[FOM] Infinitesimal calculus

Charles Silver silver_1 at mindspring.com
Sat May 23 17:31:56 EDT 2009

	I think the most important reason infinitesimals aren't used in  
calculus books is simply that epsilon-delta was given formal  
justification over a hundred years ago, so we stick to it.   Well,  
sort of.   Since the e-d proofs are too hard for most beginning  
calculus students, they've been dropped from almost all the texts (at  
least in the US).
	As far as I know, every proof using e-d is easier using  
infinitesimals.   So, if you want proofs back in intro. calculus  
texts, infinitesimals are the way to go.  But, apparently historical  
momentum trumps sensible thinking.
	Whoever said there are lots of infinitesimal calculus books around is  
wrong.   I know of Martin Davis's very fine book, Keisler's carefully  
developed book online, and a nice, simple one you could read (and  
understand) in an hour or so by Jim Henle.  These books are  
essentially all dead: Keisler's not in print, Martin's in Dover, and  
so is Henle's.  (I'm sure I must be missing a couple others.)
	The only downside I can see is that infinitesimals are not ordinary  
numbers, but neither were negative numbers once upon a time.
Charlie Silver

On May 21, 2009, at 5:54 PM, Monroe Eskew wrote:
> 2) Aren't the epsilon-delta notions of limits practical?  I know that
> in experimental science, one wants to approximate and compute all the
> time, and it is also of interest to do so within a margin of error.
> This means a FINITE margin of error, since infinitesimal error is not
> available in the real world.
> Best,
> Monroe

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