Nov 172010

In a letter to William James on November 25, 1902, Peirce spoke of “the completely developed system, which all hangs together and cannot receive any proper presen­tation in fragments,” and he went on to describe synechism as: “the keystone of the arch.”[1] Now synechism, according to Peirce, is just “that tendency of philosophical thought which insists upon the idea of continuity.”[2] Hence, in order to make sense of Peirce’s synechism, and its role in his “completely developed system”, it is essential first to understand what Peirce meant by the idea of continuity.

Peirce was far from reticent on the topic:

If I were to attempt to describe to you in full all the scientific beauty and truth that I find in the principle of continuity, I might say in the simple language of Matilda the Engaged, “the tomb would close over me e’er the entrancing topic were exhausted” . . .[3]

Yet, even though much of Peirce’s writing was devoted to this idea, there is not much in the secondary literature on his technical definitions of continuity.[4] In this paper we will show how these definitions changed as Peirce’s thinking on con­ti­nu­ity evolved. This should be valuable not only to scholars expressly concerned with Peirce’s work in the foundations of mathematics, but also to those mainly interested in other aspects of his thought.

Peirce did not have a single completed understanding of continuity extending through­out his writing. On the contrary, from about 1880 to 1911 his attempts to construct a pre­cise mathematical formulation of continuity show a definite development, marked by se­veral significant changes and, overall, a steadily increasing sophistication. This de­vel­opment is most easily located by reference to Peirce’s changing stance vis-à-vis the Ger­man mathematician Georg Cantor. There are four main periods:

  1. Pre-Cantorian: until 1884
  2. Cantorian: 1884-1894
  3. Kantistic: 1895-1908
  4. Post-Cantorian: 1908-1911.

Pre-Cantorian Period

In his famous 1878 essay, The Doctrine of Chances, Peirce described continuity as “the passage from one form to another by insensible degrees.”[5]  In a note from 1893 he corrected this description by saying that he meant to refer to an idea that continuity suggests, “that of limitless intermediation; i.e., of a series between every two members of which there  is another member of it,” implying that there may also be other ideas involved in the notion of continuity.[6] This correction seems to indicate that Peirce recognized a possible confusion in the original version of his essay, a confusion between the notions of continuity and infinite divisibility.  As a matter of fact, in an article written for The American Journal of Mathematics in 1881, Peirce expressly identified these notions. He said, for example, that “a continuous system is one in which every quantity greater than another is also greater than some intermediate quantity greater than that other.”[7] This definition is clearly inadequate.  It does not guarantee, for instance, the simple requirement of our geo­metrical intuition — that continuous lines should hit one another when they cross. It gives a property of the rational number system, infinite divisibility or, in Russell’s terminology, compactness which is not adequate for the real number system. But this confusion was apparently characteristic of Peirce’s Pre-Cantorian period.  And it is likely that the misidentification of continuity with infinite divisibility persisted until around 1884 when Peirce first read Cantor’s Grundlagen Einer Allgemeinen Mannig­faltigkeitslehre in volume II of Acta Mathe­matica.[8]

Cantorian Period

It is difficult to fix precisely the date of the transition to Peirce’s Cantorian period because it is not easy to determine how long it took for Cantor’s ideas to sink in and begin to affect Peirce. But there is persuasive evidence that Cantor did exert considerable influence on Peirce. In fact, Peirce denied almost every possible influence upon his work on multitudes and continuity except that of Cantor.  In a later manuscript Peirce wrote:

I, of course, depend much upon Cantor, although my own habits of thinking about multitudes were somewhat fixed before I ever made my first acquaintance with Cantor’s work, in Vol. II of the Acta Mathematica, or had so much as heard of Bolzano’s celebrated definition.  By the time Whitehead’s and other works had appeared, I preferred to postpone reading those works until my own ideas were in a more satisfactory condition, so that I do not know in how much of what I have to say I may have been anticipated.[9]

