FOM: Another question on infinitesimals
JSHIPMAN@bloomberg.net
JSHIPMAN at bloomberg.net
Thu Nov 13 11:23:09 EST 1997
There seems to be some agreement that the nonconstructive nature of NSA
infinitesimals is a problem. It also seems clear that constructive
infinitesimals can be introduced in a variety of ways, but that they can't give
the elementary equivalence that Robinson gets with his ultraproducts. This may
not be such a big deal. Consider the following "radical" suggestion:
As an alternative to teaching rigorous calculus via epsilons and deltas (note
I do NOT say as a "replacement for"), we should rigorously (and constructively)
develop an expanded real number system which includes infinitesimals, that is
both concrete enough that you can give examples of infinitesimals and flexible
enough that you can do calculus in it. (The theorems of calculus so developed
should of course specialize to the familiar ones when you restrict to the
standard reals.) Conway shows you can go quite a long way in this direction.
My question: given that you can't go "all the way" with constructive
infinitesimals, how far can you go? Could you get all the way through first
year college calculus, rigorously replacing epsilondelta with them?--Joe Shipman
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