathematically precise concepts as close as we can to their intuitive roots. Peirce did want to do that, for his continuum at any rate (Moore 2009, pp. 656–658), and in giving pride of place to reflexivity I mean to slight neither the philosophico-mathematical interest of a purely geometrical definition, nor Peirce’s sustained interest in finding one. I do maintain, though, that outside the philosophy of continuity itself it is reflexivity that does the heavy lifting in Peirce’s later system. And by now I hardly need to say that, when it comes to his quest for a purely geometrical account, we must take care not to confuse aspiration with achievement. (I am indebted to Jérôme Havenel for alerting me to the need for this clarification.)
For some preliminary discussion of these writings, and the connection with continuity, see (Moore (2010), pp. 345–355).
On the geometrical impulse in Peirce’s mature thinking about continuity, see (Moore (2009), pp. 656–660).
Those qualms were very real. But for them Peirce might have tried to muddle through with the Super-M conception, despite his deeply held conviction that “number cannot possibly express continuity” (Peirce 1897, p. 196). In any case the doubts were gathering strength in the months preceding the composition of the main topical texts in the Spring of 1908. The long and intricate manuscript for the abortive fourth entry in the “Amazing Mazes” series, which appears to have been written around October of 1907, contains two highly revealing mentions of the problem of linear ordering. One is in a footnote beginning on p. 225 of volume 6 of the Collected Papers. Peirce claims to “know no question of metaphysics so pressing as this of whether or not there be a maximum multitude capable of linear arrangement,” and then laments “the extreme liability to fallacy of reasoning concerning this question.” This is on (standard) page number 21 of the manuscript. On pp. 104–105 he writes:
But if there be a multitude which is beyond linear arrangement (which it is virtually said has been proved false; but I somewhat doubt it, owing to the extreme liability to fallacy on this point, as to which I have had some striking experience. There are those among profound thinkers on this matter who are almost confident that what is generally thought to have been proved false is really true. My own leaning is the other way. Yet I doubt.)—if that be so, then between any two times between which there is anytime, there is room for the highest multitude of times that is susceptible of linear arrangement.
This is a more openly skeptical variant of the agnosticism about linear orderings that Peirce espouses in the published note to which he added Version 3 (Peirce 1908c, p. 214). (I am grateful to Giovanni Maddalena for suggesting that I look at this text, and to André De Tienne for his explanation of the evidence for its date.)
Skeptical comments like this are easily misconstrued, so let me be as explicit as I can about what I am, and am not, casting doubt on here. I have already acknowledged, in the main body of the text, the mathematical interest that might well attach to a full dress technical development of Peirce’s hints at a topical theory of continuity. Nor do I doubt the philosophical interest that very likely would attach to a reconstruction of the analysis of parts and wholes and other metaphysical scaffolding that Peirce begins to build for the properly mathematical development. What I do doubt is that the mathematical development is either necessary or sufficient for the philosophical advances Peirce hopes his topical theory will clear the way for.
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This paper originated as a talk at the International Summer School for Semiotics and Structural Studies in Imatra, June 2010. Thanks to Ahti-Veikko Pietarinen for organizing a very stimulating series of sessions at the Summer School on Peirce’s mathematics, for inviting me to contribute, and for his warm hospitality throughout my time in Finland. The interchange with other participants in the session, especially Pietarinen and Frederik Stjernfelt, greatly helped me in revising the paper for publication. Thanks also to Cornelis de Waal and Jérôme Havenel, whose comments on an earlier draft helped me improve both the substance and the presentation of the paper. This is dedicated to Emily Michael.
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Moore, M.E. Peirce’s topical theory of continuity. Synthese 192, 1055–1071 (2015). https://doi.org/10.1007/s11229-013-0337-6
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DOI: https://doi.org/10.1007/s11229-013-0337-6