Jeff, your post seems to head in directions I'm unable to follow, so I'll just mention this: the final two selections in Moore's "Philosophy of Mathematics" collection are probably the best tools for "filling in the gap" in Peirce's thinking between arrival of the proofs of the article and the addendum that was printed was printed it. Both of those selections were written before the published version and show the train of thought Peirce was following as he abandoned his "supermultitudinous" view of continuity. The second one (the last selection in Moore) is especially interesting, to me anyway, although both of them break off at the end, as presumably Peirce decided to start over with the next draft.
I'm happy that Matthew Moore's collection is available relatively cheaply, because it's very helpful for understanding Peirce's philosophy (not just his mathematics). I might like it even better if it were presented in chronological order . Gary f. From: Jeffrey Brian Downard [mailto:jeffrey.down...@nau.edu] Sent: 23-Feb-17 04:39 Ben, Gary F, Jon S, List, Thanks to Ben for reminding me that Matthew provides some comments about this addendum in his collection, and also to Gary F. for supplying the relevant passage. I'd like to respond to a concern that Matthew raises about the way Peirce is explaining what he is trying to accomplish: "It is perhaps an ominous sign that Peirce devotes too much space to what appears to be a somewhat manufactured objection: since he does not explain what he means by `immediate connection,' it would hardly have occurred to the reader that time was bound up with such connection, had Peirce himself not brought it up." Here is a hypothesis that Peirce puts forward in "A Guess at the Riddle" about the real nature of time as it first evolved in the cosmos: Our conceptions of the first stages of the development, before time yet existed, must be as vague and figurative as the expressions of the first chapter of Genesis. Out of the womb of indeterminacy we must say that there would have come something, by the principle of Firstness, which we may call a flash. Then by the principle of habit there would have been a second flash. Though time would not yet have been, this second flash was in some sense after the first, because resulting from it. Then there would have come other successions ever more and more closely connected, the habits and the tendency to take them ever strengthening themselves, until the events would have been bound together into something like a continuous flow. We have no reason to think that even now time is quite perfectly continuous and uniform in its flow. The quasi-flow which would result would, however, differ essentially from time in this respect, that it would not necessarily be in a single stream. Different flashes might start different streams, between which there should be no relations of contemporaneity or succession. So one stream might branch into two, or two might coalesce. But the further result of habit would inevitably be to separate utterly those that were long separated, and to make those which presented frequent common points coalesce into perfect union. Those that were completely separated would be so many different worlds which would know nothing of one another; so that the effect would be just what we actually observe. But Secondness is of two types. Consequently besides flashes genuinely second to others, so as to come after them, there will be pairs of flashes, or, since time is now supposed to be developed, we had better say pairs of states, which are reciprocally second, each member of the pair to the other. This is the first germ of spatial extension. These states will undergo changes; and habits will be formed of passing from certain states to certain others, and of not passing from certain states to certain others. Those states to which a state will immediately pass will be adjacent to it; and thus habits will be formed which will constitute a spatial continuum, but differing from our space by being very irregular in its connections, having one number of dimensions in one place and another number in another place, and being different for one moving state from what it is for another. Pairs of states will also begin to take habits, and thus each state having different habits with reference to the different other states will give rise to bundles of habits, which will be substances. Some of these states will chance to take habits of persistency, and will get to be less and less liable to disappear; while those that fail to take such habits will fall out of existence. Thus, substances will get to be permanent. In fact, habits, from the mode of their formation, necessarily consist in the permanence of some relation, and therefore, on this theory, each law of nature would consist in some permanence, such as the permanence of mass, momentum, and energy. In this respect, the theory suits the facts admirably. The substances carrying their habits with them in their motions through space will tend to render the different parts of space alike. Thus, the dimensionality of space will tend gradually to uniformity; and multiple connections, except at infinity, where substances never go, will be obliterated. At the outset, the connections of space were probably different for one substance and part of a substance from what they were for another; that is to say, points adjacent or near one another for the motions of one body would not be so for another; and this may possibly have contributed to break substances into little pieces or atoms. But the mutual actions of bodies would have tended to reduce their habits to uniformity in this respect; and besides there must have arisen conflicts between the habits of bodies and the habits of parts of space, which would never have ceased till they were brought into conformity. (W, Vol. 6, 209-10; CP 1.412-416) The idea that I find striking here is that Peirce is offering an explanation of how things might occur simultaneously--at the same time but in two different