FOM: Jurassic pebbles (more on Davis/Hersh)

[Vladimir Sazonov]

Martin Davis wrote replying to Reuben Hersh:

> I claim that the fact about pebbles mentioned above was just as true of
> pebbles on a jurassic beach as on a contemporary beach.
>
> *Do you agree?
>
> Since I can hardly imagine that you would dispute this, I assume your answer
> is "yes". Then I ask, "How do you know?" For me, the answer is clear: it is
> a consequence of Lagrange's theorem which was as true then as it is now.

Dear Professor Davis,

Your position is not sufficiently clear for me because you seems do not 
present sufficiently full picture. Does the following, which I consider
as 
a possible explication of the above citation, as I see it, correlate in 
some way with your point of view?

I would rather say that it was always true that Lagrange's theorem is 
PROVABLE in a formal system.  It is even easier than the experimental 
truth of its pebble version. It is crucial problem how do we CHOOSE 
axioms (and proof rules) for arithmetic to deduce this and other 
theorems having consequences in the real world. To which degree this 
process is determinate? Are there alternatives? Is the situation in 
arithmetic completely different from that in geometry (Fifth
Postulate)?  
On the one hand, we do have alternatives. We can freely play with axioms 
and proof rules to get some versions of arithmetic which have true 
consequences in the real world and reflect this world under another 
angle than Peano Arithmetic.  On the other hand there may be (and 
actually are) some objections against this freedom, that only one way to 
choose axioms is "true" (or fundamental, or the like).  The only coming 
to my mind reason for the full determinateness of this process is the 
God who knows (or shows us via some prophets?) the correct way so that 
we are forced to follow it without understanding how and why. (I do not 
know how this corresponds to you point of view. I just try to 
extrapolate as I can.)

But I am very unsatisfied with this irrational explanation because this 
explains almost nothing.  This seems to be the best way to find 
ourselves in the prison of our own illusions (about the absolute truth, 
about the standard model for PA, and the like). 


Vladimir Sazonov
--
Program Systems Institute,  	| Tel. +7-08535-98945 (Inst.),
Russian Acad. of Sci.		| Fax. +7-08535-20566
Pereslavl-Zalessky,		| e-mail: sazonov at logic.botik.ru
152140, RUSSIA			| http://www.botik.ru/~logic/SAZONOV/