Date: Jun 21, 2013 9:43 AM
Author: LudovicoVan
Subject: Re: Matheology § 288
"Sam Sung" <no@mail.invalid> wrote in message

news:kq1k8g$u5d$1@dont-email.me...

> Julio Di Egidio write:

>> "Sam Sung" <no@mail.invalid> wrote in message

>> news:kq1ica$juk$1@dont-email.me...

>>> 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...

Julio