"fom" <fomJUNK@nyms.net> wrote in message news:X5CdnU5BCcfKQlnMnZ2dnUVZ_hadnZ2d@giganews.com... > On 6/21/2013 7:44 AM, Julio Di Egidio wrote: >> "Julio Di Egidio" <firstname.lastname@example.org> wrote in message >> news:email@example.com... >>> "Alan Smaill" <smaill@SPAMinf.ed.ac.uk> wrote in message >>> news:firstname.lastname@example.org... >>>> "Julio Di Egidio" <email@example.com> 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> > > For better or for worse, I have spent a great deal of time > trying to understand "logical foundation" as, apparently, > rejected by Brouwer. <snipped>
I am coming from studies in philosophy, then philosophy of language, then logic, finally I am getting to mathematics. My actual knowledge of mathematics is basic, I am a software engineer, and my knowledge of the history of mathematics (and mathematical logic) is even more limited. But I am someway strong in logic and philosophy, but, again, not so much in their history.
I concur with intuitionism: mathematical objects and systems are conceptions of the mind, intuition is non-mechanical, mathematics cannot be reduced to the application of mechanical rules.
In any case, I think a mathematician should be a pragmatist and pick up the approach that is best representative but not over-representative, so to speak: namely, no need to get into properly philosophical issues. In this sense, if intuitionism looks like it is the right tool for the job, and if the philosopher approves, it's a deal.
> It may be that there is some *given* system of names -- I > could accept that if I knew them. But, I do not.
Well, can you count?
> 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".
Numbers (and their systems) are creations of the mind, in fact numbers are a cultural product and we communicate and share them, so there is *also* a mathematical language. The objection to the formalist then is that he forgets the reference of the language to its object, and that intuition cannot be reduced to the mechanics of language, etc. So, it's not really that the formalist "assumes natural numbers", it's that he forgets them altogether.
> Logic involves priority, and Nelson's statement > is debatable on these grounds.
Priority is in question here, and, in this sense, I'd concur with Nelson's criticism: the "classical conception" does not provide satisfactory answers.
> 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.
I think that's beside the point. Anyway, those results do not (necessarily) hold in a non-standard setting.