FOM: Feferman's 10 theses

[Vladimir Sazonov]

I am Vladimir Yu. Sazonov, the head of Computer Logic Lab. in 
Program Systems Institute of Russian Acad. of Sci. 
(Pereslavl-Zalessky, to the north from Moscow).  
My present interests concern mainly with Logical Foundations of 
Computability with Bounded Resources (descriptive complexity and 
finite model theory, bounded arithmetic and arithmetic for 
feasible numbers, bounded set theory and BST as an approach 
to database theory). 
For more information: http://www.botik.ru/~logic/SAZONOV/


Jon Barwise wrote:
> 
> It is wonderful to see Sol's clear list of theses nailed to the door for
> all to see.  There is much I agree with in the list.  Not everything,
> though.  I would like to take exception to (parts of) Sol's 1, 5, and 6
> (which I repeat here with emphasis):
> 
> Sol's 6.  "While REASONING IS OUR ONLY KNOWN PATH TO SECURE MATHEMATICAL TRUTH,
> there are objective questions of truth and falsity."
> 
> Re 6: I don't believe the first part:  "While reasoning is our only known
> path to secure mathematical truth,..."  Why?  Well, think about how we
> first come to know basic mathematical facts, like
>         5 x 5 = [whatever it is].
> It is not through pure reasoning, at least not for my children, but through
> examples in the world.  Psychologists have shown that between the ages of 2
> 1/2 and 4 1/2 most children develop the ability to abstract from concrete
> situations to abstract numbers and so learn abstract facts from instances
> of them.  My son, for example, went to a Montessori school where they had
> these little blue blocks and by stacking counting he learned that five
> groups of five gave you a total of 25.  May be it is baby mathematics, but
> it is where it all gets started.

I think that taking "five groups of five" is just a formal rule of
manipulating 
with quasi-syntactic objects (blue(?) blocks). This also may be
considered as
a definition of multiplication on the concrete example or as a rule of 
calculation or even as a rule of a formal system supporting a kind of
REASONING (proving that 5 x 5 =25). Actually, I also do not completely 
agree with the *pure* term REASONING. Mathematical reasoning (in a 
final ideal form, as the result of an "intimate" process) is essentially 
FORMAL, according to some rules (like above or like chess), in contrast 
to other kinds of (non-mathematical) reasonong. 

Also, I rather agree with the first part of "Sol's 6" than with the
second. I will present my point of view on this subject in a separate 
posting. 


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/