In his aforementioned exposition of the continuity of Peirce's thought, Kelly Parker devotes an important section[8] to discussing the twin notions of true and analytical continuity. Although it can seem, writes Parker, and as has also been pointed out previously[9], as if Peirce seriously misinterpreted and misunderstood Cantor's theory of ordinals in his work on continuity, what really bothered him at that particular moment in time was his doubts regarding the legitimacy of Cantor's identification of the set of real numbers, two to the power of aleph-null, as the continuum. This problem, claims Parker, was more fundamental for Peirce than the as yet unproved debate as to whether or not the cardinal two to the power of aleph-null is equal to the cardinal of the well-ordered set whose limit is larger than any of the members of aleph null. Peirce was namely convinced, and remained so for all of his life, that no collection of discrete points could constitute a true continuum, whereas Cantor continued to conceive of transfinite multitudes - as collections of discrete elements or numerical points. This was, paraphrasing Parker, the bone Peirce wished to pick with Cantor.
Peirce's own theory of the transfinite is based solely on the power-set method of generating cardinals, and as such, is correctly worked out, while Cantor, on the other hand, based his continuum hypothesis on two independent methods for the generation of transfinite cardinals, both the power-set method and the number-class method[10]. Peirce's criticism of Cantor, a criticism which incidentally is equally applicable to both methods, is that since no collection of discrete points can in any case constitute a true continuum, then Cantor's hypothesis, postulating a specific kind of relationship between series of transfinite cardinals generated by both methods, is not actually based on a valid notion of the continuum at all, but rather on what Peirce termed `pseudo-continua'.
The distinction between analytical and true continua has its historical roots in two alternative solutions to the well-known motion problem exemplified in Zeno's Paradoxes, one of which was proposed by Karl Weierstrass[11] (1815-1897) and the other by Peirce himself. Weierstrass' `analytic' approach was based on the mathematical theory of limits[12], and it is this approach which subsequently has become the most widely accepted one among mathematicians and others working in this particular area. Peirce, however, saw a number of unfortunate disadvantages in using the theory of limits, and often used Zeno's paradoxes himself as a starting point for developing and explicating his own theories about the nature of continua. Peirce's objection to Zeno's form of argumentation was always with regard to the latter's reference to space and time as static series of points or moments.
In this connection Peirce wrote: "All the arguments of Zeno depend upon supposing that a continuum has ultimate parts. But a continuum is precisely that every part of which has parts, in the same sense. Hence he makes out his contradictions only by making a self-contradictory supposition. In ordinary and mathematical language, we allow ourselves to speak of such parts - points - and whenever we are led into contradiction thereby, we have simply to express ourselves more accurately to resolve the difficulty."[13] He then goes on to exemplify a model of continuity based on topology, or `topical geometry' as he called it, which he uses to argue for rejection of a definition of continuity as a set of relations among discrete parts. Zeno's error is, in Peirce's view, the substitution of a mathematical abstraction such as a point, line or a plane for the true topological nature of the continuum.
He continues: "Suppose a piece of glass to be laid on a sheet of paper so as to cover half of it. The, every part of the paper is covered, or not covered; for `not' means merely outside of, or other than. But is the line under the edge of the glass covered or not? It is no more on one side of the edge than on the other. Therefore it is either on both sides, or neither side. It is not on neither side; for if it were it would not be on either side, therefore not on the covered side, therefore not covered. It is not partly on one side and partly on the other, because it has no width. Hence, it is wholly on both sides, or both covered and not covered.
