On 6/21/2013 7:44 AM, Julio Di Egidio wrote: > "Julio Di Egidio" <email@example.com> wrote in message > news:firstname.lastname@example.org... >> "Alan Smaill" <smaill@SPAMinf.ed.ac.uk> wrote in message >> news:email@example.com... >>> "Julio Di Egidio" <firstname.lastname@example.org> writes: >>>> "fom" <fomJUNK@nyms.net> wrote in message >>>> news:C6KdnZCccIjltVzMnZ2dnUVZ_t6dnZ2d@giganews.com... >>>> >>>>> "Numbers count themselves" >>>> >>>> Indeed, how else? >>> >>> What does pi count? >>> Isn't it a number? >> >> pi counts pi, of course... > > << 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> > > Julio > >
For better or for worse, I have spent a great deal of time trying to understand "logical foundation" as, apparently, rejected by Brouwer.
In "logic" there is a problem with denotation and reference.
For "logic" to be applicable, there is a problem of unique reference. While 'x=x' may have ontological interpretation, 'x=x' also has a semantic or metalinguistic interpretation according to which the singular term 'x' has a consistent usage.
It may be that there is some *given* system of names -- I could accept that if I knew them. But, I do not.
So, I am left with generating a system of names (If they denote, I am speaking of a consistent global labeling. Otherwise, I am speaking of a collection of distinct inscriptions.).
I do not see how to divorce an ordinal sequence from a sequence of names introduced for the purpose of "logic". That is, after the first "naming symbol" is utilized, it is removed from possible naming symbols because of the uniqueness criterion.
This is the problem of merely "purporting reference" without considering what is presupposed. One may have a geometry or a linear order. In either case, verification of the semantic requirement involves serialization.
One may reject a "logical foundation" personally. But, that does not remove the need to prove that someone who does accept "logical foundation" is assuming natural numbers when all they are doing is "naming".
If I accept that mathematics reduces to the logical species of natural numbers, then perhaps Nelson (and his predecessors, the Brouwerian "pre-intuitionists" who gave the same criticism) are correct.
For what this is worth, Frege *defined* zero. The role of the Dedekind-Peano axioms had been that of a necessary condition to be fulfilled. Nelson's criticism merely applies to views where the axioms are taken to be "definitions-in-use".
Logic involves priority, and Nelson's statement is debatable on these grounds.
I do not disagree that the naming of numbers has a correspondence with the count of names. But, because of Cantor's insights and investigations of infinity, ordinality and cardinality are not taken to be the same thing.