[FOM] Indispensability of the natural numbers

[Timothy Y. Chow]

On Tue, 18 May 2004, Vladimir Sazonov wrote:
> Whatever are our beliefs, the mental concept of the natural numbers
> is something vague, anyway. It is only illusion of something solid.

Do you agree that our mental concepts of symbols and rules are also vague
illusions of something solid?  We cannot directly observe schoolchildren
applying *rules*; we only observe them doing specific things like making
marks on paper, and it requires an act of mental abstraction to interpret
our observations of children by saying, "Ah!  These children are applying
*rules*!"  A rule cannot be weighed on a scale or poked with a thermometer.
I see nothing more solid about this mental abstraction than about the 
mental abstraction from "1, 2, 3, ..." to the natural numbers.

> Your further considerations (which I will not quote) are actually
> about some AWFUL WORLD DISASTER if a contradiction would appear.

No, if you think this, then you have not read my article carefully enough,
but have merely assumed on a superficial reading that it is just like many
other articles on the topic that are superficially similar.

> The problem would be only how to change our intuition on natural
> numbers. The same problem on sets was resolved quite efficiently (even
> if only temporarily) after Russel's paradox. I see no essential
> difference.

My remark about nonstandard models of arithmetic was an attempt to 
illustrate an essential difference.  In the case of set theory, there
are many candidates for replacing any particular version of set theory
that we might use temporarily.  In the case of the natural numbers,
there is no candidate in sight.  If you disagree, name one.

Tim