[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 mydissertation many year ago (1966) on CSP's conception of representation):"Peirce indicates in several places that he regardsthe 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
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
Thomas, list, Peirce's version of the proof for Cantor's theorem can be mapped in a quite straightforward way to the structure of the New List of 1867. At the same time the proof of Cantor's theorem can be extended by continued diagonalization (which latter, by the way, Peirce discovered not later than 1867 and under a different name and in a much more general form than afterwards discovered and used by Georg Cantor, Kurt Goedel and Alan Turing) to a derivation of the system of Existential Graphs, which can thus be seen, as Peirce himself said, to be expressive of the properties of the continuum and fulfills the criteria Peirce gave for true continuity, namely Kanticity and Aristotelicity. I could probably show in strict terms what the above means, but this does not seem to me to make any sense in an email forum, since it involves a lot of logic and mathematics and is by no means impossible, but difficult to express in words. Anyway, I've written it down and so maybe one day... . One of the main difficulties is perhaps generally, that it is impossible to understand Peirce from a set theoretical point of view (even if this be only used as a language and however implicitly) and it seems to me equally and definitely impossible to understand Peirce's continuum in terms of any form of nonstandard analysis. This sounds perhaps complicated, but it is in fact simple and only difficult to understand, as it seems. Anyway, this is the end of the road for me, since I surprisingly found what I have been looking for over long years and Peirce, according to my understanding, is not so much about a body of knowledge, but what he found out is meant to be used and that's the only meaning it has. So I leave it at this point and shall now do something completely different. I hope that you do pass your notes to another mathematician rather than just letting the issue vanish! If true, your ideas could be incredibly valuable. Let me finish with two concluding remarks: What regards a fourth category, this means for me to simply go into the wrong direction. A reduction to two categories might be progress, but Kant already tried that, as is well known, and he failed. Its meaning for you of simply to go in the wrong direction is even more simply an unconfirmed interpretant, and in a sense makes my point for me. For my part, I will trust to truth, and not my preconceived notions of good dependent on hidden presumption of what is true, as to what will constitute progress, since evaluations of what is good or bad among ideas are pending what is true or false apart from what you or I think of them. Certainly there are four-folds for which it would create rather than remove complications to reduce to three, such as the Square of Opposition, various related logical structures, the structure of source-encoding-decoding-recipient, the light cone's four zones of causal determination, and sets of relations many-to-many, one-to-many, many-to-one, one-to-one. Certainly, saying that such--such would be good or bad not only fails to say anything about whether it would be true, but it also makes a rhetorical presumption that it would be false. Presumably one means that it would be the wrong direction not in spite of its being true but rather on account of its being false. One's meaning does not, however, prove anything at all. And, to be sure, if reality is what it is apart from what you and I think of it, then it would be presuming a great deal, to say that four-folds, if true, would be the wrong direction. Secondly, Douglas Adams once described how flying works: You throw yourself at the ground, and miss it completely. This seems to me to apply beautifully to induction in particular and signs in general, too;-) Bye, Thomas. P.S. I might be completely wrong of course. In that case, never mind! Best of luck, Ben --- Message from peirce-l forum to subscriber archive@mail-archive.com
[peirce-l] Re: on continuity and amazing mazes
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. On 3/13/06, Benjamin Udell [EMAIL PROTECTED] wrote: Thomas, list, Peirce's version of the proof for Cantor's theorem can be mapped in a quite straightforward way to the structure of the New List of 1867. At the same time the proof of Cantor's theorem can be extended by continued diagonalization (which latter, by the way, Peirce discovered not later than 1867 and under a different name and in a much more general form than afterwards discovered and used by Georg Cantor, Kurt Goedel and Alan Turing) to a derivation of the system of Existential Graphs, which can thus be seen, as Peirce himself said, to be expressive of the properties of the continuum and fulfills the criteria Peirce gave for true continuity, namely Kanticity and Aristotelicity. I could probably show in strict terms what the above means, but this does not seem to me to make any sense in an email forum, since it involves a lot of logic and mathematics and is by no means impossible, but difficult to express in words. Anyway, I've written it down and so maybe one day... . One of the main difficulties is perhaps generally, that it is impossible to understand Peircefrom a set theoretical point of view (even if this be only used as a language and however implicitly) and it seems to me equally and definitely impossible to understand Peirce's continuum in terms of any form of nonstandard analysis. This sounds perhaps complicated, but it is in fact simple and only difficult to understand, as it seems. Anyway, this is the end of the road for me, since I surprisingly found what I have been looking for over long years and Peirce, according to my understanding, is not so much about a body of knowledge, but what he found out is meant to be used and that's the only meaning it has. So I leave it at this point and shall now do something completely different. I hope that you do pass your notes to another mathematician rather than just letting the issue vanish! If true, your ideas could be incredibly valuable. Let me finish with two concluding remarks: What regards a fourth category, this means for me to simply go into the wrong direction. A reduction to two categories might be progress, but Kant already tried that, as is well known, and he failed. Its meaning for you of simply to go in the wrong direction is even more simply an unconfirmed interpretant, and in a sense makes my point for me.For my part, I will trust to truth, and not my preconceived notions of good dependent on hidden presumption of what is true, as to what will constitute progress, since evaluations of what is good or bad among ideas are pending what is true or false apart from what you or I think of them. Certainly there are four-folds for which it would create rather than remove complications to reduce to three, such as the Square of Opposition, various related logical structures, the structure of source-encoding-decoding-recipient, the light cone's four zones of causal determination, and sets of relations many-to-many, one-to-many, many-to-one, one-to-one.Certainly, saying that such--such would be good or bad not only fails to say anything about whether it would be true, but it also makes a rhetorical presumption that it would be false. Presumably one means that it would be the wrong direction not in spite of its being true but rather on account of its being false. One's meaning does not, however, prove anything at all. And, to be sure, if reality is what it is apart from what you and I think of it, then it would be presuming a great deal, to say that four-folds, if true, would be the wrong direction. Secondly, Douglas Adams once described how flying works: You throw yourself at the ground, and miss it completely. This seems to me to apply beautifully to induction in particular and signs in general, too;-) Bye, Thomas. P.S. I might be completely wrong of course.In that case, never mind!Best of luck, Ben---Message from peirce-l forum to subscriber [EMAIL PROTECTED] --- Message from peirce-l forum to subscriber archive@mail-archive.com