Peirce claimed that he was not familiar with Dedekind’s Stetigkeiten und Irrationalen Zahlen, of 1872, and suggested that Dedekind’s Was sind und was sollen die Zahlen? depended heavily upon his own 1881 paper.[10]

Cantor, on the other hand, Peirce acknowledged willingly and frequently: he called him “the immortal Dr. Georg Cantor”, and referred to his “extraordinary genius” and “pene­trating logic”.[11] “I pay full homage to Cantor,” Peirce remarked in 1905, “He is indisputably the Hauptförderer of the mathematico-logical doctrine of num­bers.” [12]  A passage from the manuscript for the eighth pragmatism lecture, given at Harvard in 1903, illustrates both the deference Peirce showed Cantor and the im­por­tance he attached to this entire field:

. . .Mathematicians always have been the very best reasoners in the world; while metaphysicians always have been the very worst. Therein is reason enough why students of philosophy should not neglect mathematics. But during the last thirty years, there has been an extraordinary mathematical development of the general doctrine of multitude, including of course, infinity, and of continuity.  Philosophers would fall short of their well earned reputation as dunces if they paid much attention to this until it begins to ring in their ears from all quarters. . . . The leader in this investigation, with whom no other ought to be put into comparison, is Dr. Georg Cantor.  ((R# 316a-s))

By 1889, at any rate, Cantor’s impact upon Peirce was clearly visible. It can be seen in the definition of continuity that Peirce wrote for the Century Dictionary:

[Continuous means] in mathematics and philosophy a connection of points (or other elements) as intimate as that of the instants or points of an interval of time: thus, the continuity of space consists in this, that a point can move from any one position to any other so that at each instant it shall have a definite and distinct position in space. This statement is not, however, a proper definition of continuity, but only an exemplification drawn from time. The old definitions — the fact that adjacent parts have their limits in common (Aristotle), infinite divisibility (Kant), the fact that between any two points there is a third (which is true of the system of rational numbers) are inadequate. The less unsatisfactory definition is that of G. Cantor, that continuity is the perfect concatenation of a system of points. . .[13]

Peirce’e definition followed closely the ideas set forth in Cantor’s Grundlagen. Cantor, for instance, had claimed that the notion of continuity must be treated independently of our conceptions of time and space; and he had discussed and discarded definitions proposed by various figures in the history of philosophy. He had also given a technical presentation of the idea of perfect concatenation, including examples, which Peirce cited later in his article, of systems that are perfect but not concat­enated and vice versa.[14]

Over the next four or five years, Peirce proposed modifications to Cantor’s definition by perfect concat­e­nation. It will give insight into these to describe, first, what Cantor meant by the property of perfection;  second, Peirce’s criticism of Cantor’s defini­tion; and third, the alternative that Peirce pro­posed.

First, Peirce did not initially understand exactly what Cantor intended by the notion of per­fection. In the con­tinuation of his Century Dictionary definition, Peirce remarked:

Cantor calls a system of points concatenated when any two of them being given, and also any finite distance, however small, it is always possible to find a finite number of other points of the system through which by successive steps, each less that the given distance, it would be possible to proceed from one of the given points to the other. He terms a system of points perfect when, whatever point belonging to the system be given, it is not possible to find a finite distance so small that there are not an infinite number of points of the system within that distance of the given point.[15]

