Christoph Zwerschke <[EMAIL PROTECTED]> wrote: > Alex Martelli wrote: > > > An alternative theory, of course, is "God made the natural numbers; all > > else is the work of man" -- and that one is by a German, too (Kronecker, > > if I recall correctly). > > Yes, it was Kronecker. But even natural numbers are usually constructed > with sets using Peano's axioms.
Peano's axioms are perfectly abstract, as far as I recall. Russell and Whitehead did try to construct naturals from sets (defining, e.g., '5' as "the set of all sets with five items"), but that was before the inherent contradictions of set theory were widely known (though Russell himself had destroyed Frege's attempts at theorization by pointing out one such contradiction, the one wrt the "set of all sets that don't include themselves as a member" if I recall correctly). Later, Goedel showed that any purely formal theory that's powerful enough to model natural arithmetic cannot be both complete and consistent... > > The hope to found all of mathematics on set theory was primarily a > > _British_ effort, as I see it (Russell and Whitehead), and failed a > > long time ago... I'm not sure what, if anything, a mathematician of > > today would propose as the foundational theory... > > Perhaps "string theory" ;-) So probably strings should become the Some physicists, maybe, surely not mathematicians though! Alex -- http://mail.python.org/mailman/listinfo/python-list
