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MathGroup Archive 2002

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Re: Re: Question about Replace

  • To: mathgroup at smc.vnet.net
  • Subject: [mg35761] Re: [mg35753] Re: Question about Replace
  • From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
  • Date: Tue, 30 Jul 2002 07:22:09 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

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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