John, List: 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.
I concur with the first two sentences, but I am puzzled by the third; would you mind clarifying what you mean? 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. As I noted before, Moore made no attempt to assess whether Peirce's later "topological" conception would have gotten the job done, nor whether it is likewise dispensable for understanding his thought as a whole. Again, my own answers to those questions are "yes" and "no," respectively. 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. CSP: But let us compare it [the Universe] rather with a painting,--with an impressionist seashore piece,--then every Quality in a premiss is one of the elementary colored particles of the painting; they are all meant to go together to make up the intended Quality that belongs to the whole as whole. That total effect is beyond our ken; but we can appreciate in some measure the resultant Quality of parts of the whole,--which Qualities result from the combinations of elementary Qualities that belong to the premisses. (CP 5.119, EP 2:194; 1903) The total effect of the "impressionist seashore piece" is the *aggregate *of the individual effects of the discrete "colored particles" that comprise it. At and beyond a certain distance, those particles are sufficiently *indefinite *that our minds can "fill in the gaps" between them; likewise the pixels of a digital photograph. Under the "supermultitudinous" conception (what I call "bottom-up"), the painting itself would be a continuum if the number of such particles were to exceed all multitude, such that they were *really *welded together. However, under the "topological" conception (what I call "top-down"), the painting is always and only a *representation *of a hypothetical instantaneous state of a continuum--the real seashore--similar to how the Cantorian pseudo-continuum of real numbers serves as a mathematical *model *of continuity. The "colored particles" are then artificial creations *for that purpose*, like positions for describing motion or propositions for describing thought. Consider, then, Peirce's subsequent sentence. CSP: But I shall endeavor to make this clearer in the next lecture. (*ibid* ) The next lecture was on "The Three Normative Sciences" and includes the following relevant statement. CSP: In the light of the doctrine of categories I should say that an object, to be esthetically good, must have a multitude of parts so related to one another as to impart a positive simple immediate quality to their totality; and whatever does this is, in so far, esthetically good, no matter what the particular quality of the total may be. (CP 5.132, EP 2:201; 1903) Again, this reflects the "supermultitudinous" conception of continuity, since the whole has "a positive simple immediate quality" that is *derived *from its parts and their relations to each other. The "topological" conception would instead hold that "the particular quality of the total" is real, while the parts with their relations to each other and their associated qualities are artificial creations. Recall how Peirce's Categories apply to a true continuum--3ns as its own *fundamental *nature, 1ns as the indefiniteness of its *potential *parts, and 2ns as the discontinuity of any *actual *parts that are "marked off." I suggest accordingly that esthetic goodness is an *ideal *quality of a whole, such that any arbitrary part thereof is esthetically good to the extent that it *embodies *that quality. This seems consistent with what Peirce said in his first Lowell Lecture on "What Makes a Reasoning Sound?" later the same year. CSP: The very being of the General, of Reason, *consists *in its governing individual events. So, then, the essence of Reason is such that its being never can have been completely perfected. It always must be in a state of incipiency, of growth ... So, then, the development of Reason requires as a part of it the occurrence of more individual events than ever can occur. It requires, too, all the coloring of all qualities of feeling, including pleasure in its proper place among the rest. This development of Reason consists, you will observe, in embodiment, that is, in manifestation. The creation of the universe, which did not take place during a certain busy week, in the year 4004 B.C., but is going on today and never will be done, is this very development of Reason. I do not see how one can have a more satisfying ideal of the admirable than the development of Reason so understood. The one thing whose admirableness is not due to an ulterior Reason is Reason itself comprehended in all its fullness, so far as we can comprehend it. Under this conception, the ideal of conduct will be to execute our little function in the operation of the creation by giving a hand toward rendering the world more reasonable