List,
In a footnote to "The Logic of Mathematics; an attempt to develop my categories from within," the editors indicate that the first four pages of the manuscript (MS 900) are missing. The Robin catalogue indicates the same thing. Has anyone, by chance, come across any references to what might be contained in the missing pages in other things that Peirce wrote--perhaps in the secondary literature or in other essays, notes or letters that Peirce write around 1896 when this essay was composed? I'm guessing the editors of the Collected Pages spent some time looking for the missing pages, but I thought it might be worth asking if anyone has heard stories about what happened to them. If I were to hazard a guess, the topic of the first four pages under Article 1 might have been similar to what is found in the first section of "The Logic of Mathematics in Relation to Education", which was written in 1898. Does that seem plausible? Thanks, Jeff PS While it is quite a long shot, let me know if you might have come across something in the archives (or elsewhere--e.g., the personal collection of a philosopher who did graduate work at Harvard back in the 40's) that might fit the description of the pages that are missing. The remaining pages are vertically oriented, unlined, neatly written, with "LofM" in the top left corner and numbered on the top right. Given the fact that the title of the piece appears to be pasted onto the top portion of a torn cover of a notebook and page five starts with Art. 2, the first page probably starts with "Art. 1." and then the text (perhaps with no title at the top). Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354 ________________________________ From: Jeffrey Brian Downard Sent: Friday, February 24, 2017 10:58 AM To: peirce-l@list.iupui.edu Subject: Re: [PEIRCE-L] Cyclical Systems and Continuity Ben, Gary F, Jon S, List The reference Ben makes to mathematical singularity theory is interesting. The general idea of turning to work in algebraic metrical and projective geometry--as well as algebraic topology--is the kind of approach I find attractive when faced with a puzzling discussion like we have in the "First Curiosity" and the Addendum. Given the fact that I don't feel at home in the larger world of mathematics in which Peirce lives, I find that a map to the larger terrain is helpful. As such, I think of the Elements of Mathematics and the New Elements of Geometry as a kind of map to understanding how Peirce thinks of the elementary parts of mathematics as fitting together. The general point I was trying to make earlier is that it is not helpful to think of the study of multitudes, on the one hand, and the study of topology, optics and metrics, on the other, as two separate approaches to understanding the continuum. Peirce does consider the study of systems that are (i) discrete and infinite and systems that are (ii) continuous and infinite as two different branches of mathematics. Having said that, it is clear from his work on the elements of mathematics that the study of number systems is part of topology. In the Elements of Mathematics, the study of one leads directly into the study of the other with no break. In the New Elements of Mathematics, the study of the different systems of number is an integral part of his discussion of the fundamental properties of space. On Peirce's account, the old division between algebra and geometry is not a separation between different branches of mathematics. Rather, they are two different approaches that employ different sorts of systems of representation--one more iconic, the other more symbolic (but both systems employing both kinds of signs in robust ways)--that can be translated from one the to other when studying the different systems of mathematics. Now, for the more particular suggestions that Ben seems to be pointing us to about mathematical singularity theory. Having skimmed through the contents of the Elements of Mathematics and the New Elements of Geometry this morning, I am not able to make out whether Peirce is drawing on the kinds of ideas about the intersection of curves that are proven in Bézout's theorem. Having said that, I do see that the wiki entry on the topic refers to the "most delicate part of Bézout's theorem and its generalization to the case of k algebraic hypersurfaces in k-dimensional projective space<https://en.wikipedia.org/wiki/Projective_space>." Without worrying about the technical parts of the discussion, we can see that the formulation of the theorem within the framework of projective geometry and a system of homogeneous coordinates yields some nice diagrams of intersecting conics. https://en.wikipedia.org/wiki/B%C3%A9zout's_theorem It is surprising, at least to me, that the curves in the diagrams have intersections with multiplicity of 2, 3 and 4. In his discussion of "The Fundamental Propositions of Graphics in Chapter 2, Peirce seems to consider why it is that, under different projective hypotheses, the possible points in the lines of different curves "rush together" in the way that they do when they meet at a tangent (see Art. 101, pp 363-5 and following). He asks what happens when a ray intersecting the curve of a conic as a secant become a tangent and says, in response to the question, "The mathematician strives in vain to