Cantor’s property of concatenation — zusammenhängend, which is also translated as “connectedness” or “cohesiveness” — Peirce expressed accurately.[16] The property of perfection, however, was only partly correct. In his Grundlagen Cantor had called a system perfect when it is equivalent to its first deriva­tive. [17] And the first deriva­tive of a system is simply its collection of limit-points, or, as Peirce put it, the collec­tion of points such that it is not possible to find a distance so small that there are not an infinite number of points of the system within that distance of any par­ticular one of them.[18] But the equivalence between a system and its first derivative must be read in both directions. Cantor emphasized this in his later definition, ac­cording to which a system is perfect just in case it is both closed — abgeschlossen — and con­densed-in-itself — insichdicht.[19] Thus a system is closed when every limit-point of the system is contained within the system and, conversely, a system is condensed-in-itself when every point of the system is a limit-point of the system.[20] In the above passage, Peirce has only described the property of being condensed-in-itself, saying, in effect, that whatever point of the system be given, it must be a limit-point.  In The Law of Mind, in 1892, Peirce apparently made the reverse error, that of assimilating per­fection to closure, saying:

By a perfect series, he [Cantor] means one which contains every point such that there is no distance so small that this point has not an infinity of points of the series within that distance of it.[21]

Peirce eventually provided a satisfactory formulation of perfection — although he did not interpret it formally in the way Cantor later did, i.e., by defining limit-points in terms of coherent fundamental series.[22]

Second, despite his initial difficulties with the property of perfection, Peirce made a significant criticism of Cantor’s definition. He remarked, in 1892, that Cantor’s defini­tion, “ingeniously wraps up its properties in two separate parcels, but does not dis­play them to our intelligence.”[23]  Perfection and concatenation, Peirce claimed, are not distinct components of our conception of continuity. Peirce was apparently cor­rect in this claim, to the extent that every concatenated system can be shown to be condensed in itself.[24] This is why Peirce later wrote in the margin of his personal copy of the Century Dictionary that:

[Cantor] defines [a perfect system] such that it contains every point in the neighborhood of an infinity of points and no other.  But the latter is a character of a concatenated system; hence I omit it as a character of a perfect system.[25]

Peirce observed that, given concatenation, the property of being condensed-in-itself is redundant.

Third, the alternative by which Peirce attempted to avoid this redundancy did not im­prove matters. In 1892, 1893, and again in 1903, Peirce proposed that continuity be defined by the properties of “Kanticity” and “Aristotel­icity”.[26] Aristotelicity, in these formulations, is analogous to Cantor’s property of closure, i.e., it is the requirement that a continuum contain its limit-points. Kanticity, however, is just com­pactness or infinite divisibility — it is, Peirce said, “having a point between any two points.”[27] Except under special assumptions of completeness, this is still not strong enough to provide concatenation.[28]  Concatenation assures that there are no finite gaps in the system, and infinite divisibility by itself cannot do this.

In his 1895 memoir, Beiträge zur begründung der transfiniten Mengenlehre, Cantor replaced concatenation with an even stronger property, the postulate of linearity, which is necessary in order to keep all continua similar.  But Peirce did not read this memoir until much later, and by 1895 his criticisms of Cantor’s definition had taken on a new dimension.[29] There was no longer the basic agreement in spirit which had seemed to prevail until 1893-94.

Kantistic Period

We have called the new position which Peirce began to formulate around 1895 Kantistic because Peirce discovered one of its important ingredients in Kant’s definition of a continuum as “that all of whose parts have parts of the same kind”.[30] This should not be confused with the earlier property of Kanticity, which was merely infinite divisibility. Rather, this definition implies that a continuum cannot have point-like parts at all.  In order to understand Peirce’s new position fully, though, we must first look at his doctrine of “postnumeral multitudes,” or “abnumeral dignities” as he came to call them.

