[8.5.] The Modal Conception of Continuity.

 

Before continuing our examination of his 1903 presentation of scholastic realism, we need to return to his synechism, his doctrine that whatever is, is continuous.

 

Recall that at the time he articulated his evolutionary cosmology (1891-93), he defined continuity as follows:

 

continuity (Peirce’s df.₁₈₉₂): continuity consists of Kanticity and Aristotelicity:

Kanticity: having a point between any two points

Aristotelicity: the property of any series that “contains the end point belonging to every endless series of points which it contains.” (EP 1:321, CP 6.123)

 

Shortly after Peirce’s adoption of strong modal realism (in 1896, when he came to believe that substantial possibility could not be adequately defined in terms of states of information) he revised this definition of continuity.[1]

 

Reflecting his new, strong modal realism, he began to explain continuity in modal terms: His new approach was to define continuity in terms of possibility.

 

He stated this new conception of continuity in a set of lectures he presented in 1898:

 

... a continuum is a collection of so vast a multitude that in the whole universe of possibility there is not room for them to retain their distinct identities; but they become welded into one another.  Thus, the continuum is all that is possible, in whatever dimension it be continuous. [2]

 

To understand Peirce’s new, modal conception of continuity, we need to consider Peirce claim that  mere “possibles” are not distinct individuals.

 

Actually existing objects are distinct from one another, and a collection of such objects is characterized by discreteness among its members. For example,

·         the collection of students registered in this class is a collection of distinct individuals;

·         were I to place five ice cubes in a cup, the collection of those ice cubes would be a collection of five distinct, individual objects (at least until they begin to melt, at which point two or more of them might merge into one).

 

But the same is not true for merely possible objects, i.e., objects that are possible but not actual, e.g., the drops of water in the ocean, or the New Yorkers who will kill themselves the year after next.

 

The water and suicide examples come from 1897’s “Multitude and Number” (which, according to the editors of the CP, may have been written as a lecture). In that work, Peirce wrote:

 

When we say that of all possible throws of a pair of dice one thirty-sixth part will show sixes, the collection of possible throws which have not been made is a collection of which the individual units have no distinct identity. It is impossible so to designate a single one of those possible throws that have not been thrown that the designation shall be applicable to only one definite possible throw; and this impossibility does not spring from any incapacity of ours, but from the fact that in their own nature those throws are not individually distinct. (CP 4.172, emphasis added)

 

When it comes to collections of “mere possibilities, the individuals merge together.” (4.172)

 

So merely possible objects, i.e., objects that are possible but not actual, are not individual. They have no distinct identity of their own. The possible points “on” the real number line, the drops of water “in” the ocean, and those unfortunate New Yorkers who will kill themselves in two years... all of these mere “possibles” lack individuality.

 

A collection of mere possibilities is non-discrete, i.e., it does not consist of distinct individuals.  The members of such a collection “lose their individual identity because the collection exceeds every positive existence of the universe.” (4.175) In other words, such a collection is continuous.

 

As an illustration, consider a continuous line. On Peirce’s new, modal understanding of continuity, a line does not consist in any number of actual individual points, however numerous; rather,

 

…a line is nothing but a collection of points of a particular mode of multiplicity, yet in it the individual identities of the units are completely merged, so that not a single one of them can be identified, even approximately, unless it happen to be a topically singular point, that is, either an extremity or a point of branching, in which case there is a defect of continuity at that point. (4.219, “Multitude and Number,” 1897)

 

... a line ... contains no points until the continuity is broken by marking the points. In accordance with this it seems necessary to say that a continuum, where it is continuous and unbroken, contains no definite parts; that its parts are created in the act of defining them and the precise definition of them breaks the continuity. (6.168, 1903)[3]

 

So the real number line, which is continuous, is not a collection of actual, individual points. Instead, the continuity of the line consists in the fact that an actual point could possibly be distinguished anywhere along the line.

 

But note that if an actual point is distinguished, the continuity of the line is broken or breached at that point. Any actual point is a breach of the line’s continuity, but the bare possibility of a point being distinguished on the line is not. In fact, it is the possibility that a point can be distinguished anywhere on the line that is the line’s continuity, on Peirce’s new, modal conception of continuity.

 

 

[8.5.] Real Generality Requires Real Possibility.

 

Having considered Peirce new, modal conception of continuity, we are ready to return to his 1903 lecture “The Seven Systems of Metaphysics” and see how his scholastic realism has expanded.

