FOM: Re Pratt and Butz on truth and models

[Shipman, Joe x2845]

 Tait: 
> >he is rejecting a claim that there is _no_ absolute truth in math and 
> >surely his argument is right. The axioms of group theory are true of all 
> >groups, the axioms of real closed fields are true of the ordered field of 
> >real numbers, etc. Any such example suffices for Mayberry's point.
Pratt: 
> Unfortunately *every* example passes this test.  Your position if I
> understand it is that axioms are true in those worlds that satisfy
> those axioms.  With that notion of truth every sentence S would be an
> absolute truth, because S would be true in every model of S.

Mayberry is correct because the point is not simply that any model of 
the axioms of real closed fields satisfies the axioms, it is that 
"the" real numbers, given independently of the axioms of real-closed 
fields, satisfy these axioms and therefore the theory of real-closed 
fields (so e.g. any polynomial factors into terms of degree at most 
2).  Going to a simpler type, the point is not simply that any model 
of Peano's axioms satisfies Euclid's theorem on the infinitude of 
primes, it is that "the" integers satisfy Peano's axioms and 
therefore there ARE infinitely many primes.  If you admit that the
statement "There are infinitely many primes" is *meaningful*, you are
making a commitment to the existence of "the" integers (or at least 
to the proposition that even if there is no distinguished model of 
your favorite set of axioms for integers, nonetheless all models 
deserving to be called "integers" will have the same truth value for 
the statement "There are infinitely many primes").

 It is possible to go down down even further in type and say, for example, 
"there is a simple group of order 95,040".  This is absolutely true, no?
If you're not familiar with the Mathieu group M12 then replace 95,040 
with 60 and consider the group A5 of even permutations of 5 objects.

If you are not disagreeing with Mayberry's assertion contra Butz that 
there is indeed  absolute mathematical truth to be had, but instead are 
interpreting him to say that "all proved theorems are *true*" and 
disagreeing with that, I won't quibble because the situation is clear 
to everybody and Butz is refuted.  Though Butz might want to chime in 
and say his question "how can anybody claim there is absolute truth out 
there?" was referring to statements of a higher type where there is 
no preferred model (he did after all allow that 1+1 not = 3 was a 
truth). 

Butz's invocations of physics are irrelevant in this context. 
Newtonian mechanics is not "true" about the physical universe we live 
in but that has nothing to do with the truth of the axioms of set 
theory with respect to the integers.  If some statements in the 
language of physics have no truth value according to quantum 
mechanics that can indeed be interpreted as saying that the universe 
is insufficiently determinate, and by analogy one can say that "the 
universe of sets" is insufficiently determined for some statements to 
have truth values, but we are asserting just that a significant 
initial segment of that universe is determinate and that *some* 
theorems therefore merit the unqualified adjective "true".  The 
analogy with relativity is also wrong because new mathematical 
methods have not superseded old ones in the same way that relativity 
superseded Newtonian mechanics--one is a supplementation, the other 
is a substitution.  Newtonian mechanists were wrong about the solar 
system and admitted it when Einstein explained the anomaly in the 
perihelion of Mercury; no amount of new mathematics will force a 
revision in the truth value we ascribe to Euclid's theorem.

Joe Shipman