> ... the following words of ... Vladimir Uspensky ...
> "The place {AZ: of NSA in modern mathematics} is quite modest.
> Non-standard analysis does not intend to abolish the standard one. All
> existing "standard" results remain valid. Moreover the non-standard
> analysis does not pretend to obtain fundamentally new results: all
> results obtaining by means of its methods can be proved by means of the
> habitual methods."
I think the conventional term for this is that nonstandard analysis
is a "conservative extension" of standard analysis -- i.e., any
theorem that can be stated without nonstandard analysis, can
also be proved without it. That fact is a *theorem* (or metatheorem)
which can be proved. The only new thing that nonstandard analysis
can hope to offer is new insight, intuition, understanding -- or
perhaps a clarification of the old insight, intuition, understanding
originally held by Pascal, Leibniz, Newton, et al. My questions have
now been clarified a bit, thanks to messages from others. I now see
these questions:
(1) How does nonstandard analysis compare with the views of
Pascal, Leibniz, Newton, et al.? I believe that Walter Felscher's
email of November 13 has addressed that question.
(2) This one is not really a history question, so perhaps the
replies should go off-list. Does nonstandard analysis actually
offer an improved insight or intuition into some parts of analysis?
Of course, this question is a bit open-ended: We may get different
answers for different parts of analysis (and not just analysis, but
other parts of math too); we may get different answers for different
levels of math (undergraduates, graduate students, researchers),
and ultimately what is an "improvement" is a matter of opinion.
So I don't know if this is a question that is precise enough
to be answerable.
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Eric Schechter *** http://www.math.vanderbilt.edu/~schectex/
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