[peirce-l] evolving universe
This is probably old news to most of you but since i just found it, i
thought i'd share it --
Reading some musings by physicist Lee Smolin on Edge.org, i thought "Hey,
sounds like Peirce." And sure enough, toward the bottom he quotes Peirce.
http://www.edge.org/q2006/q06_4.html#smolin
gary
}If we can see (as we once saw very well) that our conversation with the
planet is reciprocal and mutually creative, then we cannot help but walk
carefully in that field of meaning. [David Suzuki]{
gnusystems }{ Pam Jackson & Gary Fuhrman }{ Manitoulin University
}{ [EMAIL PROTECTED] }{ http://users.vianet.ca/gnox/ }{
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[peirce-l] Re: on continuity and amazing mazes
Thomas:Your thoughts on the potential relation between Peirce's continuity and mathematical history were fascinating. I must confess that I am a bit of a skeptic when it comes to the possibility of a sensible relation between logic, any logic, and a philosophy of mathematics.Nonetheless, I remain puzzled by the concept of the "form" of logic, .Should logic be grounded in the logos? That is, in the sentences of the language?What is it that would trigger the jump to forms? Roughly speaking, the abstract conceptualization of mental motion from sentences to geometry?I note in passing that Waismann's concept of number as the root of mathematics avoids this particular issue as the concept of "number" already exists in the natural language and does not acquire a sense of geometry in ordinary usage, in ordinary day to day communication.CheersJerryOn Mar 15, 2006, at 1:08 AM, Peirce Discussion Forum digest wrote:Subject: Re: on continuity and amazing mazes From: "Thomas Riese" <[EMAIL PROTECTED]> Date: Tue, 14 Mar 2006 13:39:29 +0100 X-Message-Number: 2 On Mon, 13 Mar 2006 19:37:14 +0100, Marc Lombardo <[EMAIL PROTECTED]> wrote: Thomas, If you don't mind my asking, what's wrong with the "nonstandard analysis" approach to illustrating continuum, so long as that approach is VERY nonstandard? I was quite convinced by Hilary Putnam's introduction to "Reasoning and the Logic of Things." Putnam suggests that rather than understanding infinitesimals as deriving from major points, instead we understand all points as themselves infinitesimals and all infinitesimals as points, such that any infinitesimal point names another infinity of infinitesimals. It's difficult to express things in a few useful words, Marc, but I'll try. I know what Hilary Putnam writes. I believe that he extremely underestimated what a black belt master logician like Peirce can do with these seemingly simplistic, "childish" syllogistic forms. And it is very important to understand thst Peirce's logic is primarily focused on "forms". Another master in this way of thinking was the mathematician Leonhard Euler and in fact Peirce perhaps received his idea for the "cut" from Euler (in his Letters to a German Princess). John Venn later "amended" this form, but he misunderstood it completely. Euler wasn't childish. Neither was Peirce. Euler could work miracles in analysis, but he had no explicit logical theory. He simply knew what he did. Later then others came, working more or less by rule of thumb and that often landed them in the ditch. They simply did not know what they were doing. So there was a crisis in mathematics. To save mathematical logic there had to come Cauchy and Weiertrass, Dedekind and Cantor etc. Secure foundations were needed. But that also closed the door to a lot of possibilities. Peirce found the logic behind what Euler has been doing, I believe. But now we have "Bourbakism" in mathematics, i.e. set theory as a language, which is by no means "neutral". Just an example: in mathematics, if you have discovered an "isomorphism" you have made a discovery, you have "reduced" things and then you are finished with these things. They are just simply "the same thing". The equivalence relation is so to speak the primary mode of _expression_. Peirce is exactly interested in the relation between isomorphous forms. His primary relation is the general form of transitivity. The difference has far reaching, profound implications. So in nonstandard anylysis as soon as you base things on "point sets", however generally understood, you have already missed the point (no pun intended) of Peirce's continuity. Peirce can represent it in that form (and then mathematical points split etc), but I don't believe it's possible the other way round. But what I here say, can be only very loose talk indeed of course. Just to give you a vague idea what I mean. Cheers, Thomas. Jerry LR ChandlerResearch ProfessorKrasnow Institute for Advanced StudyGeorge Mason University --- Message from peirce-l forum to subscriber archive@mail-archive.com
[peirce-l] Re: on continuity and amazing mazes
Arnold says: I would venture to suggest (subject to the better sense of those on the list who have greater experince with the MSS than I have) that the notion of a Sign contains the concept of a transitive function, making a very strong case for what Thomas has said on this subject. Other transitive functions in Peirce can be found in Vols III and IV of the CP (see especially 3.562)RE RESPONSE: You won't get any objections from me on that, Arnold. Let me quote myself (from my dissertation many year ago (1966) on CSP's conception of representation): "Peirce indicates in several places that he regards the nota notae as the generic inference principle (see esp. 5.320 and 3.183). [Nota notae est nota rei ipsius: the mark of the mark is the mark of the thing itself.] Further, he identifies this with the dictum de omni (4.77) [which is in Aristotle], and with what De Morgan called the principle of the transitiveness of the copula. (2.591-92). The latter is in turn identified with the illative relation (3.175), and this, again, is explicitly said to be the "primary and paramount semiotic relation." (2.444n1). I suggest, therefore, that all of Peirce's statements of the representation relation may thus be taken as so many variant expressions of what he understands to be expressed by the nota notae, the dictum de omni, the notion of the transitivity of the copula, or the principle of illation." (Charles Peirce: The Idea of Representation, 63) Joe Ransdell No virus found in this outgoing message. Checked by AVG Free Edition. Version: 7.1.375 / Virus Database: 268.2.1/278 - Release Date: 3/9/2006 --- Message from peirce-l forum to subscriber archive@mail-archive.com
