Jeff, List: Thanks for your comments and questions.
JD: Are you focusing on a definition adequate for a mathematical conception of continuity? Or, do you take yourself to have offered a definition of continuity that is adequate for philosophical inquiry in phenomenology, the normative sciences and metaphysics--as well as for mathematics? As I state in my last introductory paragraph, the primary objective of my article is to reinterpret Peirce's late writings on continuity in light of two manuscripts that have not previously been mentioned at all in the relevant secondary literature--R 144 (c. 1900) and RS 30 (c. 1906). As such, it is a product of the methodology that I spelled out in the recent List thread on "The Logic of Interpretation"; namely, a retroductive hypothesis about how we might better understand the various quoted texts, especially taken together. The resulting definition is mathematical in the broad sense that the subject matter of mathematics consists entirely of hypotheses, but I am cautiously optimistic that it is also amenable to philosophical applications, consistent with Peirce's preference for "synechism" as the name of his overall system. Only time will tell whether and to what extent it is successful. Since I am proposing a new formulation of what is necessary and sufficient for a specifically *Peircean* continuum, I join him in acknowledging that it "as yet has not received that sanction which can only come from the critical examination of the most powerful and exact intellects," such that admittedly it might turn out to be "all wrong" (RLT 242, 1898). JD: On its face, I find your suggestion that Zalamea uses terms not found in Peirce's texts to be puzzling. Let me focus on two components of his account that are not found in your list: genericity and modality. Is there some reason that you've set genericity and modality to the side in your account? As I state in my conclusion, I find Zalamea's entire terminology a bit inscrutable, and "supermultitudeness" is clearly a vestige of the bottom-up conception that Peirce ultimately abandoned. I know what generality and modality are, but "genericity" is unfamiliar, and despite multiple readings of Zalamea's writings it is frankly still unclear to me exactly what it means to *define *a continuum as "generic" and "modal." As far as I know, Peirce never uses "reflexive" or "inextensible," and he only once employs "plastic" in the relevant sense, urging his 1903 Lowell Lectures audience to imagine that the continuum of all true propositions as represented by the sheet of assertion "can be deformed in all sorts of ways without the continuity and connection of parts being ever ruptured" (CP 4.512). I hope that this clarifies my purposes in writing the paper and how they are different from Zalamea's, which is why I reach a different outcome. My intention is not at all to detract from the fruitful work that he and others have been and still are doing within the more technical fields of contemporary mathematics, inspired by and building upon Peirce's insights that were indeed "remarkably prescient." Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt On Sat, Jul 25, 2020 at 2:42 PM Jeffrey Brian Downard < [email protected]> wrote: > Jon Schmidt, List, > > Congratulations on the publication of the paper. Here are some initial > questions. > > > 1. In the opening pages of his discussion of continuity in the last > Cambridge Conferences lecture published in *Reasoning and the Logic of > Things*, Peirce distinguishes between the task of arriving at a > conception of continuity adequate for mathematical inquiry and arriving > at a conception adequate for philosophical inquiry. As he points out, once > we have arrived at the former, that is where the real difficulties begin. > Having read through your article, most of what you say pertains to the > mathematical conception of continuity. Are you focusing on a definition > adequate for a mathematical conception of continuity? Or, do you take > yourself to have offered a definition of continuity that is adequate for > philosophical inquiry in phenomenology, the normative sciences and > metaphysics--as > well as for mathematics? > 2. At the end, you compare the components of your definition of > continuity to the list of components provided by Fernando Zalamea. Much to > my surprise, you brush aside the differences by saying Zalamea uses terms > not found in Peirce's writings and that many of the conceptions he employs > are only familiar to working mathematicians. If the aim is to provide an > account of continuity that is adequate for inquiry in mathematics, then it > would have made sense to compare the competing accounts and to engage > in a discussion with what Zalamea says. > 3. On its face, I find your suggestion that Zalamea uses terms not > found in Peirce's texts to be puzzling. Let me focus on two components of > his account that are not found in your list: genericity and modality. Is > there some reason that you've set genericity and modality to the side > in your account? > 4. In your discussion, you refer to Peirce's writings on topology, and > you focus on the language used in the Articles set forth in the chapter on > topics in the *New Elements of Mathematics*. One of the very first > points that Peirce highlights in italics is the importance of > *generation*--e.g., the process of *generating* something such as a > continuous line. That is the conception Zalamea is focusing on under the > heading genericity. > 5. One of the later manuscripts you consider is on the EGs. As the > distinction between the beta and gamma graphs highlights, modality is > central in his own work in mathematical logic. > 6. Let me take a step back from the details and focus on a larger > question. What is needed for an adequate analysis of the concept of > continuity? One way of coming at the matter is to provide a definition that > will, as they say, stand the test of time. Another way of coming at the > matter is to provide an account that is adequate for the time being. My > sense is that you are looking for the former. In my view, Peirce is > offering an account that is more limited: i.e., it is meant to be adequate > for mathematical inquiry and practice up to his time. > 7. The future of mathematics is hard to see--even for the best of > mathematicians. As such, he is surveying the most important ideas that > have surfaced in mathematical inquiry ranging from the classical writings > of Euclid right up to the 19th-century writings of Gauss and Riemann, and > he is trying to identify the most important components in the current > conception of continuity as it has arisen in the research of mathematicians > working in areas including--but not limited to--topology, calculus and > mathematical logic. > 8. In the last lecture of RLT, Peirce provides a quick historical > survey of modern work in topology (e.g., Möbius), projective geometry > (e.g., Desargues, Poncelet), metrical geometries (e.g., Riemann and > Lobochevsky) and group theory (e.g., Cayley and Klein). It will be > difficult to see whether or not one definition of continuity or another > (e.g., yours or Zalamea's) is adequate for these different areas without > digging into paradigmatic examples of mathematical reasoning--as Peirce > does time and again. > 9. As far as I can see, Zalamea is following Peirce's lead. That is, > he is drawing on Peirce for the sake of developing an account of continuity > that is adequate for the additional work in mathematics that has been done > in the 20th century--and he is pointing out that Peirce was remarkably > prescient in his philosophical analysis of the mathematical conception--as > recent work in category and topos theory shows. > 10. In the concluding remarks, you claim that the thick account you > provide is more faithful to the common notion of continuity. Our > common-sense notion is something Peirce turns to as a starting point for > philosophical inquiry. As Peirce says in the passage you cite, Kant's basic > idea is a pretty good starting point as far as our common notion goes. As > such, it looks to me like you may be moving the target. Instead of starting > and ending with a discussion of what is needed for an adequate definition > of continuity for the sake of mathematics, you are claiming that Zalamea's > account misses the mark and yours is better as an explication of what is > involved in our common conception. > > There is more to be said, but I'll stop here for the sake of giving you a > chance to reply to these initial questions and comments about your article. > > Yours, > > Jeff > Jeffrey Downard > Associate Professor > Department of Philosophy > Northern Arizona University > (o) 928 523-8354 >
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