Jon, John, List, Thanks very much to both of you for this interesting exchange (the entire thread to date) which clarified a great deal which had been murky for me regarding Peirce's evolving views of continuity. There's, no doubt, much more to be said on the topic, perhaps even in this thread. Yet, having read analyses of Peirce on continuity for years, if not decades, and often coming away shaking my head, your exchange went very far in helping me, a non-mathematician, "get a handle" on the topic. For example, I now see much more clearly the distinction being drawn between top-down vs bottom-up conceptions of continuity
JFS: None of the examples cited depend on the supermultitudinous claim. CP 5.119 and CP 5.132 can be interpreted with just a finite number of parts. Whether the others are finite or infinite is not obvious, but there is no need to consider anything that the topological methods can't handle -- no need for anything supermultitudinous. JAS: I agree; again, I quoted and commented on those passages because I think that they helpfully *illustrate the difference* between the two conceptions of continuity--bottom-up vs. top-down, a (supermultitudinous) collection where the parts are real and the whole is an *ens rationis* vs. a (topological) continuum where the whole is real and the parts are *entia rationis*. I might add, somewhat parenthetically, that this thread represents in my view an example of the best--i.e., most productive--kind of exchange that can occur on peirce-l. Best, Gary *Gary Richmond* *Philosophy and Critical Thinking* *Communication Studies* *LaGuardia College of the City University of New York* On Tue, Aug 6, 2019 at 12:56 PM Jon Alan Schmidt <[email protected]> wrote: > John, List: > > JH: Peirce's topology considers as similar two objects that can be > continuously deformed into one another, provided that during such a > deformation "no parts are separated which were at first continuously > connected" (MS 1170, article "topical"). (p. 13) > > > JFS: I believe that this definition is adequate for Peirce's uses of the > term 'continuous'. > > > It is a definition of topological similarity, not continuity. Havenel > explains Peirce's "topological" conception of continuity later in the same > paper. > > JH: Peirce distinguishes a perfect continuum and an imperfect continuum, > this last being a continuum "having topical singularities" (CP 4.642). He > maintains that according to his concept of continuity, "a topical > singularity ... is a breach of continuity". This idea is likely to be an > evolution from his previous idea that a continuum cannot contain actual > points since in a continuum every part consists of parts, but an actual > point in a continuum is a part "which does not consist of parts" (NEM > 3.748, 1900). But if there is in a continuum no topical singularity, then > it is a perfect continuum for which hold [*sic*] the law that "all the > parts of a perfect continuum have the same dimensionality as the whole" (CP > 4.642) ... > The new aspect in this definition of a perfect continuum is "that every > sufficiently small part has the same mode of immediate connection with > others as every other has"; and this idea is related to Peirce's work on > topology. (pp. 27-28) > > > CP 4.642 is dated 1908 May 26, and Havenel (2008 > <https://www.jstor.org/stable/40321237>, pp. 117-125) seems to think that > it initiated Peirce's "Topological Period," but he was already moving in > this direction several years earlier. > > CSP: Kant's real definition implies that a continuous line contains no > points ... Hence a point or indivisible place really does not exist unless > there actually be something there to mark it, which, if there is, > interrupts the continuity ... On the whole, therefore, I think we must say > that continuity is the relation of the parts of an unbroken space or time. > The precise definition is still in doubt; but Kant's definition, that a > continuum is that of which every part has itself parts of the same kind, > seems to be correct. This must not be confounded (as Kant himself > confounded it) with infinite divisibility, but implies that a line, for > example, contains no points until the continuity is broken by marking the > points. In accordance with this it seems necessary to say that a continuum, > where it is continuous and unbroken, contains no definite parts; that its > parts are created in the act of defining them and the precise definition of > them breaks the continuity. (CP 6.168; c. 1903-1904) > > > Moreover, Havenel seems unaware of Peirce's explicit definition of a > continuum that I quoted in the very first post of this thread. > > 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) > > > The parts of a continuum "have the same dimensionality as the whole" and > are *indefinite*, unless and until they are "marked off." They then > become definite "topical singularities," which is why they can be "of > lesser dimensionality down to absolute simplicity"; i.e., points. This > procedure is an "operation of thought," by which we can distinguish "parts > of