By the term multitude Peirce meant essentially what Cantor had called the power (Mächtigkeit) of a collection. This is now commonly called the “cardinal number” of a set. Neither Peirce nor Cantor was apparently aware of Frege’s famous abstractive definition of number, given in his Grundlagen der Arithmetik in 1884.[31] Peirce simply said, “I shall use the word multitude to denote that character of a collection by virtue of which it is greater than some collections and less than others, provided the collection is discrete. . .”[32] In one manuscript, Peirce tried to describe this char­acter as “a certain scheme of otherness” which subsists among the members of a collection; but he added that this is extremely vague and that the whole question needs further study.[33]

It is important to realize that Peirce’s theory of multitudes preceded, by many years, his first attempts to construct a theory of transfinite ordinals.  Peirce relied upon Bolzano’s technique of defining an order relation among multitudes by the use of one-to-one correspondences.[34 Hence, the series of transfinite or “postnumeral” multi­tudes which he developed are confined to those obtained by repeated application of Cantor’s theorem, that 2n > n.[35] So Peirce’s entire series of multitudes, in contem­porary notation, would look like this:

0, 1, 2, . . . ℵ₀, 2ℵ₀, 22ℵ₀, 222ℵ₀, . . .

Peirce sometimes confused the “postnumeral” portion of this series with Cantor’s series of cardinals, ℵ₀, ℵ₁, ℵ₂, . . ., corresponding to the ordinal number classes. This is why he called the “primipostnumeral” multitude, 2ℵ₀, the “smallest multitude which exceeds the denumerable multitude”.[36] More often, though, Peirce was just puzzled by the continuum problem. He lacked the ordinal machinery necessary to understand the difficulty. In a typical manuscript, from around 1897, Peirce first claimed to prove the continuum hypothesis but then apparently had second thoughts, the word “prove” being crossed out and replaced with the word “argue”.[37]

By 1896, Peirce’s theory of multitudes was sophisticated enough that he could begin to describe true continuity by reference to it.  A true continuum, he said, is a multi­plicity, “greater than any discrete multitude”.[38] This is what Peirce meant by calling a continuous collection “super-multitudinous”, and by saying that “the possibility of determining more than any given multitude of points, or in other words, the fact that there is room for any multitude at every part of the line, makes it contin­uous.”[39]

There were several motivations behind this Kantistic approach to defining continuity.  On an intuitive level, it must have been disquieting for Peirce, the synechist, to discover that the putative power of the continuum was only 2ℵ₀. If continuous collections can be distinguished from “compact” collections by their size, Peirce must have reasoned, why shouldn’t this same characteristic distinguish true continuity from every discrete collection.  The notion of a discrete collection more multitudinous than a continuum must have seemed odd to Peirce.

But also, as early as 1892, Peirce had been concerned with the non-metrical pro­perties of continua.  One of his passing criticisms of Cantor had been that Cantor’s defi­nition, “turns upon metrical considerations; while the distinction between a contin­uous and discontinuous series is manifestly non-metrical.”[40] And in 1893 Peirce had asked himself the question: How can continua be colored when their proper parts, points, cannot be colored?[41] By locating true continuity beyond the series of post­numeral multitudes Peirce thought he had solved this kind of problem — since points could no longer be regarded the actual constituents of a continuum at all.[42]  Every part of a true continuum must be continuous. Yet, Peirce still retained the relation “greater than” for comparison, to allow for the possibility of determining any multi­tude of points whatsoever on a continuum. The attempt to fuse these strains is the central theme of Peirce’s Kantistic period.

It has been argued that Peirce’s definition of continuity during this period falls prey to Cantor’s paradox — the paradox of asserting the existence of a greatest transfinite cardi­nal.[43] But this is not the case: by 1895 Peirce was completely aware that such an assertion leads to a contradiction.[44]  In 1896 he observed that “there cannot be any multitude of infinite dignity; for if there were, the multitude of ways of distri­buting it into two houses would be no greater than itself.”[45]  Again, in 1900 Peirce stated unequivocally: “There is no maximum multitude.”[46]  So how did such a mis­un­der­standing of Peirce come about?