master such a hypothesis. He finds it much easier to suppose there is some other part of space, unseen by us, which those points enter." In this way, Peirce seems to be contemplating a move from one system of homogeneous coordinates (e.g., rational numbers) into another system of coordinates (e.g., imaginary numbers) which enables us to have a space with higher dimensions and with different kinds of properties. The parallels between what he is saying here about (1) moving to a mathematical perspective in which we suppose there is some other part of space that is unseen by us and (2) what he says in the illustration of the person in the the steamboat at night who is able to understand how the disjointed scenes illuminated on the shore are, from another perspective, part of a continuous unfolding of time involving parts of the scene that are unseen by him, is--at least to my ear--quite suggestive. As such, one of the ideas that stands out to me in Peirce's discussion in the addendum is the movement to a different perspective that enables us to treat as continuous what had seemed, from the earlier perspective, to be discontinuous. We have a similar kind of shift in perspective when we move from thinking of systems numbers as fields that are spread in a "flat" diagrammatic space (i.e., as with the Sylvester matrix representation in the explanation of the projective properties of the conic curves that are meeting with different orders of multiplicity in the wiki entry on Bézout's theorem) to thinking of systems of numbers as cyclical systems. The shift from one perspective to another involves a movement to a different sort of topological configuration in the relations between the numbers in the system--i.e., from linear sequences of ordered relations that don't return to those that do. Making sense of the implications of such shifts in perspective and how they might enable us to better see what has heretofore been unseen in common experience and mathematics, as well as in philosophical inquiry in the normative sciences and in metaphysics, is what I'm hoping to gain. --Jeff Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354 ________________________________ From: Benjamin Udell <baud...@gmail.com> Sent: Friday, February 24, 2017 8:06 AM To: peirce-l@list.iupui.edu Subject: Re: [PEIRCE-L] Cyclical Systems and Continuity Jon S., Jeff, Mathematical singularity theory may be relevant here. I've just now read some things on it including Bézout's theorem. If it is relevant, then I would wonder why Peirce didn't mention Bézout's theorem (he does mention Bézout a few times) or the like. Decades ago a topological singularity theorist told me that the theory provided ways to distinguish points that one would think were not distinguishable (or discernible). For example, think of a string that loops over itself on the floor, or a curve that loops through itself on a plane. The point of self-intersection is a point with "multiplicity two", since it corresponds to two different points along the curve. They both correspond to a single point on the plane (or floor). They are ordered, one coming earlier or later than the other, depending the direction in which the points along the curve are ordered. Another case of multiplicity two is that of the factor 3 in the equality 3×3=9. Deforming the string or curve so that it does not pass across itself does not result in our needing to attribute the former intersection point to just part of the loop or the other. Has the intersection point "exploded" into two points? But it always had multiplicity two. Peirce's example involves neither such a loop nor a cusp, but a simple circle. The circle gets cut by a line segment that is not a part of the circle. The loop cuts itself by intersection of two parts that "locally" are not part of each other. The multiplicity of the circle's cut point is not the multiplicity of an intesecting line segment's point coinciding with a point on the circle. Instead, Peirce cuts the circle show that any point on it is a kind of potential multiplicity of ordered points. He cuts the circle and widens it into a C, and if one closes it again, there is nothing to stop us from considering the rejoined end points as distinct but coinciding at a point on the plane, as long as we can find them again. But in making it back into a simple circle, we have melted them into each other. Instead, some re-interruption of the circle, some decision as to how to cut it, is required in order to revive the singularity and determine its actual multiplicity or, to put it another way, to determine what multiplicity has been actualized. I hope that makes some sense. It's still kind of early in the morning for me. Best, Ben On 2/23/2017 10:28 PM, Jon Alan Schmidt wrote: Jeff, List: JD: I'm wondering if anyone can explain in greater detail what Peirce is suggesting in this passage in making the comparison between the atomic weight of oxygen and the continuity of Time--or if anyone knows of clear reconstructions of what he is doing in the secondary literature? Kelly Parker discusses the same passage on pages 116-117 of The Continuity of Peirce's Thought. The continuum of time is a species of generality, and is present in any event whatever. Moreover, time is the most perfect continuum in experience. Accordingly, when