FOM: Re Pratt and Butz on truth and models
Shipman, Joe x2845
shipman at bny18.bloomberg.com
Thu Feb 5 08:14:42 EST 1998
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
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