ONT Re: Logic As Semiotic -- Still Quasi After All These Years
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John Collier wrote (JC):
Jon Awbrey wrote (JA):
JC: Current logic has proven that neither notion can be formalized in
the syntax sense, which implies that neither can be formalized
in the formal semantics sense (which many of my benighted
colleagues take to be the only sense of semantics).
JA: I do not think that the young Peirce, or even very many of his mid-century
contemporaries, would have been terribly surprised to hear about our late
blooming discovery that our fantasy ache for syntactic sole sufficiency
has turned out to be a hysterical pragnanz -- from the classical POV
of Pythagoras and Plato, this is simply understood from the get-go.
As to the fact that science outpaces ab-ante-cedental inference,
that most notorious empiric/physic Aristotle already knew that.
I have never understood how we post*moderns manage to acquire
so much credit for remembering what we had, just a moment ago,
forgot.
JC: I don't think it was forgotten so much as doubted.
Successes, for example in the formalization of first
the transcendental numbers and then the real numbers gave
good reason to doubt earlier worries, such as Zeno's paradox.
The real advance was a technological proof that technique alone
was not sufficient for mathematics.
Sigh ...
No one, not even the Greeks, doubted the possibility of motion.
Do the experiment, gedanken or otherwise, and Achilles surpasses
the tortoise nine times out of ten -- Our Hero was, after all, prone
to a touch of foot trouble from time to time -- and so everybody always
understood that the sense of the puzzle was to account for the actuality
and thus the possibility of the familiar phenomenon of motion -- where the
assignment "to account for" means to come up with a mathematical image that
is adequate to the thing. A mathematical image befits the thing in question
by capturing a relevant aspect of its form. Thus be it ever a "formal" model.
It is only on the scarce occasions of the Pythagorean Sabbath that I contemplate
a faith in the formula "matter = form", and so my platonically real thinking tells
me that the Reality surpasses a "fallible and mortal finite information creature"
(FAMFIC) like me more times than I can count out of more times than I can reckon.
I am puzzled by the persistence of this image of form.
All of the mathematical circles that I used to run in
have long ago forgotten about the peculiar notions of
that tiny parochial school of self-styled "formalists"
and have returned to the form of platonic realism that
was always their native code, no matter what a few of
them say to student reporters. This was the attitude
of Peirce, the attitude of Gödel, and calling the bug
that Gödel found in Russell's gear a techne-logical
advance is like calling denatured alcohol a medical
miracle. For the mathematics that I know, "form",
"pattern", "structure", as revealed by invariants
and morphisms, is what it's all about. Period.
So I continue to wonder at this passing strange turn of affairs,
and I feel at times a little bit like Zorro, being called on to
defend the good name of the Latin Beauty "Forma" from the evil
designs of the Viceroy's bumbling besmirchers.
JA: Still, by posing things in all-or-none terms,
saying "X cannot be formalized" when we need
to say "X cannot be exhaustively formalized",
we characteristically miss the whole point
of the exercise, partially to formalize X,
and deny ourselves the practical benefits
of doing just that. I see a certain type
of psychodynamic here that needs to be
interpreted in a therapeutic remedium.
JC: Granted, but there is a tendency to resort to technique
when it is available to the ignorance of everything else.
This has led many contemporary philosophers of logic, some
of them friends of mine, to argue that truth and validity are
restricted to what we can constructively define through one or
another specific technique. I believe that, much as formalization
is useful, it must systematically push out issues of truth and validity
for any logic as strong as 1st order. Truth functional logic and modal logic,
on the other hand, are complete. As I have said before on this list, that is one
of the reasons I favour an information based logic. It gives us both sides of the
coin within the logic itself, as Greg Chaitin argues admirably, in my opinion, but not
in the opinion of many logicians who do not see intuitively how distinctions are relevant
to logic. I do think that they need therapy, and Wittgenstein is as good as Peirce for this,
I think.
Logical systems can be "complete" in their own terms,
but sign systems are never complete when you wake up
and recall their due function in describing a world,
a fact that Peirce's sign relations keep constantly
before our minds, not to mention under our gnosis.
The whole point of my current slow skippy reading
of Peirce's 1865 Lectures is to examine the way
that he develops a theory of information out of
logic as a theory of inquiry by exploiting the
very forms of continuity between the two areas.
Taking Wittgenstein as a cure for Russell
is like morphine as a remedy for laudanum.
Just my opuniom ...
Jon Awbrey
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