FOM: correction Re: consistent/contradiction-free?
Robert Tragesser
RTragesser at compuserve.com
Thu Feb 18 20:53:06 EST 1999
--------------- Forwarded Message ---------------
From: Robert Tragesser, 110440.3621
To: FOM, INTERNET:fom at math.psu.edu
Date: Thu, Feb 18, 1999, 08:18 PM
RE: Re: consistent/contradiction-free?
Correction applies to (A) below -- "have a model" read 'have no model" and
dropping (a'), (b').
In reponse to Steve Simpson, a preface and then some questions.
I'm wanting to propose some FOM/f.o.m. research programs which it
is odd have not been undertaken (or they have and I missed out on the
literature or didn't know how to read the literature rightly.)
My interest is in the interface, or extended zones between,
informal mathematics (or "mathematics"), whether in mind or text, and its
(rigorously) formal-logical counterparts.
In Saunders MacLane's book (figuratively and literally) and essays
related to the AMS-Bulletin value-of-rigorous-proof-controversy, he seems
to think that informal mathematics is something preliminary to, but as yet
not fully achieved, mathematics. Fine; I'm not interested in splitting
hairs, drawing lines in the dust, or whatever. I';m satisfied is that
there is a before or a not-quite-having-made-it-to mathematics in his
sense, and it's the transition or prior phase I'm interested in
understanding.
In this spirit, I am interested in the family of notions of
"consistency" before the "identification" of inconsistency with not
harboring a formal-logical contradiction. If this "identification" did not
leave something behind, no one would have made it;--of course what is left
behind might be garbage well lost (but what is someone's garbage may be
someone else's healthy lunch -- what is the physicist's lunch seems often
enough to be the mathematician's garbage; together keeping the
intellectual ecosystem working, however much they hiss and sneer at one
another).
Some questions:
[1] What can be said in general for logics in which Go"del's completeness
theorem fails, about theories in those logics which
(A) have no model, but
(B) an explicit logical contradiction is not derivable?
Bill Tait and Eliot Mendelsohn provided contrived theories (as in
the appendix the the new edition of Mendelsohn); one wonders if there are
"real" theories. The last seems to me an important Friedman type question.
In general, I can't find anything in the literature that treats
these matters systematically; nor can I find anything like a recent
discussion of the significant ways of choosing formal expressions for
Con(T) and there virtues and failings.
Should we nevertheless say of such theories [I mean: (A) and (B)]
that they are _consistent_? More exactly, what logical value can be
conferred on "having a model" in these circumstances. (If one insists on
an identification between consistency and contradiction-free, then I think
something subtle must be said about "has a model"/"has no model" beyond
what is conveyed by "ocontradiction-free, at least in the case (A)(B) ?)
[2] It does seem that w..r.t. Bolyai-Lob. geometry, "conmsistency" meant
finding an interpretation in significant/current/substantial mathematics,
which integrated B-L geometry into mathematics in a significant/fruitful
way, such as the upper-half plane model vividly does. So here consistency
comes to more than having a model (such as a nonconstructive countable
model guaranteed by the S-L theorem). Is this now an irrelevant sense of
"consistency" -- or does the activity "integrating significanlty into
accepted mathematics" have no interest now? [[N.B., the point of this is
to try to suggest that "consistency" could have a richer sense than the
standard identification might suggest, and that it is worth looking into
this).
MY final query might be better off placed in another posting, though it
continues the central theme,
[3]The logical function of conservation principles in physics.
Reverse mathematics gives us much surprising information about _logical
equivalences_ and or especially _logical equivelences mod..._
But it does seem that we do not understand very well in f.o.m. what
is lost through these equivalences (when one substitutes one for the
other). . .crudely called the inensional content
I am especially interested in this content since it seems to be
exploited deeply in solving problems in mathematics and all but
ubiquitously in physics. Crudely(again): the key to solving many problems
is "equating" (or otherwise taking together) two or more different ways of
characterizing the same thing. This indeed seems to be the principal
logical/mathematical value of conservation laws in physics -- taking
advantage of the "identity" of the conserved in order to calculate a
quantity.
So: Is there work in fom -- could there be work in fom?????!!! --
which is directly relevant, concretely valuable, for understanding and
articulating how expressions which a logically equivalent mod something or
other are fruitfully different (from the point of view of enabling us to
understand things or solve problems not otherwise really solvable by us
with some understanding intact?)
There is a sense in which constructive mathematics can be placed
here, but constructive mathematics fails, I think, in its obsession with
computation/calculations, and it has no interest in the logical structure
of the intensional content [_not_ to be confused with intensionality as its
manifests itself in constructive systems). I don't think that constructive
mathematics could lend deep appreciation to the ways in which Riemann's
physical/geometric treament (definitely not Weierstrassian) of the
Dirichlet problem or Steiner's analogus treatment of the isoperimetric
problem are legitimate. [It is very worthwhile to read the Feyman Lectures
from the point of view of his habit of tryiny always to pair a more or less
purely mathematical treatment of a problem with a physical (or "more
elementary") treatment.
robert tragesser
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