The solution of this is, that we have supposed a part too narrow to be partly uncovered and partly covered; that is to say, a part which has no parts in a continuous surface, which by definition has no such parts. The reasoning therefore, simply serves to reduce this supposition to an absurdity."[14]
Peirce's point here is that even though it might often be functional for calculation or other purposes to consider parts in a continuum as ultimately existent, indivisible individuals, it is nonetheless mathematically and philosophically incorrect to do so. For Peirce a true continuum can have no such indivisible parts, and is indeed, characterized precisely by this lack of any such indivisibles. As he points out: "my notion of the essential character of a perfect continuum is the absolute generality with which two rules hold good, first, that every part has parts; and second that every sufficiently small part has the same mode of immediate connection with others as every other has."[15]
A key notion contained in this last citation is the phrase: mode of immediate connection, a notion which we shall explore in some more detail in the following: Peirce, then, offers two defining characteristics for the true continuum, the first being that it does not consist of an aggregate or collection of distinct parts, but is rather a kind of topological entity of "parts with parts", upon which discrete individuals may be marked, making it a "continuum of possible determinations", to use Parker's formulation. Although the continuum is not itself composed of parts, it retains a potential for division into parts wherever and whenever someone or something external to it marks a topical singularity upon it. Peirce writes in this connection: "[What] I mean by a true continuum is a line upon which there is room for any multitude of points whatsoever. Then the multitude or what corresponds to multitude of possible points - exceeds all multitude. These points are pure possibilities"[16]
Parker points out, with reference to Murphey's discussion of the same[17], that in 1854 Bernard Riemann[18] had suggested two alternative hypotheses for considering space in terms of what he called `manifoldness'[19], one of these being discrete manifoldness, and the other continuous manifoldness, noting that "...in a discrete manifoldness, the ground of its metric relations is given in the notion of it, while in a continuous manifoldness, this ground must come from the outside. Either therefore the reality which underlies space must form a discrete manifoldness, or we must seek the ground of its metric relations outside it, in binding forces which act upon it."[20] Regarding the twin notions of discrete and continuous manifolds, Riemann had previously maintained that "[m]agnitude-notions are only possible where there is an antecedent general notion which admits of different specializations. According as there exists among these specializations a continuous path from one to another or not, they form continuous or discrete manifoldness: the individual specializations are called in the first case points, in the second case, elements, of manifoldness."[21]
What is important in Riemann's observations above for our present discussion is not so much his naming of the different `specializations' obtaining in the case of the two manifolds as `points' and `elements', but rather his formulation that in the case of continuous manifoldness, the path between individual specializations is itself continuous, whereas in the case of discrete manifoldness it is not. Peirce, who was well acquainted Riemann's work and harboured a good deal of respect for it, opted for the alternative where manifoldness is considered as continuous rather than discrete, where the spatial continuum is a continuity of possible determinations upon which any number of points may be imposed from outside by the `binding force' of an action of mind.[22]
Peirce's preference for grounding his notion of the continuum in the notion of "binding forces which act upon it" from "outside", to reiterate Riemann's terms, was certainly related to the fact that he often applied the metaphor of `welding' in his definitions of the continuum. He wrote for instance that: "Breaking grains of sand more and more will only make the sand more broken. It will not weld the grains into unbroken continuity"[23], and in his famous Monist article of July 1892, The Law of Mind, also published in the Chance, Love and Logic collection[24], he wrote that "[l]ogical analysis applied to mental phenomena shows that there is but one law of mind, namely, that ideas tend to spread continuously and to affect certain others which stand to them in a particular relation of affectibility. In this spreading they lose intensity, and especially the power of affecting others, but gain generality and become welded to other ideas."
Peirce's second defining characteristic for the true continuum has then to do with the specific kind of way in which the various parts of the continuum are `welded' to one another and to the whole to which they are parts. In order to understand how individual parts of the continuum relate to each other and to the whole without being considered as finite points, and just how these relations reflect the positionings of individual parts of the continuum relative to one another in space and time, we will need to look more closely at Peirce's use of the concept of the infinitesimal.
For Peirce an infinitesimal is no more than a positive quantity which is less than any specifiable quantity. As such it cannot possibly be part of any discrete multitude which is identifiable by means of a cardinal number - i.e. such multitudes as form the basis for Cantorean transfinite sets. Cantor, indeed, had expressly rejected the idea that infinitesimals, traditionally and formally defined as the reciprocal of transfinite numbers, could have any place in mathematics. For Peirce, on the other hand, since the infinitesimal is merely one indeterminately small part of a true continuum, and since a true continuum can have no particular transfinite cardinal associated with it, then the infinitesimal cannot be defined as the reciprocal of such a cardinal. Infinitesimals are for Peirce both real and of unspecifiable extension, this taken to mean that they are not discernibly - in Leibnizian terms - different from one another. They have identical properties but are nonetheless distinct real entities, a condition Peirce characterizes as being "like two ideal rain drops, distinct but not different"[25].
Returning then, to the notion of mode of immediate connection mentioned previously: in order for parts of a linear continuum to be immediately connected, Peirce's main requirement is that they must be sufficiently small so as to no longer be subject to the same kinds of constraints on the relation of identity and otherness which hold for larger parts. When this is so, and when the parts are so connected, each and every part has exactly the same internal nature, they are in other words continuous, one-dimensional, geometric entities which are infinitesimal - less than any possibly specifiable positive length. Thus, they cannot be discerned as different from one another with regard to length, and the only difference between them will be with regard to the sequencing of their positions relative to one another on the continuous line of which they are parts.
