FOM: Reply to Sazonov, Kanovei, and Black: Truth and Ultrafinitism

[JoeShipman@aol.com]

I think Professor Black represented Professor Sazonov's position very well 
(of course Sazonov is the proper judge of this).  Like Black, I don't accept 

"1.  The only things which really exist are concrete material objects, and 
there are (or for all we can know may be) only finitely many of them.", 

but I agree that given this assumption Sazonov's position is quite natural.  
My reasons for not accepting this assumption are both philosophical (which I 
won't go into here) and scientific (namely, so far we have only been able to 
make sense of the world by using theories positing infinities of many kinds 
and working finitist versions of these theories have not been found).

The other point I am trying to make to Sazonov and Kanovei is that the 
correspondence between mathematics and physics, which we can accept as an 
empirically strongly supported phenomenon, allows us to INTERPRET the results 
of physical experiments (such as running on a particular well-understood 
physical machine a computer program X on input Y and seeing its output Z) as 
evidence for statements about mathematics (in the example given, the evidence 
is for the strictly mathematical statement "on input Y, algorithm X halts 
with output Z").  

This evidence is not the same as mathematical proof, though it is closely 
related in the algorithmic case, and in my posts to Kanovei I called it 
"scientific proof".  The new twist I added is that physical theories might, 
by virtue of their infinitary ontology, allow us to obtain experimental 
evidence for mathematical statements of a higher logical type than "on input 
Y, algorithm X halts with output Z".  This evidence, unlike evidence from 
experiments consisting of the running of computer programs, would NOT entail 
the existence of a classical mathemtical proof.

Although I understand Sazonov's position, I am still having some difficulty 
with Kanovei's.  Insofar as he wants to systematically replace talk about 
models with talk about proofs and avoid making ontological commitments, so 
that "truth" entails provability, I do not see a problem, but I don't 
understand then how he can accept theorems proved from ZFC with an essential 
use of infinite sets as "true" or "proved" with the same status as 
ontologically tame theorems of PA.  Does he think the Axiom of Infinity has a 
meaning?  If not, why is it acceptable in proofs?

In other words, I think the idea that "there is no proper meaning of 'truth' 
for arithmetical sentences apart from provability" is defensible and can be 
reconciled with acceptance and use of PA in proofs, but is inconsistent with 
the use of proof systems like ZFC which contain an axiom of infinity.

When Kanovei says

Take any arithmetical sentence, say A, known to be 
unprovable in ZFC. Applying "the standard definition", we shall be left with 
another formula, say Sat_N(A), which, in correct manner, describes that A "is 
true" in N. Both A and Sat_N(A) are ZFC-formulas (A even a Peano formula), 
whose equivalence is easily provable in ZFC, hence, the new formula is as 
well dead unprovable as the original A.  Hard remainder: instead of one 
unprovable formula we have now two of them, mathematically equivalent and 
both unprovable, as clueless regarding the truth of A as earlier. 

he is equivocating.  Something cannot "known to be unprovable in ZFC" unless 
you go beyond ZFC, i.e. by assuming Con(ZFC).  While I agree that 
reformulating an arithmetical sentence A by the ZFC-sentence SAT_N(A) doesn't 
make proving A any easier, I claim that someone who accepts ZFC ought to 
regard SAT_N(A) as *meaningful*, and to see that SAT_N(A) applies to A for 
which |-_ZFC(A) (that is, "ZFC proves A") does not. 

-- Joe Shipman