[FOM] Re: Indispensability of the natural numbers

[Vladimir Sazonov]

Timothy Y. Chow wrote:
> On Wed, 19 May 2004, Vladimir Sazonov wrote:
> 
>>Timothy Y. Chow wrote:
>>
>>>Do you agree that our mental concepts of symbols and rules are also vague
>>>illusions of something solid?  
>>
>>*General* mental concepts - of course! See also below.
> 
> [...]
> 
>>First, the (highly informal, vague and floating) entity (N)
>>does not vanish. Only our understanding and intuition on it
>>may be changed in some way.

Probably I should add that by the entity N I mean just the
name "N" or wording like "one, two, three, and so on" with
various informal and formal considerations around this.
Also, N may split into various versions, each of them being
still vague.

> Thanks for your long and detailed message.  With this clarification of 
> your point of view, and with the new (to me) insight that I described in 
> my last post to FOM, I find that I disagree with you much less than I 
> originally thought.  


Is it?


A few small points remain.
> 
> For example, I now see no reason why the inconsistency of PA should 
> threaten platonism.  Platonism has successfully survived set-theoretic
> paradoxes, Goedel's theorems, and the independence of the continuum
> hypothesis.  The notion that N is a determinate, independently existing
> object will not be refuted by a discovery that we were wrong about one
> of its properties.

In which way is N so determined. There is the name "N" which
all of us are using. But the "full" meaning of this N is unclear
(even independently on any possible contradiction in PA or
even independently on the incompleteness of PA).
Is it true the sentence forall n log log n < 10 according to
your platonism (if you accept at all my, as you say,
"clarification")? Do you have a unique meaning of "and so on"
according to your platonism? In the context of our exchange
I do not understand what is your platonism?


> In fact, I would say that ZFC does not eliminate the vagueness of which 
> you speak, since it doesn't settle every first-order question about N.
> ZFC proves only the existence and uniqueness of N.

- relative to imaginary and therefore vague world of ZFC. No absolute
existence and uniqueness. Also, I did not (mean to) say that ZFC
eliminate the vagueness of N.


Vladimir