 

In this lecture he considers an old definition of generality:

 

You may, perhaps, ask me how I connect generality with Thirdness. Various different replies, each fully satisfactory, may be made to that inquiry. The old definition of a general is Generale est quod natum aptum est dici de multis [“A general is that whose expression naturally suits many things.”[4]] This recognizes that the general is essentially predicative and therefore of the nature of a representamen. And by following out that path of suggestion we should obtain a good reply to the inquiry. (EP 1:183, CP 5.102)

 

According to this traditional definition,

·         general terms like “gold,” “lithium” and even “sun” are capable of applying truly to many different individual things;

·         a general proposition like “All solid bodies fall in the absence of any upward forces or pressure” does not refer to only one, or a few, solid bodies; it applies to many solid bodies.

 

But, says Peirce, to say that a general is that the expression of which suits many things is to describe “a very degenerate sort of generality.” (EP 2:183)

 

General terms are capable of applying truly, not just to many things, but to any number of

individual objects whatsoever.

 

If sol {“sun”] is apt to be predicated of many, it is apt to be predicated of any multitude however great, and since there is no maximum multitude, those objects, of which it is fit to be predicated, form an aggregate that exceeds all multitude. Take any two possible objects that might be called suns and, however much alike they may be, any multitude whatsoever of intermediate suns are alternatively possible, and therefore as before these intermediate possible suns transcend all multitude.  In short, the idea of a general involves the idea of possible variations which no multitude of existent things could exhaust but would leave between any two not merely many possibilities, but possibilities absolutely beyond all multitude. (EP 2:183; CP 5.103; PMM p.193, 1903, emphases added)

 

Since there is no greatest multitude, there is no limit to the number of objects to which a general term can truly apply.

 

So the number of objects to which a given general term can truly apply exceeds the number of ACTUAL objects... both present actual objects and objects that will be actual in the future. A general term is predicable, not only of a multitude greater than that of the individuals which exist at any given time, but also greater than that of all individuals that will ever exist.

 

So the reality of a general, like sun, requires the reality of possible (but non-actual) suns.

 

Peirce’s example of the suns calls to mind the real number line:

·         Between any two points on that line, there is another point; in fact, there is an infinity of other points.

·         And between any two real numbers, there is another real number; in fact, there is an infinity of other real numbers.

·         Analogously, “between” any two possible suns, no matter how similar, lies another “intermediate” sun; in fact, there is an infinity of other “intermediate” suns.

 

So Peirce’s later scholastic realism (his extreme scholastic realism) requires not only the reality of generals, but also the reality of what he calls vagues.

 

... the scholastic doctrine of realism ... is usually defined as the opinion that there are real objects that are general, among the number being the modes of determination of existent singulars, if, indeed, these be not the only such objects. But the belief in this can hardly escape being accompanied by the acknowledgment that there are, besides, real vagues, and especially real possibilities. (EP 2:354, CP 5.453, 1905; selection 7 in the modality readings document)

 

So after the modal shift of 1896, by which Peirce moved to a strong version of modal realism, his scholastic realism grew, so that it required not just real generals (including real necessity) but also real vagues (including real possibility).

 

 

[8.6.] Vagueness, Determinacy, and Generality.

 

This foreshadows a more fully developed set of concepts that we will see him using in an upcoming reading, one that ties together his new extreme scholastic realism even more closely with his categories:

 

 

Firstness	Secondness	Thirdness
vagueness / vagues	determinacy	generality / generals
include real possibility.	actuality / existence	include natural laws (“all solid bodies fall in the absence of any upward forces or pressure”) and thus necessity.

 

 

[8.7.] Generality and Continuity as Thirdness.

 

Peirce closely associates generality and continuity with Thirdness. He is explicit about the association between generality and Thirdness in the 1903 lectures we’ve been reading:

 

Now Thirdness is nothing but the character of an object which embodies Betweenness or Mediation in its simplest and most rudimentary form; and I use it as the name of that element of the phenomenon which is predominant wherever Mediation is predominant, and which reaches its fullness in Representation.

                Thirdness, as I use the term, is only a synonym for Representation, to which I prefer the less colored term because its suggestions are not so narrow and special as those of the word Representation. (EP 1:183-4, CP 5.104-105)

 

But he identified Thirdness with continuity a few years before, in 1898:

 

Permit me further to say that I object to having my metaphysical system as a whole called Tychism. For although tychism does enter into it, it only enters as subsidiary to that which is really, as I regard it, the characteristic of my doctrine, namely, that I chiefly insist upon continuity, or Thirdness, and, in order to secure to thirdness its really commanding function, I find it indispensable fully [to] recognize that it is a third, and that Firstness, or chance, and Secondness, or Brute reaction, are other elements, without the independence of which Thirdness would not have anything upon which to operate. Accordingly, I like to call my theory Synechism, because it rests on the study of continuity. I would not object to Tritism. And if anybody can prove that it is trite, that would delight me [in] the chiefest degree. (CP 6.202, also in RLT)

 

 

 

Stopping point for Wednesday October 24, For next time, begin reading “What Pragmatism Is” (pp.331-38).