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

[Bill Taylor]

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.

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. When you have answered, I will have further
questions - I hope and believe you will not object to answering publicly?


->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.

->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 *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.

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

HMMMmmmmmm...

->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.


-> 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.

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

Beats me!

But anyway, I will pause until you have answered my question about the
tens-towers.

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             Bill Taylor          W.Taylor at math.canterbury.ac.nz
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             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
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