Let us first take the case of two discernible parts, call them XX1 and YY1, of a continuous line, both of which have extension and are thus of some specifiable positive length. In this case there are three possible ways in which they may have parts in common. They are as follows:
1. Full inclusion of XX1 in YY1, as in:
2. Exact coincidence between XX1 and YY1 as in:
3. Overlap of XX1 and YY1 as in:
For two indefinitely small, indiscernibly different neighbouring parts (let us for the sake of comparison refer to these as xx1 and yy1) on the same continuous line, such distinctions among possible relations between them converge and cease to be meaningful, since one can no longer say that xx1 is smaller than yy1, or that xx1 is enclosed in yy1. In fact, all that one can really say is that there is a special kind of exact coincidence between xx1 and yy1, with each one sharing all of its parts with the other. Continuing to assume that xx1 and yy1 are immediately neighbouring parts on the continuous line, how then is it then possible to take account of this in terms of their positioning relative to one another, while at the same time allowing them to remain immediately connected?
Peirce presented the following example to illustrate how this may be done in the third of his Cambridge Conferences Lectures in 1898[26]. If a point is marked on a continuous line, and the line is cut at this point, this will produce a left hand region (LhR) and a right hand region (RhR). The original marked point then has become two points, one which constitutes the end of LhR and the other which constitutes the end of RhR, points which we shall again for the sake of consistency with the discussion above, continue to call xx1 and yy1. If these two ends are then rejoined, they become a single point again. This means that the original point must previously have been capable of being divided into parts. If we now proceed along the line from LhR towards the RhR from another point zz1, not immediately connected to xx1 or to yy1 , and lying a discernible distance to the left of the site of the original cut, we will obviously arrive first at xx1 before reaching yy1. The implication of this is that the positions of xx1 and yy1 on the continuum are ordered with respect to this third point. This is the reason why when the line is initially cut at the point that after the cut and rejoining process is carried out, comes to represent the exact coincidence of xx1 and yy1 that xx1 ends up in LhR and yy1 ends up in RhR.
For Peirce, such an ordering of infinitesimals is compatible with continuing to consider xx1 and yy1 as individuals with identity, since they were indiscernibly different before the cut was made, and since the difference in their respective distances from the third not immediately connected point zz1 to their left is infinitesimal. When the cut is made, presumably due to the agency of some outside influence, the inherent sequential- positional difference becomes discernible, making the performance of cutting the continuum the determinative act which makes manifest the inherent possibility of a distinction being made between the two points.
Here, it is of course key for our whole argument that what we have been referring to as `points' in what has gone before above, are not actually points, but rather infinitesimally small parts of the continuum, which are themselves continua capable of infinite division into further continua. Since the extensions of xx1 and yy1 are infinitesimally small and objectively indeterminate, the position where they occur on the continuum can never be fully determinate, and is thus in Peircean terms a general, to which by definition the principle of the excluded middle does not apply[27].
Murphey (1993, 281) notes in this context that: "... on Peirce's theory the limit of a convergent series of intervals on the geometric line will be an interval rather than a point. [...] Accordingly all positions on a plane are not uniquely paired to number couples of Cartesian coordinates: rather these coordinates define a network covering the plane, the points of intersection of which comprise only a small part of the positions on the plane itself."[28] It is neither completely true nor completely false to say that two immediately connected parts of a continuum are identical in every possible respect including their relative locations. For Peirce this means that a true continuum must contain no discrete points at all excepting those which more or less arbitrarily become imposed upon it by us or some other outside agency in the form of topical singularities, which might be for, for instance, measurement purposes, and which, since they have been imposed from outside, come to constitute a breach of, or break in continuity, representing for the moment a place of lower dimensionality than that of the whole.
In multidimensional continuous systems, then, general logic is for Peirce applicable "providing we take care not to assume that objects have independent identity after it has been assumed that have not"[29], but the moment we attempt to actually fix or determine absolutely such things as the instant at which a moving object becomes completely still, or exactly where two crossing lines intersect, then the general logical relations of identity and otherness which normally apply to singularities or individuals cease to function. As Parker puts it, the absolute instant or point "elude(s) dyadically structured thought. Continuity and combination are essentially triadic according to Peirce"[30].
