Re: Re: Question about Replace

*To*: mathgroup at smc.vnet.net*Subject*: [mg35762] Re: [mg35753] Re: Question about Replace*From*: Andrzej Kozlowski <andrzej at tuins.ac.jp>*Date*: Tue, 30 Jul 2002 07:22:11 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

That should have been Leibniz, of course. On Monday, July 29, 2002, at 05:15 PM, Andrzej Kozlowski wrote: > 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 >> >> >> > > Andrzej Kozlowski Toyama International University JAPAN http://platon.c.u-tokyo.ac.jp/andrzej/