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.
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