FOM: Re: consistent/contradiction-free?
Robert Tragesser
RTragesser at compuserve.com
Thu Feb 18 20:20:57 EST 1999
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 Goedel's
completeness theorem fails, about theories in those logic which
(A) have model, but
(B) an explicit logical contradiction is not derivable?
[[and What about:
(a') does not have a model
(b') a logical contradiction is not derivable. Are there examples? --
I'm not sure if this is an exact enough question.)
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" beyond
what is conveyed by "ocontrdiction-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 querty 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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