any multitude we please"; as Peirce wrote elsewhere, "Whatever is > continuous has *material parts* ... Almost everything which has material > parts has different sets of such parts, often various *ad libitum*" (CP > 6.174; 1908). Hence the Cantorian pseudo-continuum and even the > "supermultitudinous" conception are among the infinitely many options for > "bottom-up" *analysis *of a continuum into parts, none of which capture > the "top-down" nature of *true *continuity in accordance with the > "topological" conception. > > JFS: I agree with Moore that Peirce never stated a definition of the > term 'supermultitudinous' that is mathematically adequate (i.e., stated > with sufficient detail and precision that another mathematician could use > the definition in a formal proof). > > > Thank you for clarifying this. What about Peirce's "topological" > conception of continuity? Is it amenable to a "mathematically adequate" > definition? Havenel seems to think so. > > JH: This means that Peirce's conception of the continuum is incompatible > with modern point-set topology which defines the continuum in a Cantorian > spirit ... But there is another way to do mathematical analysis. Within the > mathematical theory of category, in *Smooth Infinitesimal Analysis* (SIA), > the elements of a continuum are not all distinguishable. > > Moreover, there is a third way between point-set topology and algebraic > topology that could have been used by Peirce to formalize his > conceptualization of continuity. This third way is topology without points > ... > > It is a topological theory in which points as ultimate parts do not exist. > (p. 27) > > > Moore (2007 <https://www.jstor.org/stable/40321199>, p. 468 n. 45) agrees > with Herron (1997 <https://www.jstor.org/stable/40320632>, pp. 621-623) > and Havenel (2008 <https://www.jstor.org/stable/40321237>, p. 124) that > SIA is a better candidate for a rigorous mathematical analysis of Peirce's > *true > *continuum than Robinson's Non-Standard Analysis. > > JFS: None of the examples cited depend on the supermultitudinous claim. > CP 5.119 and CP 5.132 can be interpreted with just a finite number of > parts. Whether the others are finite or infinite is not obvious, but there > is no need to consider anything that the topological methods can't handle > -- no need for anything supermultitudinous. > > > I agree; again, I quoted and commented on those passages because I think > that they helpfully *illustrate the difference* between the two > conceptions of continuity--bottom-up vs. top-down, a (supermultitudinous) > collection where the parts are real and the whole is an *ens rationis* > vs. a (topological) continuum where the whole is real and the parts are *entia > rationis*. > > Regards, > > Jon Alan Schmidt - Olathe, Kansas, USA > Professional Engineer, Amateur Philosopher, Lutheran Layman > www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt > > On Mon, Aug 5, 2019 at 10:00 PM John F Sowa <[email protected]> wrote: > >> On 8/5/2019 6:22 PM, Jon Alan Schmidt wrote: >> > JFS: Peirce developed outstanding theories of logic and semeiotic >> > with just a topological version of continuity. There is no evidence >> > that anything more would be significantly better. I agree with MEM. >> > >> > JAS: I concur with the first two sentences, but I am puzzled >> > by the third... The thesis of Moore's paper is that Peirce's >> > "supermultitudinous" conception of continuity was inadequate to >> > do the specific work for which he tried to deploy it throughout >> > the Harvard Lectures in 1903, and therefore inessential to his >> > late philosophical system as a whole. >> >> For the topological version, I'll quote Havelnel's summary in >> >> https://www.academia.edu/25930950/Peirces_Topological_Concepts_Jerome_Havenel >> : >> > Peirce's topology considers as similar two objects that can be >> > continuously deformed into one another, provided that during >> > such a deformation "no parts are separated which were at first >> > continuously connected" (MS 1170, article "topical") >> >> I believe that this definition is adequate for Peirce's uses >> of the term 'continuous'. I agree with Moore that Peirce never >> stated a definition of the term 'supermultitudinous' that is >> mathematically adequate (i.e., stated with sufficient detail and >> precision that another mathematician could use the definition in >> a formal proof). >> >> JAS >> > With that in mind, I would like to call attention to a passage >> > from one of the Harvard Lectures that Moore did not specifically >> > address, because I believe that it can helpfully illustrate the >> > difference between the two conceptions of continuity. >> >> None of the examples cited depend on the supermultitudinous claim. >> CP 5.119 and CP 5.132 can be interpreted with just a finite number >> of parts. Whether the others are finite or infinite is not obvious, >> but there is no need to consider anything that the topological >> methods can't handle -- no need for anything supermultitudinous. >> >> John >> >
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