whenever, as the slang is, it is "up to us" to do so. (CP 1.615, EP 2:255; 1903) The ultimate example of *esthetic *goodness--that which is admirable in itself, the *summum bonum*--is the development of Reason, which consists in embodiment or manifestation. Accordingly, our *ethical *imperative is to seek and seize opportunities to make each *part *of the Universe--including ourselves--more consistent with that ideal quality of the *whole*. Returning to the Harvard Lecture on "The Three Normative Sciences," this is "the only possible ultimate aim" satisfying the requirement "that the esthetic quality toward which the agent's free development tends and that of the ultimate action of experience upon him are parts of one esthetic total" (CP 5.136, EP 2:203; 1903). Regards, Jon Alan Schmidt - Olathe, Kansas, USA Professional Engineer, Amateur Philosopher, Lutheran Layman www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt On Sun, Aug 4, 2019 at 12:36 PM John F Sowa <[email protected]> wrote: > Jon and Jeff, > > This is an important thread that clarifies many of the debates. > > JAS > > The new subject line is the title of the paper by Matthew E. Moore... > > Thanks for those references. The concluding paragraph at the end > of MEM's 2012 paper summarizes the issues: > > MEM > > We wind up with an attitude that should have had a lot going for > > it in advance of any detailed examination of the vicissitudes of > > the Peircean continuum; for it is the attitude we should bring to > > the study of any great figure from the past. Peirce has left us, > > not any kind of final word, but a work in progress, one eminently > > worth carrying on, in the spirit of the one who started it. Which > > is to say that we must as resolutely critical, and as ruthless in > > paring away what does not work, as Peirce was at his best. > > Peirce himself said that his oeuvre was a work in progress that he > himself could not complete. He also insisted that every statement > in science is fallible, and he would certainly include his own. > > As Moore said, Peirce left us "a work in progress, one eminently > worth carrying on, in [his] spirit." Any attempt to "harmonize" > his work would turn that living, growing spirit into a dead ghost. > > JBD > > figuring out what conception of continuity is adequate for pure > > mathematics and, in turn in applied mathematics is essential. > > No. There are infinitely many theories of pure mathematics. > Gödel's undecidability theorem makes it doubtful that a single > foundation is possible, desirable, or useful. > > In the early 20th c, Whitehead & Russell adopted set theory as > the foundation for mathematics, but Leśniewski developed mereology > as an alternative. For geometry, Whitehead and Tarski independently > developed versions of mereology as the foundation. But after Gödel > proved his undecidability theorem, the goal of a single universal > foundation for all of mathematics became doubtful. > > As for applied mathematicians (which include scientists and engineers), > they find the debates about foundations irrelevant for solving any > practical problems. Newton, Leibniz, Einstein, Bohr, and Dirac are > physicists who invented novel mathematical techniques long before > the mathematicians discovered proofs to justify the computations. > > JBD > > We seek a mathematically adequate conception of continuity for > > the sake of developing a sufficiently rich conception of continuity > > for inquiry in logic and semiotics. > > 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. > > JBD > > For the sake of applying mathematics to positive questions in > > the sciences, a proper understanding of different kinds of > > mathematical models and forms of measurement are important for > > our inquiries in philosophy and the special sciences. > > I agree. But with the qualification that "a proper understanding > of different kinds of mathematical models" does not privilege any > one as proper and make all the others improper. > > As Peirce said, all mathematics is based on diagrammatic reasoning. > Even by Cantor's standard, a countable infinity is Aleph 0, the > infinity of real numbers is Aleph 1, and the infinity of possible > diagrams is Aleph 2. > > You don't need a supermultitudinous infinity to realize that > Aleph 2 is unimaginably huge. To claim that any finitely > specified theory could serve as a foundation for all those > diagrams is almost certainly false. > > JBD > > 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. > > Actually, Abraham Robinson (whom MEM mentioned) showed that the > theory of infinitesimals (as Newton, Leibniz, and Peirce used it) > is mathematically sound. > > But Robinson's proof does not require a supermultitude. Aleph 1 > (the infinity of real numbers) is sufficient. > > John >
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