The difficulty derived from not recognizing the contrast, which Peirce emphasized in this period, between a “multitude” and a “multiplicity.”  Multitude can only charac­terize discrete collections, but multiplicity, Peirce said, refers to “the greatness of any collection, discrete or continuous.”[47] The following passage, for example, has been cited as evidence that Peirce was enmeshed in Cantor’s paradox:

I now inquire, is there any multitude larger than all of these [postnumeral multitudes]?  That there is a multitude greater than any of them is very evident.  For every postnumeral multitude has a next greater multitude. Now suppose collections one of each postnumeral multitude, or indeed any denumerable collection of postnumeral multitudes, all unequal. As all of these are possible their aggregate is ipso facto possible. For aggregation is an existential relation, and the aggregate exists . . . by the very fact that its aggregant parts exist. But this aggregate is no longer a discrete multitude, for the formula 2n > n which I have proved holds for all discrete collections cannot hold for this.[48]

But when Peirce said “That there is a multitude greater than any of them is very evi­dent,” he was not suggesting that there is a multitude greater than all postnumeral multitudes.  On the contrary, he was saying that there is a multitude greater than any particular postnumeral multitude.  This is clear from the argument he adduced: “For every postnumeral multitude has a next greater multitude.”  Only after this did Peirce return to the original question, whether there is a multitude greater than all post­numeral multitudes.  And his answer is that there is a multiplicity greater than all of them but not a multitude.[49] Peirce recognized that the only way in which a con­tinuum can be greater than every discrete collection is if continua do not have size in quite the way that discrete collections do.  So it is incorrect to interpret the above passage (or other passages after 1895) as asserting the existence of a greatest post­numeral multitude.

Post-Cantorian Period

Peirce’s Kantistic period extended to about 1908. His post-Cantorian period developed gradually out of instabilities in the Kantistic approach.   First, there was the problem of how to interpret the relation “greater than” when it is applied to multiplicities.  For a while Peirce tried to maintain that Bolzano’s technique of defining order relations among multitudes is also applicable between multitudes and multiplicities.[50] But a correspondence between a point and a possible point tends to turn into a possible correspondence which can establish only a possible order relation.  And, in 1908, Peirce did discard the notion that a continuum is actually greater than every discrete multitude.

Second, Peirce began to question the sense in which a continuum can be thought of as a collection.  This doubt is present as early as 1900, when Peirce suggested, in a letter to the editor of Science, that, because collections have multitude and obey Cantor’s theorem, a continuum is not really a collection.[51] But if a continuum is not a collection, Peirce would need to find some other way of explaining how the parts of a continuum come together as a whole — for a continuum clearly does have parts.  His solution was anticipated in a passage from The Bedrock beneath Pragmatism, written in 1906.  Peirce proposed the definition: “Whatever is continuous has material parts,” emphasizing that a continuum should not be thought of as a collection of points. He then explained:

I begin by defining these thus: The material parts of a thing or other object, W, that is composed of such parts, are whatever things are, firstly, each and every one of them, other than W; secondly are all of some one internal nature (for example, are all places, or all spatial realities, or all spiritual realities, or are all ideas, or are all characters, or are all relations, or are all external representations, etc.); thirdly, form together a collection of objects in which no one occurs twice over and, fourthly, are such that the Being of each of them together with the modes of connexion between all subcollections of them, constitute the being of W.[52]

The second condition here, that material parts must be all of some one internal nature, seems to be a reformulation of the requirement of Kant’s definition, that in a continuum all of the parts must have parts of the same kind.  The other conditions were an attempt to spell out how these parts can constitute a whole. The most important point is clearly that the mode of connection between the parts contributes to the nature of the whole.  Thus, Peirce said:

It will be seen that the definition of Material Parts involves the concept of Connexion, even if there is no other connexion between them than co-being; and in case no other connexion be essential to the concept of W, this latter is called a Collection . . .[53]

Now, it is important to notice that Peirce did not say that everything with material parts is continuous.  In fact, the mode of connection between the parts turns out to be the real key to the difference between a collection and a continuum.

