[FOM] Re: Indispensability of the natural numbers

[Steven Ericsson-Zenith]

Vladimir Sazonov wrote:
>>>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.

After giving this further thought I realise this is not an 
unfamiliar problem.

I have been recently considering essentially the same issue 
in the context of proofs of the Pythagorean theorem -  I am 
particularly interested in two cases.

	     1. Which demonstrations provide a communication 
that the theorem is correct for all possible values of c?  
This requirement reduces the available proofs considerably 
... and 

	     2. From any proof how does one predict (other 
than by the known demonstrations) that there are in fact 
Diophantine solutions.

Question one here is essentially equivalent to the question 
of how one predicts that what is correct for N and N + 1 is 
correct for all N.

Question two has a fundamentally different nature and relates 
to how one abducts/inducts/deducts certain properties of a 
proof. 

However, the first should be familiar as the question of how 
one moves from special cases to general cases. This question 
is as old as mathemathics itself and has been attacked 
formally a number of times. (I will reference here only Peirce's 
"Exact Logic" again, only because it is my most recent reading 
on the subject).

Is there some insight in your observation that I have missed?

With respect,
Steven

--
Dr. Steven Ericsson-Zenith