[FOM] Query for Martin Davis. was:truth and consistency

Vladimir Sazonov V.Sazonov at csc.liv.ac.uk
Mon Jun 2 17:04:41 EDT 2003

Bill Taylor wrote:
> This query is addressed to Martin Davis, re the excerpts below, but OC,
> anyone else who cares to, please join in.
> > If the underlying system is PA (first order number theory), I believe that
> > one can claim that the evidence for its consistency is simply overwhelming.
> > Even the trivial consistency proof based on the standard model uses far
> > less than much well-accepted ordinary mathematics.
> Exactly.  This is what I've been pointing out for some time in sci.logic.
> Essentially that the standard model N is so crystal-clear-cut and plain,
> that attempts to "justify" it using arguments based on the theory PA
> and its proof-theoretic properties are just trying to justify the simple
> by taking the complex on trust.

I will tell nothing about consistency of PA. I am much more 
concerned about the statement "the standard model N is so 
crystal-clear-cut" and that PA has only a minor role in 
clarifying the idea of natural numbers. 

Long time ago I stated in FOM the question what does it mean "the 
standard model of PA". I do not remember that there was any 
convincing (let me say, for me) answer. 

I ABSOLUTELY do not understand what does it mean and by which 
way it is possible to assert what was cited above. For me, just 
vice versa, N is rather vague concept. Only by fixing things 
in the form of a formalism like PA (or some sufficiently formal 
simple rules of reasoning on natural numbers like those which we 
studied at school) we are coming to a sufficiently solid 
ground which is a basis for our illusions that we have some 
"crystal-clear" N. 

I also have no idea why should we consider our illusions, like 
mentioned above, more seriously than they deserve. 

Let us consider, for example, natural numbers which are 
provably (say, in PA) the halting moments of various 
Turing machines starting with the empty tape (for explicitly 
presented Turing machines). Let us call these numbers provably 
computable (or provably existing). It seems evident that 
the stronger is theory extending PA the more we will have 
provably existing numbers. (At least, for some weak theories 
we can demonstrate that even 2^1000 is not provably existing.) 
As I understand, the work of Harvey Friedman is related in some 
way with these considerations. This is an (informal) evidence 
that the "length" of N and therefore N itself is not fixed. 

The whole philosophy is very simple, and, I believe, the most 
appropriate and honest for mathematics: take any interesting 
vague idea, formalize it, if possible at all (making it thereby 
somewhat different from the original one), get some illusion of 
the resulting "crystal-clear" concept and, finally, use (consciously 
or not) the formalism together with our (already new) intuition. 

Also, I would never identify our illusions with the reality. 
However, they can be related - then this is an applied mathematics. 

Vladimir Sazonov

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