Services & Resources / Wolfram Forums
-----
 /
MathGroup Archive
2002
*January
*February
*March
*April
*May
*June
*July
*August
*September
*October
*November
*December
*Archive Index
*Ask about this page
*Print this page
*Give us feedback
*Sign up for the Wolfram Insider

MathGroup Archive 2002

[Date Index] [Thread Index] [Author Index]

Search the Archive

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
>
>
>






  • Prev by Date: RE: Animation looping glitch (confirmed)
  • Next by Date: More weird integration issues...
  • Previous by thread: Re: RE: RE: Question about Replace
  • Next by thread: Installing package "SpreadOption`"