"Sam Sung" <email@example.com> wrote in message news:firstname.lastname@example.org... > Julio Di Egidio write: >> "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.
The point in fact was "numbers count themselves, how else?"
>>> Take any (uniquely operated) iteration and count the >>> steps by whichever method you like - the result is isomorphic >>> to N in whichever 'theory'. >> >> No, > > Why not? > >> 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.
Indefinitely large yet (ever) finite: there just is no set N in the strict finitist view...