FOM: Re: ultrafinitism again and true =? provable

V. Sazonov V.Sazonov at
Sun Nov 12 10:53:17 EST 2000

Robert Black wrote:

> Professor Sazonov seems happy with the appellation, 

I should present my position myself. 

Probably there might be some repetitions with what have been 
said by some participants, including myself, almost from the 
very beginning of FOM list, but misinterpretation of anyone's 
opinion proves to be so usual thing in FOM, that I am forced 
to do that. 

If I am 'ultrafinitist' then it is only in the sense that this 
(potential mathematical) subject is interesting to me (as some 
other more traditional mathematical subjects). Some prefer 
Algebra, others Analysis. Only in this sense. It is one of 
subjects of my interests, not a kind of "quasi-scientific 
religion" or the like. I never could say that ZFC is false 
[(unless a contradiction will be found; even in that case it 
is sufficiently meaningful for me) or reject it as somebody 
in FOM assigned to me this position - I never gave any occasion], 
because allegedly I think that 'ultrafinitism' is true and the 
only true doctrine. Ultrafinitism, how I understand it, is 
even not a philosophy. For me it is a mathematical idea (like 
continuity) which deserves formalization and mathematical 
investigation. Of course each new idea needs in a philosophical 

My general position in Philosophy of Mathematics (without any 
relation to ultrafinitism) is based mainly on the strong distinction 
between really existing material objects of the real world and 
whichever ideals or illusions we could have. By the way, the real 
world could be infinite. (Moreover, who knows what does it mean 
precisely? It is a mixture of physical and mathematical considerations 
and concepts and facts.) In any case, there is no reason to think that 
``all'' the imaginary natural numbers of PA may be denoted by numerals 
SS...S0 which we write physically (and which correspond intuitively 
to feasible numbers). This is ABSOLUTELY TRIVIAL NOTE, but some 
participants of FOM sometimes assert something looking as opposite.  
I consider that it is a rough philosophical mistake to identify 
any theoretical (mathematical) world with the reality. 

As to ultrafinitism, I do not think that there is some general 
agreement on the meaning of this word. For me it consists in 
(i) an intuition on feasible numbers (cf. above), (ii) attempts 
to formalize this intuition as rigorously as mathematics requires 
and, therefore, (iii) to include this concept into mathematics as 
a reasonable and useful concept both from theoretical and 
practical point of view (as any mathematical applications). 

That is all. 

The fact that our real world is probably bounded (but, as I very 
informally think, nevertheless, infinite) is just an additional 
argument in favor of considering ultrafinitism, no more. 

Even if there are infinitely many of material objects (whatever this 
would mean), the physical meaning of the word "infinitely" may be 
such that even Peano Arithmetic may be not realizable in the real 
world (full induction axiom schema could fail). However, strictly 
speaking, what does it mean to interpret PA in the real world, 
really? I am afraid that this is meaningless. We could say only 
on SOME relations between theory and reality, 

> (Contrast here the
> nominalism of Hartry Field, for example, for whom although mathematical
> objects are fictions, mathematical theories can be applied to the real
> world in virtue of isomorphisms between the fictional realm and the real
> realm.)

He really says that? Is it serious? Say, abstract real numbers 
or set-theoretic cardinals can be isomorphically embedded in 
the real world? I am asking even independently on the assumptions 
on (in)finity of the universe. 

> The other problem which I raised is the possibility that accepted
> mathematical proofs might fail to be formalizable in ZFC in the sense that
> the full formalization would be too long to fit into the universe. The
> obvious reply to this, that one introduces definitions, doesn't seem to me
> to work. The reason is that standardly the distinction between a definition
> and an axiom is that the former is conservative. But once we have a finite
> bound on the lengths of proofs, 'definitions' will no longer be
> conservative, and must therefore be regarded as axioms. And what then
> justifies them as axioms, once the usual considerations which justified
> them as definitions are no longer available?

Of course, if we decide to be completely rigorous in mathematics 
(let us put aside for a while whether it is necessary or reasonable) 
and write proofs as formally as programmers write programs for 
computers, then we hardly could work in pure ZFC without explicitly 
formulated abbreviation mechanism, be them given as new axioms or 
in any other technical way. We can also abbreviate proofs (not only 
formulas) having a recurrent character. The resulting version ZFC' 
of ZFC (or of any other formal system T) depends on which concrete 
abbreviation mechanisms we will choose. Say, we could allow
of formulas, but not terms. Therefore abbreviations could play even a 
role as new NONLOGICAL or SPECIAL principles (like induction axiom in 
PA is considered as nonlogical one, specific to the given subject 
matter). It is possible, IN PRINCIPLE, (for some unusual T) that some 
version T' could be even contradictory while T will be consistent. 
I mean here that some abbreviations could allow to REALLY infer a 
contradiction which in original T would be so long that nobody 
would be able to write it physically. This possibility can be 
considered as a pathology of T, but it might be also a source 
of introducing some new formalisms allowing to formalize what was 
otherwise non-formalizable. 

Finally, I would like to reply to Prof. Martin Davis

> Of course for someone who imagines that using the word "myth" will demolish
> ordinary mathematical practice, none of this counts.

I think this discussion was on a principal questions of philosophy 
of mathematics and by no means demolished ordinary mathematical 
practice. Say, it was about informal interpretation of Goedel's 
theorem, but not about rejection of it in ANY rigorous mathematical 
sense. There was a lot of misunderstanding and therefore repetitions 
and other, sometimes unpleasant moments during the discussion, but 
the question proves to be too difficult or the approaches considered 
too unusual (is it bad?). 

Let me recall that some recent posting of Prof. Friedman was also 
devoted to mathematical truth in a way seemingly related (if not 
equivalent?) with one of the positions presented in this discussion:

     From: Harvey Friedman <friedman at> 
     Date: Mon, 4 Sep 2000 20:31:03 -0400 

But the justification for the general math community is not along any
of truth. It is rather along the lines of coherent pragmatism. Because
possibility of experimental confirmation is nonexistent - or at least
inconceivable at this point - truth plays no significant role in the
equation. Only coherent usefulness.

A simple and precise metamathematical result was also presented 
in one of my posting in favor of 

	true => provable 

and some related informal comments even in favor of 

	true = feasible provable. 

related with the main topic of discussion. 

Vladimir Sazonov

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