[FOM] Re: Indispensability of the natural numbers
Vladimir Sazonov
V.Sazonov at csc.liv.ac.uk
Fri May 21 15:51:11 EDT 2004
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
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