> "Sam Sung" <email@example.com> wrote in message > news:firstname.lastname@example.org... >> Julio Di Egidio wrote: >> >>> << Edward Nelson criticizes the classical conception of natural numbers >>> because of the circularity of its definition. In classical mathematics >>> the >>> natural numbers are defined as 0 and numbers obtained by the iterative >>> applications of the successor function to 0. But the concept of natural >>> number is already assumed for the iteration. >> >>> >>> <http://en.wikipedia.org/wiki/Ultrafinitism#Main_ideas> >> >> So what? > > You have snipped the context.
You refer to:
>> What does pi count? >> Isn't it a number? > > pi counts pi, of course...
? Sorry then.
>> Take any (uniquely operated) iteration and count the >> steps by whichever method you like - the result is isomorphic >> to N in whichever 'theory'. > > No,
> confront strict finitism.
"Classical finitism vs. strict finitism In her book Philosophy of Set Theory, Mary Tiles characterized those who allow countably infinite objects as classical finitists, and those who do not allow countably infinite objects as strict finitists. Historically, the written history of mathematics was thus classically finitist until Cantor invented the hierarchy of transfinite cardinals in the end of the 19th century."
They may 'not allow' whatever they want, but cannot 'forbid' appliyng endlessly an operation like a successor function to 0.