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

Vladimir Sazonov V.Sazonov at csc.liv.ac.uk
Wed Jun 4 17:23:27 EDT 2003

Bill Taylor wrote:
> Vladimir Sazonov <V.Sazonov at csc.liv.ac.uk> writes:
> -> I am much more
> ->concerned about the statement "the standard model N is so crystal-clear-cut"
> Hi Vladimir; I have seen your articles here from time to time, expressing your
> formalist views, or, as many may prefer to say, your ultrafinitist views.

"...ists" (ituitionists, ultrafinitists, etc.) usually do not accept 
other approaches and corresponding formalisms. This is not about me. 
I accept any mathematical (or pre-mathematical) ideas and their 
formalizations. However, I am interested in some subjects more 
than in others. The idea behind ultrafinitism is interesting for 
me and seems promising. 

> As you are here now, please allow me to quiz you on one or two matters.
> Just one for now.
> Do you admit the reality of 10^10^10^10 in the same sense (whatever that is)
> that you admit the reality of 3 and 7 ?   Would you put these three numbers
> on the same footing, ontologically. 

In a sense yeas, in a sense not. 3 and 7 have some real counterparts 
(3 apples, etc.), unlike 10^10^10^10. On the other hand, 
like any mathematician, I am able to "put these three numbers
on the same footing", say, via PA. Also I am able to separate them 
by a weak arithmetical theory. 

> ->and that PA has only a minor role in clarifying the idea of natural numbers.
> No, I would say it has a very major role, but perhaps not so much in
> clarifying them, as in describing them.

I could add - to strengthen or "mechanize" our thought in reasoning 
about numbers. Without formalisms (like school algorithms for 
multiplication of decimal numbers) we would have the idea of natural 
numbers something like "one, two, many". 

> ->Long time ago I stated in FOM the question what does it mean "the
> ->standard model of PA".
> I recall the occasion.  My feeling was that most people took the view,
> "if you have to ask the question then you wouldn't understand the answer".
> I often wonder how much people like you and Kanovei and Edward Nelson and
> Yessenin-Volpin 

I am not sure that all mentioned persons have the same views. 
As to understanding, I am not sure that I was understood well. 

*really* don't understand N, and how much it is a posture
> of principle.  I'm sure there WAS a time, back in high school, say, that
> you understood it perfectly well.  It would be interesting to know what
> it was that "disinformed" you.

I would start with the beginning school, or even earlier, 
for children  from about 4-8 years. The teacher is for them 
the highest authority who actually "disinformed" them (and 
of course, gave some concrete knowledge about some algorithms 
and (semi-)formal way of reasoning). 

On the other hand, what is this "to understood it perfectly well"? 
To be able to manipulate with arithmetical expressions and be able 
to do some reasoning about numbers? Or to have some quasi-religious 
beliefs concerning the standard model N? Anyway, such beliefs 
are absolutely unnecessary to work successfully with natural 
numbers (obeying, say, axioms of PA). The normal intuition, 
imagination (recall imaginary geometry of Lobatshevsky, or 
imaginary unit - the square root of -1) or illusions 
(not beliefs) are quite sufficient for that. 

What (as you say) "disinformed" me? Some deeper that in the school 
things like Goedel's theorems, especially on incompleteness and  
Goedel/Cohen proof on independence of CH, what demonstrated (to me)  
that both N and continuum are vague concepts, Parikh's formalization 
of the (vague) idea of feasible numbers, however with 2^1000 still 
(provably) feasible, my own attempts to obtain an independence 
result of a statement close to P=NP (in which some unusual concept 
of constructive/non-constructive finite(!) binary strings was used) 
and, later, to obtain some formalization of feasible numbers with 
2^1000 not feasible. 

Also, I could mention some Russian papers of Esenin-Volpin
(which, strictly speaking, I do not consider as rigorously 
mathematical), some papers of Mycielski (quite rigorous). 

> -> For me, just vice versa, N is rather vague concept.
> HMMMmmmmmm...

Yes, yes! What are we ever talking about when discuss about 
this N? I know - this is some illusion. This is, I believe, 
a more honest position than to assert that there exists 
some idealized unique N. By which way unique? Illusion or 
idealization are something about which uniqueness makes 
no real sense. This is rather from religion-like views 
on which science should not rely. 

> ->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
> I would agree if you had said real numbers, or sets, or geometry,
> or group theory, or most other mathy things.  But for the naturals?   
> Fie.

I feel the difference, but in a sense they are equally ullusive, 
i.e. ALL of them are illusive. 

> -> This is an (informal) evidence
> ->that the "length" of N and therefore N itself is not fixed.
> This is merely a restating of the fact that you are talking about X-wise
> computable numbers for a nonfixed X.  If you want to identify N with
> N_(X-comp) we can't stop you, but no-one else is going to play ball.

You want to say that there is a limit of all these Ns? 
I see no reason in believing on this limit. 

> As I see it, the real problem with formalism, is that you deny the reality
> of all of N, but somehow re-admit the reality of all derivations from
> a system of logic.  

Not all (I do not know what means `all' here), but only those 
we can really do. 

> The latter is more complicated than the former,
> and just as big, so why regard it as more fundamental?

Not so complicated. Say, PA consists of several axioms 
and one axiom schema. It is based on FOL based on several 
(schematic) rules. We easily understand how to use these rules. 
All of this is quite concrete, unlike N, which is both illusive 
and not determined enough (as ANY illusion). 

I am not sure that I will convince many participants of FOM 
as the previous experience shows. 

>              And God said
>              Let there be numbers
>              And there *were* numbers.
>              Odd and even created he them,
>              He said to them be fruitful and multiply
>              And he commanded them to keep the laws of induction

Really God? Back to Kronecker? May be induction axiom, 
as well as axioms of set theory, were created by 
mathematicians? If so, mathematicians can also choose 
other alternatives if there will be some reason. 

Vladimir Sazonov

More information about the FOM mailing list