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 ________________________________ From: Jon Alan Schmidt <jonalanschm...@gmail.com> Sent: Saturday, July 25, 2020 11:15:20 AM To: peirce-l@list.iupui.edu Subject: [PEIRCE-L] Peirce's Topical Continuum List: I adapted some of my posts from last summer in threads like "Is Synechism Necessary?" (here<https://list.iupui.edu/sympa/arc/peirce-l/2019-08/msg00029.html>), "Vargas on Continuity" (here<https://list.iupui.edu/sympa/arc/peirce-l/2019-09/msg00025.html>), and "Defining Continuity" (here<https://list.iupui.edu/sympa/arc/peirce-l/2019-09/msg00041.html>) to write a paper, "Peirce's Topical Continuum: A 'Thicker' Theory." I submitted it to Transactions of the Charles S. Peirce Society, and it is included in the current issue of that journal (https://www.jstor.org/stable/10.2979/trancharpeirsoc.56.1.04). The following is the abstract. Although Peirce frequently insisted that continuity was a core component of his philosophical thought, his conception of it evolved considerably during his lifetime, culminating in a theory grounded primarily in topical geometry. Two manuscripts, one of which has never before been published, reveal that his formulation of this approach was both earlier and more thorough than most scholars seem to have realized. Combining these and other relevant texts with the better-known passages highlights a key ontological distinction: a collection is bottom-up, such that the parts are real and the whole is an ens rationis, while a continuum is top-down, such that the whole is real and the parts are entia rationis. Accordingly, five properties are jointly necessary and sufficient for Peirce's topical continuum: rationality, divisibility, homogeneity, contiguity, and inexhaustibility. The article thus attempts to meet Matthew E. Moore's challenge<https://link.springer.com/article/10.1007/s11229-013-0337-6> to provide "a well-defined conception [of continuity], fully equipped to do all the philosophical work Peirce thought that it could do." Here are my one-sentence summaries of the five properties, employing Peirce's terminology of "portion" for a part and "limit" for a connection between parts (R 144, c. 1900). * Rationality - every portion conforms to one general law or Idea, which is the final cause by which the ontologically prior whole calls out its parts. * Divisibility - every portion is an indefinite material part, unless and until it is deliberately marked off with a limit to become a distinct actual part. * Homogeneity - every portion has the same dimensionality as the whole, while every limit between portions is a topical singularity of lower dimensionality. * Contiguity - every portion has a limit in common with each adjacent portion, and thus the same mode of immediate connection with others as every other has. * Inexhaustibility - limits of any multitude, or even exceeding all multitude, may always be marked off to create additional actual parts within any previously uninterrupted portion. I welcome feedback on this hypothetical/mathematical analysis of continuity, as well as suggestions of its potential implications for other fields of study. For example, over the last several months I have been exploring its applications in phenomenology/phaneroscopy, logic/semeiotic, and metaphysics, especially as they contribute to a comprehensive Peircean philosophy of time. Contrary to Moore's negative answer to his own question<http://revistas.pucsp.br/cognitiofilosofia/article/download/16603/12457>, my hope is that this formulation (or something like it) might facilitate demonstrating that synechism is as central to Peirce's entire system of thought as he consistently maintained. Thanks, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt<http://www.LinkedIn.com/in/JonAlanSchmidt> - twitter.com/JonAlanSchmidt<http://twitter.com/JonAlanSchmidt>
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