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 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
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