On 1/29/2013 1:04 PM, WM wrote: > On 29 Jan., 19:54, fom <fomJ...@nyms.net> wrote: >> On 1/29/2013 12:21 PM, WM wrote: >> >>> Done several times. Nevertheless logic and analysis can get along >>> without sets. And that better than with. >> >> Have you any references for the presentation >> of analysis without sets? >> >> I mean, here, textbooks. I would love to >> look at one. > > Is that a real question? > A very good book without any sets: > Euler: Introductio in analysin infinitorum > Or newer and at least without actually infinite sets, i.e., without > *trans*finite sets my book: > http://www.oldenbourg-verlag.de/wissenschaftsverlag/mathematik-ersten-semester/9783486708219
Sadly, I do not read German.
I may, perhaps, be able to find some translation of the Euler text.
And, yes, it is a real question.
I am fully aware of a number of historical issues concerning the foundation of mathematics. I accept certain logical considerations that lead to the present state of affairs. By the same token, I do not find your objections to actual infinity disconcerting in any way. Kant defined infinity as "plurality without unity." One does not get around that by introducing the transfinite since the language must carefully speak of the transfinite or carelessly distinguish the absolute infinite.
As for those "logical considerations," I mean that one can develop a hierarchy of definitions that depend on actual infinity. To say that mathematics is "logical" is to concede to such a framework. I do not believe that mathematics is logical at all. It uses logic to investigate structure in relation to sensuous experience (geometric incidence). That is quite different from the religious issue of justifying the material beliefs of modern scientists.
There is no greater trash philosophy than the arithmetization of mathematics followed by the subsequent logicism (which Frege and Whitehead retracted).
As for "counting," long before the British transformed their economic successes into mathematical philosophy, an Italian mathematician developed *double-entry* bookkeeping. That is a geometric concept at its heart.