Peirce defined a continuum as that which, first, has no ultimate parts, and second, exhibits immediate connection among sufficiently small neighboring parts, he appealed to the experience of time to illustrate the notion of immediate connection. Time serves as the experienced standard of continuity, through which we envisage all other continua (CP 6.86). There is an apparent circularity in using time to define continuity. The definition of continuity involves the idea of immediate connection, immediate connection is clarified by an appeal to the concept of time, and time is conceived as continuous (CP 4.642). In chapter 4, I sought to clarify the notion of immediate connection without an appeal to time, so as to break this circle. With the mathematical account of immediate connection in hand, though, we can see that Peirce's appeal to time identifies it as the experiential standard of continuity. Peirce asserts that to say time is continouos is "just like saying the atomic weight of oxygen is 16, meaning 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, just nothing at all" (CP 4.642). Time, the standard of continuity, is the "most perfect" continuum in experience, but should not be taken as an absolutely perfect continuum. The perfect true continuum is only described hypothetically in mathematics. Peirce observed that time is in all likelihood not "quite perfectly continuous and uniform in its flow (CP 1.412). Phenomenological time does exhibit the properties of infinite divisibility and immediate connection, but is probably not best conceived as an unbroken and absolutely regular thread. The only constant we have noted in time is the regularity of development or change, but change is not smooth. Changes differ from one another. The "regular" phenomenon of change consists, on closer examination, of numerous (perhaps infinite) parallel courses of development with different patterns and histories that interweave and diverge. Here is what Parker says in chapter 4, page 89, about "immediate connection." Immediate connection is a kind of relation, but a rather peculiar kind. That the connection is immediate suggests that there is no need of a third, C, which mediates the relation of A to B. Not only does this exclude the possibility that there is anything located between A and B, it also excludes the possibility that A and B are related only in virtue of belonging to some class of objects. In the case where A and B are connected only in co-being, for example, the third is that which brings the two independent things together in a collection. My dog and my hat are "connected" in the sense that both are elements of the set of things in the house as I write this sentence. Suppose that two things, A and B, have some part in common. If that part is the region of intersection of A and B, then the region acts as a third and the relation is mediated by that third. Now suppose that A and B have everything in common. This means that A and B are the same. In fact, the relation of immediate connection appears to be tied up with the relation of identity. Roughly speaking, the "mode of immediate connection" between sufficiently small neighboring parts of a continuum is that such parts are in some sense identical. If I read Peirce correctly, he is here flirting with what would appear to be a disastrous contradiction: neighboring parts of a continuum may be the same in every respect, including location. But if all neighboring parts of a continuum are thus identical, there seems to be no way to explain how they make up a continuum rather than collapsing into a single point. How is it that we get from the identity of neighboring points to a line with a left hand and a right hand region? Peirce escapes the paradox by denying that the relations of identity and otherness strictly hold in respect to sufficiently small neighboring parts of a continuum (NEM 3:747). This assertion demands some elaboration. Parker goes on to cite "a loophole in Leibniz's Law," since "difference is equated with discernable difference, that is, a difference that in principle could become apparent to a mind." If "A and B are neighboring parts of a continuous line" that "are sufficiently small to be immediately connected," they must be "shorter than any specifiable positive length"; i.e., infinitesimal. "The only difference between A and B must be in their location on the continuous line of which they are parts. Because they are neighboring parts and are connected, they have parts in common; because they are immediately connected, they have all their parts in common." Parker then invokes Peirce's illustration from the third RLT lecture of cutting a line at a point, such that it becomes two points. "It is clear that if the original 'point' thus admits of being divided, it must be divisible into parts of the kind we have been discussing. This example illustrates that the immediately connected neighboring parts A and B (which are at the same 'point' before the cut) must be ordered." Regards, Jon Alan Schmidt - Olathe, Kansas, USA Professional Engineer, Amateur Philosopher, Lutheran Layman www.LinkedIn.com/in/JonAlanSchmidt<http://www.LinkedIn.com/in/JonAlanSchmidt> - twitter.com/JonAlanSchmidt<http://twitter.com/JonAlanSchmidt>
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