From: gw7rib@aol.com   
      
   Hi all   
      
   Recently there was a discussion in comp.theory about infinitesimals. It seems   
   I can't post to that group but can post to this one, so hopefully people will   
   not mind too much and some kind person might even post a link there to my post   
   here?   
      
   I wanted to point out that Ian Stewart had written an article called "Beyond   
   the vanishing point" in which he discusses the strange situation in which   
   doing calculus by using very small numbers and then treating these numbers as   
   zero after you've divided    
   through by them is not valid but nevertheless seems to work. Here is some of   
   the article as a taster:   
      
      
   As far as we know, the first people to ask questions about the proper use of   
   logic   
   were the ancient Greeks, although their work is flawed by modern standards. And   
   in about 500BC the philosopher Zeno of Elea invented four famous paradoxes to   
   show   
   that infinity was a dangerous weapon, liable to blow up in its user’s hands.   
   Even so,   
   the use of "infinitesimal" arguments was widespread in the sixteenth and   
   seventeenth   
   centuries, and formed the basis of many presentations of (for example) the   
   calculus.   
   Indeed it was often called "Infinitesimal Calculus". The logical   
   inconsistencies involved   
   were pointed out forcibly by Bishop Berkeley in a 104-page pamphlet of 1734   
   called   
   The Analyst: A Discourse Addressed to an Infidel Mathematician. The trouble   
   was,   
   calculus was so useful that nobody took much notice. But, as the eighteenth   
   century   
   wore on, it became increasingly difficult to paper over the logical cracks. By   
   the middle   
   of the century, a number of mathematicians including Augustin-Louis Cauchy,   
   Bernard   
   Bolzano and Karl Weierstra8, had found ways to eliminate the use of infinities   
   and   
   infinitesimals from the calculus.   
      
   The use of infinitesimals by mathematicians rapidly became "bad form", and   
   university students were taught rigourous analysis, involving virtuoso   
   manipulations   
   of complicated expressions in the Greek letters epsilon and delta imposed by   
   the traditional definitions. There is even a colloquial term for the process:   
   epsilontics.   
   Despite this, generations of students in Engineering departments cheerfully   
   used the   
   outdated infinitesimals; and while the occasional bridge has been known to   
   fall down,   
   nobody to my knowledge has ever traced such a disaster to illogical use of   
   infinitesimals.   
      
   In other words, infinitesimals may be wrong - but they work. Indeed, in the   
   hands of an experienced practitioner, who can skate carefully round the thin   
   ice, they   
   work very well indeed. Although the lessons of this circumstance have been   
   learned   
   repeatedly in the history of science, it took mathematicians a remarkably long   
   time to   
   see the obvious: that there must be a reason why they work; and if that reason   
   can   
   be found, and formulated in impeccable logic, then the mathematicians could   
   use the   
   "easy" infinitesimal arguments too!   
      
   It took them a long time becauseit’s very hard to get right. It relies on   
   some deep   
   ideas from mathematical logic that derive from work in the 1930s. The   
   resulting theory   
   is called Nonstandard Analysis, and is the creation of Abraham Robinson.It   
   allows the   
   user to throw real infinities and infinitesimals around with gay abandon.   
   Despite these   
   advantages, it has yet to displace orthodox epsilontics, for two main reasons:   
      
   * The necessary background in mathematical logic is difficult and, except for   
   this   
   one application, relatively remote from the mathematical mainstream.   
   * By its very nature, any result that can be proved by nonstandard analysis   
   can also   
   be proved by epsilontics: it’s just that the nonstandard proof is usually   
   simpler.   
      
   You can get the article (published in "Eureka") by going to http   
   ://www.archim.org.uk/eureka/archive/index.html and downloading Issue 50	-   
   April 1990. Enjoy!   
      
   Paul.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)