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.