places--by suggesting that the habits governing such dyadic relations evolved from the habits governing a different sort of dyadic relation that is involved in ordered relations in time. He says: "Different flashes might start different streams, between which there should be no relations of contemporaneity or succession. So one stream might branch into two, or two might coalesce. But the further result of habit would inevitably be to separate utterly those that were long separated, and to make those which presented frequent common points coalesce into perfect union." The idea of "common points coalescing into a perfect union" would seem to be the root of the idea of "immediate connection," would it not? One thing I find particularly puzzling about the addendum is the rapidity with which Peirce moves from more phenomenological points about our experience of time to more metaphysical points about the real nature of time and then back again. The key idea in making such a rapid movement back and forth is to draw a contrast between our idea of time and the facts we seek to explain by introducing and using such an idea. How does the suggestion that time can function as a kind of standard for measuring degrees of continuity akin to way that oxygen can serve as a standard for the measurement of atomic weights help us to better understand how the idea of time first arose as a hypothetical explanation of some sort of surprising set of facts? The puzzling phenomena that are offered in the example he provides are the disjointed experiences associated with looking out of the window of a steamboat at night at the shore as the lightning periodically illuminates scenes on the riverbank. He claims that the surprising and apparently contradictory character of phenomena we observe can be explained if we "suppose them to be mere aspects, that is, relations to ourselves, and the phenomena are explained by supposing our standpoint to be different in the different flashes." The point of this illustration, I take it, is to help us see more clearly what otherwise might be obscure in the newly amended conception of continuity that he is trying to articulate. As such, how does the illustration help to clarify how it is possible for a continuum to have room for a denumeral multitude of points, or an abnumeral multitude, or any multitude whatsoever? What might, from one point of view, appear to be disjointed might, from another point of view, appear to be continuous-just like the relations between the dimensions in space. From the perspective of a person living on a one dimensional line, a point is a kind of discontinuity. Similarly, from the perspective of person living in a two dimensional space, a one dimensional line appears as a kind of discontinuity.and so on. Having said that, it still isn't clear to me how the illustration helps to clarify the puzzling features of the amendments he is making to the conception of continuity. What is more, it isn't clear what role the point about the continuity time being a sort of standard for measure is supposed to play in the account. The insights that seem to have prompted the revisions in his account of continuity appear to have grown from reflections on the character of cyclical systems and the light that such systems shed on the relations between a perfect continuum and those that are imperfect. Following this line of thought, I tend to think that the continuity of time can, on his account, serve as a standard for measuring the degree to which different sorts of systems are more or less perfect in their continuity. His point, I take it, is that it doesn't really matter whether one or another thing (the connections between the parts of space, the connections between shades of the hue of a color, the connections between parts of time etc.) are entirely perfect as continua. Instead, time is like oxygen in the scale of atomic weights in that it supplies us with a sufficiently reliable standard that we are thereby enabled to make relative comparisons--even if we don't (yet) have an absolute standard. Matthew makes a further remark to the effect that Peirce moves in this addendum from an account of continuity that is based on the size of collections to an account that is grounded on topological relations-and that this seems to represent a dramatic shift in the way he is thinking about continuity. The last lecture of RLT makes it clear, I think, that Peirce has been reflecting in rather deep ways on the relationships between these two different mathematical approaches to understanding different aspects of the conception of continuity. By the time that he is writing the addendum, he has been reflecting on the relationship between more arithmetic approaches that start with what is discrete and more topological approaches that start with what is continuous. As such, I tend to think that the insights that sparked the revisions in the conception of continuity in the year that intervened between the time that he received the proofs for the "First Curiosity" of "Some Amazing Mazes" and the addendum might stem from something that can been seen when one thinks about the topological character of different systems of number--especially when one experiments with diagrams involving cyclical systems. My hunch is that Peirce was drawing on cyclical systems in his exploration of different sorts of multitudes, and that he was thinking quite deeply about the different sorts of formal relations (e.g., symmetries) that hold between the different systems of numbers. In doing so, he was working in the same spirit as contemporary topologists when they use what is called the Farey diagram to explore the relations between the number systems of the rationals, the reals, the imaginaries, etc. See, for example, Allen Hatcher's The Topology of Numbers. (https://www.math.cornell.edu/~hatcher/TN/TNbook.pdf_ Pursuing this line of thought further would take considerable time and space, so I'll stop here. --Jeff
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