Peirce’s post-Cantorian definition, then, relied upon describing the mode of connection between continuous parts, without reference to the size or “roominess” of such a whole. This new approach was first made explicit in an addendum, dated May 26, 1908, to a note on continuity in a paper in The Monist.[54] Peirce wrote, “In going over the proofs of this paper, written nearly a year ago, I can announce that I have, in the interval, taken a considerable stride toward the solution of the question of continuity.”[55] Peirce first presented a variant of Kant’s definition, according to which “all of the parts of a perfect continuum have the same dimensionality as the whole,” explaining that this requires not only that all of the parts must have parts of the same kind, but also that sufficiently small parts must have a uniform mode of immediate connection. Then he added:

In endeavoring to explicate “immediate connection,” I seem driven to introduce the idea of time.  Now if my definition involves the notion of immediate connection, and my definition of immediate connection involves the notion of time; and the notion of time involves that of continuity, I am falling into a circulus in definiendo.  But on analyzing carefully the idea of time, I find that to say it is continuous is just like saying that the atomic weight of oxygen is 16, meaning that that shall be the standard for all other atomic weights.  The one asserts no more of time than the other asserts concerning the atomic weight of oxygen; that is, nothing at all.[56]

Peirce has come full circle since 1889 when he agreed with Cantor, that the notion of continuity should be defined independently of the notion of time.  His post-Cantorian definition of continuity was stated completely in terms of the time-like mode of im­mediate connection which obtains between sufficiently small time-like parts:

Following out this idea, we soon see that it means nothing at all to say that time is unbroken.  For if we all fall into a sleeping-beauty sleep, and time itself stops during the interruption, the instant of going to sleep is absolutely unseparated from the instant of waking; and the interruption is merely in our way of thinking, not in time itself.  There are many other curious points in my new analysis.  Thus, I show that my true continuum might have room for only a denumeral multitude of points, or it might have room for just any abnumeral multitude of which the units are themselves capable of being put in a linear relationship, or there might be room for all multitudes, supposing not multitude is contrary to a linear arrangement.[57]

Peirce’s post-Cantorian continuum has become indifferent to multitude and thoroughly dimensional — time-like.  Peirce apparently still held this view as late as 1911.[58]

This outline has touched upon important areas of Peirce’s thought — mathematical and philosophical — that need more detailed consideration.  Our intention has been to provide an accurate overview of this neglected topic in the hope that it will encourage related research, and serve as a guide to philosophers who have reason to be in­ter­ested in Peirce’s de­finitions of continuity.

Vincent G. Potter, S. J.
Paul Shields

[This paper originally appeared in the Transactions of the Charles S. Peirce Society, 13:1, Winter, 1977, pp. 20-34, and has been posted here with permission of IU Press. An impor­tant paper by Jérôme Havenel, "Peirce's Clarifications of Continuity", Trans­ac­tions, 44:1, Winter, 2008, pp. 86-133, pro­vides more extensive context, and more precise chro­no­logy, for the changes in Peirce's thinking.]

