[FOM] Indispensability of the natural numbers

[Vladimir Sazonov]

Steven Ericsson-Zenith wrote:
> Vladimir Sazonov said:
> 
>> Mathematicians do not need to have
>>a metatheory. They just deduce their theorems (with the help
>>of some intuition, of course).
> 
> 
> ...
> and then said:
> 
> 
>>Whatever are our beliefs, the mental concept of the natural numbers
>>is something vague, anyway. It is only illusion of something solid.
> 
> 
> I would like to understand how you sustain this position since the 
> first appears - at least to me - to contradict the second position.

I do not see any contradiction.

> 
> A metatheory would seem essential if you are to resolve the 
> inadequacies of "illusion" - for example, what *exactly* do you mean 
> by "illusion"?

I have no *exact* understanding of what illusion is.

According to Webster's English Dictionary in the Internet:
illusion is "perception of something objectively existing in such
a way as to cause  misinterpretation of its actual nature".

The actual nature of natural numbers (I mean - in general - not
in concrete examples) is our thoughts of them. Our thoughts are
sufficiently objective. We can communicate on them one with another.
But our perception of our thoughts on N that this imagined entity
is solid is our illusion.

I do not need any metatheory (as you suggest) to realize this,
although I do not exclude the possibility that some metatheoretical
considerations might illuminate some questions discussed here.


See also my answer to Timothy Y. Chow which probably should
clarify my position for you.

Kind regards,

Vladimir

> 
> Sincerely,
> Steven
> --
> Dr. Steven Ericsson Zenith