Jon, List,
In the opening remarks of the last lecture in RLT, Peirce frames three questions. Let me restate them in my own words. 1. What conception of continuity is needed for mathematics? 2. What conception of continuity is needed for a philosophical theory of critical logic and the larger theory of semiotics? 3. What conception of continuity is needed for metaphysics? We could add, what conception of continuity is needed for the special sciences, including the physical sciences as well as the human sciences? Focusing on the first question, I don't think the ongoing disputes about what conception of continuity is needed, for instance, to ground the hypotheses that lie at the basis of the calculus are "much ado about nothing". In the late 19th and early 20th century, many mathematicians and philosophers tried to defend a general approach that grounds the calculus on set-theoretical notions concerning the limit. Peirce, on the other hand, argued that the conception of the infinitesimal is more fundamental and less problematic--logically speaking. Those questions have continued to call out for more inquiry in the latter part of the 20th century and the first part of the 21st. I was in the audience when Matthew Moore delivered that presentation in Bogota. He seemed to be suggesting that set-theoretical notions involving certain types of infinity will suffice for all of mathematics. Given the fact that much inquiry in topology from the time of Poincaré to the present has moved in the direction of treating continuous paths and surfaces as point-sets, there are strong arguments that can be marshaled in support of this general approach to analyzing the mathematical conception of continuity as being sufficient for topology as well as for the calculus. I don't buy those arguments, but they are probably the majority opinion in most mathematics departments in the U.S. --Jeff Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354 ________________________________ From: Jon Alan Schmidt <[email protected]> Sent: Saturday, August 3, 2019 3:55 PM To: [email protected] Subject: [PEIRCE-L] Is Synechism Necessary? (was Lecture by Terrence Deacon) List: The new subject line is the title of the paper by Matthew E. Moore that I quoted below. It turns out that there are two versions available online--the original, delivered as a conference presentation in 2012 (https://www.pucsp.br/pragmatismo/dowloads/lectures_papers/mattew-moore-paper.pdf), and a somewhat longer text that appeared in a 2013 issue of Cognitio (http://revistas.pucsp.br/cognitiofilosofia/article/download/16603/12457). Moore gives a negative answer to his own question, provocatively suggesting that continuity can be eliminated as a core component of Peirce's late philosophical system. However, he focuses specifically on the "supermultitudinous" conception of continuity, rooted in Peirce's mathematical theory of collections, as deployed in the 1903 Harvard Lectures. My response is that this is effectively much ado about nothing--as I already noted in a recent post<https://list.iupui.edu/sympa/arc/peirce-l/2019-07/msg00072.html>, Peirce himself abandoned (or at least significantly modified) the "supermultitudinous" conception around 1906, adopting instead the "topological" conception as dubbed by Jerome Havenel in his 2008 paper, "Peirce's Clarifications of Continuity" (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.473.9336&rep=rep1&type=pdf). Again, the difference between the two is what I have in mind when contrasting a "bottom-up" approach with a "top-down" one--for any discrete collection, the parts are real and the whole is an ens rationis; but for any true continuum, the whole is real and the parts are entia rationis. The latter obviously cannot be derived from the former; a continuum is not properly characterized as an aggregate of parts, even if those parts are considered potential rather than actual. CSP: I conceive that a Continuum has, IN ITSELF, no definite parts, although to endow it with definite parts of no matter what multitude, and even parts of lesser dimensionality down to absolute simplicity, it is only necessary that these should be marked off, and although even the operation of thought suffices to impart an approach to definiteness of parts of any multitude we please. *This indubitably proves that the possession of parts by a continuum is not a real character of it. For the real is that whose being one way or another does not depend upon how individual persons may imagine it to be. It shows, too, that Continuity is of a Rational nature. But it conveys no gleam of evidence that Continuity itself is Unreal ... (S30 [Copy T:6-7]; c. 1906) Although Moore briefly acknowledges this later "topological" conception of continuity, he makes no attempt to assess whether it might have been able to do the work in 1903 for which he finds the "supermultitudinous" conception inadequate. My sense is that it would have gotten the job done, and that it is as indispensable for properly understanding and applying Peirce's overall thought as Peirce himself clearly considered it to be. CSP: After that, readers might be prepared to take an interest in a proof that the doctrine [of pragmaticism] is true,--a proof which seems to the writer to leave no reasonable doubt on the subject, and to be the one contribution of value that he has to make to philosophy. For it would essentially involve the establishment of the truth of synechism. (CP 5.415, EP 2:335; 1905) CSP: We here reach a point at which novel considerations about the constitution of knowledge and therefore of the constitution of nature burst in upon the mind with cataclysmal multitude and resistlessness. It is that synthesis of tychism and of pragmatism for which I long ago proposed the name, Synechism, to which one thus returns; but this time with stronger reasons than ever before. (CP 4.584; 1906) Regards, Jon S. On Fri, Aug 2, 2019 at 8:33 AM Jon Alan Schmidt <[email protected]<mailto:[email protected]>> wrote: Gary R., List: Your comments about Deacon reflect remarks that I came across recently in a 2012 paper<https://www.pucsp.br/pragmatismo/dowloads/lectures_papers/mattew-moore-paper.pdf> by Matthew E. Moore--"We who are already convinced of Peirce’s greatness tend to forget how impenetrable he can look to the rest of the human race. The conceptual overhead of Peirce’s philosophy is almost prohibitively high ..." I suspect that this contributes to at least some of the recurring disputes here on the List--each of us has absorbed different amounts and different portions of that vast "conceptual overhead"--and I have found it to be a formidable obstacle to overcome whenever I want to explain Peirce to a non-Peircean. The "curse of knowledge<https://en.wikipedia.org/wiki/Curse_of_knowledge>" comes into play; I tend to forget that others are utterly unfamiliar with certain ideas and principles that I have come to take for granted. Moore went on to suggest that "we will need at least something answering to his categories and his theory of signs if we are to get much philosophical mileage out of Peirce," and the purpose of that particular paper is to question whether "the continuum" also needs to be included, as Peirce seemed to believe. What else (if anything) qualifies as bare minimum for understanding and appreciating Peirce's thought overall? What resources (if any) have you and others found especially helpful for enabling the uninitiated to get up to speed? 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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