  1. Charles Hartshorne and Paul Weiss, eds. The Collected Papers of Charles Sanders Peirce, vols I-VI, Belknap Press, Harvard, 1931-1935, and Arthur W. Burks, ed. vols. VII, VIII, Belknap Press, Harvard, 1938,  8.255-257. The Collected Papers will be referenced by the conventional volume and paragraph number. []
  2. 6.169 []
  3. 1.171 []
  4. Some important works are Murray G. Murphey, The Development of Peirce’s Philosophy, Harvard U. Press, 1961, and George A. Benedict, The Concept of Continuity in Charles Peirce’s Synechism, Ph.D. Dissertation, SUNY Buffalo, 1973. We have found Murphey’s book invaluable, although we disagree with him on important points. We also look forward to the appearance of Carolyn Eisele’s edition of Peirce’s mathematical writing. []
  5. Peirce, The Doctrine of Chances, 2.646. []
  6. 2.646. []
  7. 3.256. []
  8. This was originally the fifth of a series of papers entitled Über Unendliche lineare Punktmannigfaltigkeiten written in 1882 and published in Mathematische Annalen, Vol. XXI, in 1883. It was reprinted with an added preface and the full title, Grundlagen Einer Allgemeinen Mannigfaltigkeitslehre: Ein Mathematisch-philosophischer Versuch in der Lehre des Unendlichen, Leipzig, 1883.  Portions of this latter were translated into French in Acta Mathematica, Vol. II, in 1884.  Translated into English by Philip E. B. Jourdain, Georg Cantor, Contributions to the Founding of the Theory of Transfinite Numbers, Dover, 1955, intro., p. 54n. []
  9. Richard S. Robin, Annotated Catalogue of the Papers of Charles S. Peirce, Cambridge, University of Mass. Press, 1967, #27. This catalogue was supplemented by Robin, “The Peirce Papers: A Supplementary Catalogue,” Transactions of the Charles S. Peirce Society, vol. III, pp. 37-57. Reference to the Peirce manuscripts will be made by giving the conventional Robin number from these catalogues. []
  10. “I have seen it stated in some book that I modified the statement of Dedekind. But the truth is that Dedekind’s Was sind und was sollen die Zahlen? first appeared in 1888. It contains not a single idea which was not in my paper of 1881, of which an extra copy was sent to him and I do not doubt influenced his work.” R# 316a-s.  Also, see 3.563 and 4.331. []
  11. 6.175,  6.113. []
  12. 4.331. []
  13. Century Dictionary, 1889 — 6.164. []
  14. Cantor, Contributions, pp. 70-72. []
  15. 6.164. []
  16. Concatenation, for Cantor, was the property of a collection P according to which if t and t′ are any two of its points, and ε an arbitrarily small posi­tive number, a finite number of points t1, t2, … tv of P must exist such that the dis­tances tt1, t1t2, … tvt′ are all less than ε. Cantor, Contributions, p. 72. []
  17. Ibid, p. 72.  See Bertrand Russell’s treatment of this issue in Principles of Mathematics, Norton, 1903, ch. xxxv, “Cantor’s First Definition of Continuity,” pp. 290, 291. []
  18. Cantor defined a limit-point (Grenzpunkt) roughly after the fashion indicated here.  This early approach may seem somewhat odd to those more accustomed to his later definition of limit-elements (Grenzelement or Hauptelement) by fundamental series. Ibid, p. 30 and p. 131. []
  19. This is Cantor’s 1895 description of perfection from Beiträge zur begründung der transfiniten Mengenlehre. Contributions, p. 132. []
  20. “If an aggregate M consists of principal elements, so that every one of its elements is a principal element, we call it an ‘aggregate which is condensed-in-itself.’ If to every fundamental series in M there is a limiting element in M, we call M a ‘closed aggregate.’ An aggregate which is both ‘condensed-in-itself’ and ‘closed’ is called a ‘perfect aggregate’.” Ibid, p. 132.  Also, see Russell, Principles, p. 297. []
  21. 6.121. []
  22. 6.167. Cantor’s description is in Contributions, pp. 128-131. []
  23. 6.121. []
  24. The property of being condensed-in-itself means that all of the points of a system must be limit-points.  Given that it is specified whether these are to be upper or lower limit-points, this is equivalent to the property of compactness. And concatenation clearly implies compactness. Without this provision it is still true that compactness would imply being condensed-in-itself, although the reverse implication would not necessarily hold, as in the case when a decimal ending in an infinite sequence of 9’s were distinguished from the following decimal in which those 9’s were replaced by 0’s and the preceding decimal place increased by one unit. See Bertrand Russell, Introduction to Mathematical Philosophy, N.Y., Macmillan, 1919, pp. 102, 103.  So, in any case, concate­nation implies being condensed-in-itself. []
