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
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