[peirce-l] Re: on continuity and amazing mazes
Dear Arnold, I believe L 224, the letter Peirce wrote to William James on 1909 Feb 26 is exceedingly important here. In print you find it in volume III/2, p.836 ff. of "The New Elements of Mathematics", ed.: Carolyn Eisele. What is important is the fact that general transitivity has a property that would in Boolean algebra amount to the "Idempotency Law", i.e. AA=A. You can see this clearly on page 838. It would be very tedious to try to cite things here in an email text, so I can only indicate what is meant. Since the idempotency expresses identity, we can see here quite clearly why Peirce in connection with the Existential Graphs spoke of "ter-identity" (as in "line of teridentity"). But this very insight in fact goes back to the 1867 paper "On the Natural Classification of Arguments" (see CP 2.461 ff) where Peirce writes: "There is, however, an intention in which these substitutions are inferential. For, although the passage from holding for true a fact expressed in the form "No A is B", to holding its converse, is not an inference, because, these facts being identical, the relation between them is not a fact; yet the passage from one of these forms taken merely as having "some" meaning, but not this or that meaning, to another, since these forms are not identical and their logical relation is a fact, is an inference. The distinction may be expressed by saying that they are not inferences, but substitutions having the "form" of inferences."(CP 2.496) And this then lead to the "On a New List of Categories" in 1867. I think we should address this as "diagonalization", since in L 224, after exploring what is meant by "Logical Analysis of a Concept and a Real Definition of it" (p.844) and addressing Johann Benedict Listing's "Barycentral Calculus" [Sorry, it is impossible to go through all this mathematics; but these things, as e.g. the Barycentral Calculus is not so much important here in itself, anyway] Peirce writes (on page 854) concerning the "theory that concepts are combined in a form substantially like that of the combination of points in the barycentric calculus": "I may remark, that it is in barycentral composition that forces are combined according to the principle of the parallelogram of forces". I remind you, what Peirce here still is talking about is general transitivity and what he found in 1867! (you'll see that later in the text of L 224/see below) Next (poor William James:-)) he gives his proof of what today is known in mathematics as the Peirce Theorem. It says that "Every linear associative algebra is isomorphic to a matrix algebra." (compare e.g. Birkhoff/ MacLane, Modern Algebra, p.398; by the way interesting to compare this to the theorem that "Any abstract group is isomorphic with a group of transformations" on page 139 of the same book;-)). In the proof Peirce gives of his theorem, it is not the isomorphism that is interesting. By way of proof analysis it is interesting to note how Peirce arranges these (A:A), (A:B) etc forms, which are part of sums, in a square and again compare this to Cantor's ¨¨Diagonalverfahren¨, secondly it is interesting to note the distinction Peirce makes on page 858 between ���substitution��� and ¨¨replacement¨ and it is interesting to note that the last formula on page 858 somehow reminds one strongly of the form of the general transitive relation back at the beginning of L 224 and in CP 3.523f. (here then the formula on top of the page 332 in the CP.) Sorry, if Peirce could do this with poor William James, then I here simply do the same with poor you :-) I could even "top" it by saying that what is important here, as in the Bernoulli series, is not the concrete system of numbers or abstract algebraic "units", but the relation beween multiplication and addition or "distributivity" and "collectivity" (just to coin two new terms to be thrown away immediately after use!). By the way, Jacob Bernoulli didn't derive his "Bernoulli numbers" algebraically. That probably would have been utterly hopeless. He SAW A PATTERN in the series he was exploring. With his very eyes. That's historical fact and not a myth, as it seems. Interesting curiosity, isn't it? What I want to say: the conception of probability and its logical foundation is not very far here. Sorry again, I don't say this here to appear "intelligent" or something, or to impress anybody with my "superior personality". I simply don't know a better way to say it and perhaps it is better so to say it, instead that it is lost for further use. In other words: this might be a good field for a research project, Arnold;-) Again, analysing Peirce's proof, keep categorial structure in mind! Cheers, Thomas. On Wed, 15 Mar 2006 08:32:12 +0100, Arnold Shepperson <[EMAIL PROTECTED]> wrote: Thomas TR: Thomas Riese AS: Arnold Shepperson TR: Peirce is exactly interested in the rel
