FOM: How real are real numbers, anyway? JSHIPMAN at
Wed Nov 12 16:38:32 EST 1997

Harvey is willing to admit that "infinitesimal reasoning" is valid without
admitting that "infinitesimals" are.  Since infinitesimal reasoning has played
a large role in obtaining results historically (this goes back further than
Euler, further than Leibniz, all the way back to Archimedes who then
rigorously proved the results he had obtained by his infinitesimal "Method"
using a technique equivalent to epsilon-delta), there is a real foundational
issue here.  WHY does this type of reasoning "work"?  To prove that it is a
conservative extension gives a little comfort but ignores the basic issue.
Either there is something about the human mind which makes the infinitesimal
type of argument "easier" or there is something "real" about infinitesimals
which our intuition captures.  If you deny the reality of infinitesimals it is
not so easy to defend the reality of arbitrary real numbers....  Our theories
of physics use real numbers but it is not clear that any physical quantities are
meaningful to, say, 2^2^1000 decimal places.  Why is "infinitesimal reasoning"
any more of a trick than the methods used in standard analysis?--Joe Shipman

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