  25. Published as 6.167 in The Collected Papers. []
  26. 6.121-124 (1892), 4.121 (1893) and 6.166 (1903). []
  27. 6.166. []
  28. While every concatenated system is compact, it is not the case that every compact system is concatenated.  For example, the system formed of 0 and 2 – m / n, where m and n are any integers such that m is less than n, is compact since it is always possible to find another member of the system between any two members. But it is not concatenated because the steps between 0 and any other point cannot all be made less than 1.  In order to assure that a compact system will also be concatenated, one must assume that the compact system is complete, i.e., that the axiom of Archimedes and generalized axiom of linearity hold for it.  See Russell, Principles, pp. 289, 290. []
  29. Cantor described the postulate of linearity (do not confuse with the axiom of linearity in the previous note) as the property according to which an aggregate M, “contains an aggregate S with the cardinal number ℵ₀ which bears such a relation to M that between any two elements m0 and m1 of M elements of S lie.”  Cantor, Contributions, p. 134.  In 1900 Peirce said that he “never had an opportunity sufficiently to examine” this 1895 memoir (3.563), but by 1911 Peirce was citing Cantor’s definition of “power” from it (3.632).  From the Peirce-Cantor correspondence it appears that Peirce first read this memoir in December, 1900. See R# L73. []
  30. 6.168 (1903). []
  31. Gottlob Frege, Grundlagen der Arithmetik, Breslau, 1884, p. 117.  Frege gave the definition: “The number which applies to the concept F is the extension of the concept ‘concept equinumerous with the concept F’”. Frege explicitly compared his “numbers” to Cantor’s “powers”.  To the best of our knowledge, Peirce did not speak of Frege, even though he apparently reviewed Russell’s Principles, which cited Frege prominently and had an appendix devoted to him, in 1903. See 8.171n.  On Cantor and Frege, see Jourdain’s notes in Cantor, Contributions, p. 202. []
  32. 4.175. []
  33. R# 27. Murphey, in Development, pp. 251ff., claimed that Peirce distinguished between the terms “multitude” and “cardinal number”. There is something to this, but the observation is likely to be misleading. There were many passages, in the period from 1905 to 1911, in which Peirce described “cardinal numbers” as the vocal or written appellatives of finite multitudes. But Peirce, himself, was aware that this was not the normal use of the term. See 3.628, 4.337, and 4.637. Murphey did not make it clear that this dispute was mostly verbal, i.e., that Peirce meant by “multitude” what Cantor meant by “cardinal number”, but objected to Cantor’s choice of words.  At any rate, there is no question but that Peirce normally used the term “multitude” in the way we use the term “cardinal number” today. Also, see 3.630. []
  34. 4.177. []
  35. Peirce seems to have discovered the general application of diagonalization independently of Cantor–see 4.204. []
  36. 4.200-213, 4.674. []
  37. R# 28. []
  38. 4.219. []
  39. R# 28, and 3.568. []
  40. 6.121. []
  41. 4.126ff. []
  42. 3.568. []
  43. Murphey, Development, p. 260. Murphey said, “But a paradox of considerable complexity here presents itself, for, Peirce argues, the proposition that there is no greatest multitude leads to an absurdity.” []
  44. R# 14. []
  45. 3.549 — this would contradict Cantor’s theorem. []
  46. 3.568. []
  47. 4.175. []
  48. 4.218.  In Development, pp. 260-263, Murphey cited this passage, and another one from R# 154 which supports his interpretation no better. He con­structed an ingenious explanation for how Peirce came to make such a mistake — by mis­placing parentheses when evaluating powers. Peirce did occasionally do this, but he needs no excuses here. []
  49. 4.219. []
  50. 4.178. []
  51. 3.568. []
  52. 6.174. []
  53. 6.174. []
  54. 4.639.  The Monist, vol. 18, April 1908, pp. 227-241. []
  55. 4.642. []
  56. 4.642. []
  57. 4.642. []
  58. See 3.631. []

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