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RE: Re: Re: Question about Replace
*To*: mathgroup at smc.vnet.net
*Subject*: [mg35791] RE: [mg35761] Re: [mg35753] Re: Question about Replace
*From*: "DrBob" <majort at cox-internet.com>
*Date*: Wed, 31 Jul 2002 01:33:31 -0400 (EDT)
*Reply-to*: <drbob at bigfoot.com>
*Sender*: owner-wri-mathgroup at wolfram.com
I agree. I had a graduate math course in non-standard analysis years
ago, and I recall that it was just as formally logical as the usual way.
Integrals became "finite" (transfinite) sums and the usual proofs of
calculus became much simpler, once the groundwork was laid.
It could be tricky to incorporate it in Mathematica, but no more so than
intervals or complex numbers. The synthesis of integration with
summation could be very powerful, too, as each can build on the other.
Ditto for differentiation and finite differences, differential equations
and difference equations, etc.
Bobby Treat
-----Original Message-----
From: Andrzej Kozlowski [mailto:andrzej at tuins.ac.jp]
To: mathgroup at smc.vnet.net
Subject: [mg35791] [mg35761] Re: [mg35753] Re: Question about Replace
While Berkeley's critique of 18th century Calculus was right at the
time, Abraham Robinson showed that ultimately that the intuition
behind the sort of thing that Leibnitz and others did was right and
could be completely formalized and turned into a very powerful tool. It
certainly would be nice to implement non-standard analysis in
Mathematica (perhaps someone has already done this?). Indeed one can in
this way turn calculus into algebra (getting rid of the concept of
Limit) and it may well be the most natural approach to calculus via
symbolic algebra.
(For more see Abraham Robinson, "Non-standard Analysis", Princeton
Landmarks in Mathematics, 1996).
Andrzej Kozlowski
Toyama International University
JAPAN
http://platon.c.u-tokyo.ac.jp/andrzej/
On Monday, July 29, 2002, at 04:13 PM, John Doty wrote:
> In article <ai06os$1f6$1 at smc.vnet.net>, "Andrzej Kozlowski"
> <andrzej at tuins.ac.jp> wrote:
>
>> Actually on second thoughts I began to suspect that this question is
>> related to another one posted by Heather, concerning simplifying
>> expressions in which x is "much larger than" y. I am not at all sure
if
>> a sensible calculus of this kind can be developed but obviously
>> Simplify
>> will not do this.
>
> It seems to me that this is essentially a (capital-C) "Calculus"
> problem,
> and unless a simple /.y->0 is what's wanted, the correct tool is
> Limit[].
> Berkeley's critique of 18th century Calculus applies here: while it
was
> essentially antiscientific, his reasoning was flawless and should warn
> us
> against trying to solve this sort of problem by mindless algebra.
>
> Of course, Limit[] is a tricky and somewhat unreliable power tool,
> requiring caution. This reflects the mathematical subtlety of this
kind
> of problem. It is generally essential to formulate the problem in such
a
> way that the direction of the approach to the limit is unambiguous.
>
> --
> | John Doty "You can't confuse me, that's my job."
> | Home: jpd at w-d.org
> | Work: jpd at space.mit.edu
>
>
>
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