In Murphey's view[31], Peirce's particular and quite unique way of reasoning places him squarely between the intuitionist and logistic schools of mathematics, since Peirce does not hesitate to allow nascent possibilities to blend into collections and to extend the operations of logic, including the law of the extended middle, to such collections of possibilities. He is willing to admit a pre-existing plurality of possibilities, but maintains at the same time that only a very small portion of these possibilities are actually actualized as existent individuals, and that this particular existence depends on them being brought into being by some outside agency.
Parker contends, and I tend to agree with this view, that Peirce's concept of true continuity is, in accordance with his general definition of the role of mathematics within his classification of the sciences as that which "merely posits hypotheses and traces out their consequences"[32], more functional, also in other ways, than Cantor's analytical continuity, in the sense that it allows for development of a more precise understanding of the notion of combination, while at the same time being broader and more generally oriented, and thus less likely to block the way of inquiry. I shall not here begin to go into his discussions of possible ways of rigorously demonstrating the logical consistency of a conception of continuity as composed of non-discrete parts[33], except to mention in passing that he brings in Abraham Robinson's (1966) theory of non-standard analysis[34] here as having a considerable potential, since it introduces analytical and formal means by which a consistent theory of infinitesimals may be developed. Peirce himself never developed such a proof, at least as far as we are aware of today, and Robinson's method, although based on the same apparatus of predicate calculus inherent in both Peirce's Logic of Relatives and his Existential Graphs, is not directly comparable with what might have come to be Peirce's developed system, since Robinson depended on Gödel's (1930) completeness theorem and Malcev's (1936) compactness theorem, both of which were not discovered until well after Peirce's death in 1914.
Also, Robinson maintains a variety of mathematical formalism which claims that working with the notion of infinite totalities is meaningless in any real sense, and that non-standard analysis only introduces new deductive procedures, rather than any new mathematical entities. Peirce, claims Parker, would most likely object to formalistic fence-sitting of this kind. The Peircean mathematician need not be concerned with metaphysics, but the Peircean physical scientist must however embrace some form of metaphysical framework. For the scientist, says Peirce, "the infinitesimals must be actual real distances, and not the mere mathematical conceptions like [radical]-1"[35]
[8] Parker 1998, 81-101
[9] See Murphey 1993, 238-288 for thorough examination of this theme
[10] See Parker 1998, 75ff and Murphey 1993, chapter 4, for discussion of some of the finer technical points involved in these two methods
[11] A concise biography of Karl Weierstrass can be found on at the following website:
http://www-history.mcs.st-and.ac.uk/history//Mathematicians/Weierstrass.html
[12] For exemplifications of the use of the theory of limits to address Zeno's paradox, see the following website: http://www.shu.edu/projects/reals/cont/index.html
[13] W: 2:256-57
[14] ibid.
[15] CP 4.642
[16] NEM 3.388
[17] Murphey 1993, 219-228.
[18] In a probationary lecture of June 10th 1854, at Göttingen. German title: Ueber die Hypothesen welche der Geometrie zu Grunde Legen. Published first in German in 1867, and translated to English in 1873 by W.K. Clifford, a friend of the Peirce family. English title: "On the Hypotheses Which Lie at the Basis of Geometry", Nature VIII, 14-17, 36f (May 1873), cf. Murphey 1993, 219.
[19] This way of considering space as a species of magnitude on Riemann's part was subsequently criticised as leading to circularity by Bertrand Russell in his Essay on the Foundations of Geometry (Cambridge 1897), pp. 63f, but this need not concern us here for the moment, since Riemann's formulations were in any case somewhat more ambiguous than Russell assumed. (Murphey 1993, 224f).
[20] Riemann 1873, 37.
[21] Ibid. 15
[22] Cf. Parker 1998, 242, 46n.
[23] CP 6.168
[24] CLL 203-204
[25] CP 4.311
[26] See Parker p. 90, who points out that the Cambridge Conference Lectures [NEM 4:342-43] have also been reprinted in full in Ketner & Putnam, 1992, and that the example mentioned discussed at length in the editors' introduction.
[27] CP 5.448, 505
[28] Murphey 1993, 281
[29] NEM 3:748
[30] Parker 1998, 92 and 243, 56n
[31] Murphey 1993, 287
[32] CP 1.240
[33] Parker 1998, 97-99
[34] See also Davis & Hersh 1972
[35] CP 3.570