FOM: Re: consistent/contradiction-free?

[Robert Tragesser]

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