[FOM] Indispensability of the natural numbers

[Vladimir Sazonov]

isles at kingcon.com wrote:
> A few comments on T. Chow's FOM email of May 17, 2004 05:16:42.
[...]
> 3. Finally, I disagree with Chow's comments in paragraph 5: 
" Formal languages are interesting mostly because they capture
some features of an entity that we're studying, and if the
entity vanishes, then why bother with the formal language anymore?"
This argument has a Platonic quality: there is an "entity" (for
example, the natural numbers), and we try to describe it
linguistically via a formal language (for example, PA.)
>  It seems to me that the following alternative picture is 
a more accurate description of what happens: We have a practice
of which we conceive a vague notion (e.g. counting.) This "entity"
we discuss linguistically and/or with pictures. The use of the
linguistic and pictorial representations suggests new aspects
of the notion AND WITHOUT THE USE OF THESE REPRESENTATIONS WE
WOULD NOT HAVE CONCEIVED OF THESE ASPECTS. These new aspects
are then incorporated as part of our notion of the "entity".
But this enriched  understanding requires a richer linguistic
framework with which to grasp and discuss it. Etc. Thus the
notion of an "entity" such as the natural numbers is not a
static one: it grows and is redefined as our experience with
it and discussions about it continue. In this interactive preocess,
notations and formalisms play a creative, not merely a descriptive, part.

Well done! I agree. I have expressed this in FOM in slightly
different terms. Any idea without a formalism is like amoeba.
Formalization is like a skeleton of a highest animal. During
and because of formalization our initial vague idea is usually
strongly changing, and an illusion appears that we had this
resulting idea from the very beginning.

As to natural numbers: in our childhood we just counted "one, two,
etc." This is too little of the idea of N satisfying PA. When
we came to using quantifiers over N, something really new appeared.
We quantify over something which, before that, even did not exist
in our minds as a whole (as actual infinity). N was something 
indefinitely growing (a vague idea). Only by this reason it is
doubtful that for any arithmetic formula F(n), especially involving
quantifiers, the (seemingly selfevident) Minimum Principle

         F(n) -> exists the least m such that F(m)

equivalent to Induction Schema should *necessarily* hold. This F
based on vague quantification is itself a vague property of numbers.
We already know the well-known paradox on "the minimal number that
cannot be defined by a phrase..." which appears essentially by the
same reason - vagueness of the definition of this minimal number.
(However, the difference is that Induction Axiom can be expressed
in a formal language.)

The fact that the above Minimum Principle and the Induction Axiom
do not happen (yet) to lead to a contradiction does not mean that
the meaning of F above and of the very concept of N is not vague
and is uniquelly existing. It seems convenient to feign that
everything is OK and to forget how things were really evolving
(like embryo) and may continue to evolve. *This* is ptatonistic
view.

It is highly strange to me that Timothy Chow seemingly agrees
with the evolution of the idea of N, but still consider that
there is no need to reject the notion of the standard integers.
What is so standard (rigidly fixed) in so vague, floating concept?

I hope it is clear that I am not against using Induction Axiom
in arithmetic once it is so strong tool. Any reasonable mathematical
formalism is a tool (accelerator, lever, etc.) to strengthen our
thought.

But once Induction Axiom is, in principle, subject to some doubts,
we can consider other versions of arithmetic - some other way
of evolving the embryo "one, two, three and so on".

What I consider especially important - we should not stop just
on informal discussions. A formalization of any mathematical
idea is necessary.

For example, it *is* possible to formalize

       1,2,3,... (with no end) < 2^1000

in a weak version of first order classical logic (thus, even with
quantifiers). This is another possibility for evolution of the
naive concept N.

> 
> Comments on the above opinions would be appreciated.
> 
>                              David Isles

Kind regards,

Vladimir Sazonov