[FOM] Infinitesimal calculus
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.
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.
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