Just to add to that discussion of the problem of metaphysical origins of continuity and quanta (integers). It’s from Jerome Havenel’s “Peirce’s Clarifications of Continuity.” First a quote from Peirce.
In Spencer’s phrase the undifferentiated differentiates itself. The homogeneous puts on heterogeneity. However it may be in special cases, then, we must suppose that as a rule the continuum has been derived from a more general continuum, a continuum of higher generality . . . If this be correct, we cannot suppose the process of derivation, a process which extends from before time and from before logic, we cannot suppose that it began elsewhere than in the utter vagueness of completely undetermined and dimensionless potential- ity. (RLT, p. 258) Note how here Peirce is still keeping with the basic ontology of neoplatonic emanations. This is prior to time and thus not cosmology in a physical sense. Time itself is regularity and thus has to develop before we can speak of time. Quoting from Havenel: Now, how could the universe have arisen from “nothing, pure zero, . . . prior to every first”? (CP 6.217, 1898). Contrary to Hegel’s logic of events, Peirce considers that no deduction “necessarily resulted from the Nothing of boundless freedom” (CP 6.219), but that the “logic may be that of the inductive or hypothetic inference” (CP 6.218). But at which stage did the “nothing, pure zero”, becomes a continuum? Whereas the zero is mere “germinal possibility” (NEM 4.345), the continuum is “developed possibility” (ibid). According to Floyd Merrell, before any existing thing could have arisen in the universe, the nothingness has become: “the continuous flux of Firstness”. Indeed, Peirce states that: The whole universe of true and real possibilities forms a continuum, upon which this Universe of Actual Existence is, by virtue of the essential Secondness of Existence, a discontinuous mark . . . There is room in the world of possibility for any multitude of such universes of Existence. (NEM 4.345) Peirce thinks that the “original potentiality is the Aristotelian matter or indeterminacy from which the universe is formed” (RLT, p. 263), and this “original potentiality is essentially continuous” (RLT, p. 262). Since the definitions of otherness and “. . . the principle of excluded middle, or that of contradiction, ought to be regarded as violated” (NEM 3.747). Although for Peirce the dimension of a continuum may be of any discrete multitude, there is an exception for the original continuum, whose number of dimensions is no longer discrete. If the multitude of dimensions surpasses all discrete multitudes there cease to be any distinct dimensions. I have not as yet obtained a logi- cally distinct conception of such a continuum. Provisionally, I iden- tify it with the uralt vague generality of the most abstract potentiality. (RLT, p. 253–254) Havenel then turns to the issues I think John is getting at. Yet, Douglas R. Anderson has pointed out the following difficulty: how can evolution be a continuous process if chance and spontaneity are discontinuous events82? According to Peirce’s Tychism, there can be no rational continuity between past events and spontaneity. The answer is that discontinuity is not absolute but is relative. For example, if one draws a new curve on a blackboard, it is a discontinuity. Nevertheless, “although it is new in its distinctive character, yet it derives its conti- nuity from the continuity of the blackboard itself ” (RLT, p. 263). Moreover, Peirce’s theory of evolution involves the notion of a final continuum. The universe evolves not only by chance and necessity, but also towards a final continuum, which is final less as a result than as a principle. For Peirce, “continuity is Thirdness in its full entelechy” (RLT, p. 190), and as a final cause, there is an end of History, but as a final result there is not. Thus, the ultimate good lies in the evolutionary process but not in individual reactions in their isolation; it lies in the growth of sympathy with others. Synechism is founded on the notion that the coalescence, the becom- ing continuous, the becoming governed by laws, the becoming instinct with general ideas, are but phases of one and the same process of the growth of reasonableness. (CP 5.4, 1902) Havenel sees a period of further development from 1908 to 1913 which he calls Peirce’s topological period. Peirce changes his mind about what is a “true” continuum. Although he maintains the idea of potentiality, the notion of continuity does not mainly rest on the notion of multi- plicity anymore, but mainly on topological considerations and on the relations between the parts of a continuum. Such a change explains why I call this last period “topological”. Going on about changes, he says, He left aside his previous distinction between the pseudo- continuum and the true continuum, for a new distinction between a perfect continuum and an imperfect continuum, this last being a con- tinuum: “having topical singularities” (CP 4.642, 1908). According to his concept of continuity, “a top[olog]ical singularity . . . is a breach of continuity”. But if the continuum has no topological singularity, then it is a perfect continuum whose essential character 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. This manifestly vague statement will more clearly convey my idea (though less distinctly) than the elaborate full explication of it could. (CP 4.642, 1908) The new aspect in this definition of a perfect continuum is: “that every sufficiently small part has the same mode of immediate connection with others as every other has”; but Peirce thinks that the theory of col- lections cannot correctly analyze this property. Indeed, since at least 1897, the idea that the mathematical theory of set is not enough to inves- tigate the continuum was growing in Peirce’s mind: “the development of projective geometry and of geometrical topics has shown that there are at least two large mathematical theories of continuity into which the idea of continuous quantity, in the usual sense of that word, does not enter at all” (CP 3.526). Moreover, we have seen that since 1900 Peirce has rejected the idea that, strictly speaking, a collection can be continuous. The shift to topological considerations I suspect most here are familiar with although I’ll fully confess I don’t feel I have a sufficient grasp of this part of Peirce’s thought. Havenel doesn’t discuss discreteness much here although in his conclusion he touches upon it. Is everything continuous for Peirce? It is clear that there are discontinuities, for Peirce states against Hegel that both Firstness and Secondness are not reducible to Thirdness. But according to Peirce, nothing is absolutely isolated or separated; hence, there is no absolute discontinuity. Anderson has pointed out that according to Peirce’s tychism, there can be no rational continuity between past events and spontaneity. But discontinuity is relative, not absolute, like a new curve on a blackboard which is a discontinuity, but which “derives its discontinuity from the continuity of the blackboard itself” (RLT, p. 263, 1898). Moreover, Peirce’s theory of evolution proceeds from an original continuum towards a final continuum, which is final less as a result than as a principle. In other words, there are different kinds of discontinuities but within different kinds of continua. This can explain Peirce’s enigmatic claim that: “the doctrine of continuity is that all things so swim in continua” (CP 1.171, Summer 1893). If I follow him correctly his argument entails that the appearance of quanta is never absolute but only at best relative. Perhaps this is what John sees as unsatisfactory. It does seem from Havenel that up to his death from cancer Peirce is still struggling to work all of this out. While the continua might never be satisfactorily concluded I do think it fair to say that any break with it is only apparent for Peirce. How that relates to Peirce’s chemistry I can’t say. Hopefully this at least gives us a ground from which to argue. I confess I just wasn’t at all clear the grounds you were coming from John, beyond the appearance of a genealogical argument.
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