Date: Jan 30, 2013 6:02 AM
Author: fom
Subject: Re: Matheology § 203
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.