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From: "Lindsey, Dr. Charles" <clindsey@fgcu.edu>
To: Phil Mahler <mahlerp@ADMIN.MCC.MASS.EDU>
Date: Mon, 15 Apr 96 10:08:00 PDT
I am having trouble posting to the list since we switched email systems, so
I am responing directly (maybe I need to resubscribe?). Feel free to forward
this to the list if you think others would be interested.
Calculus has its own way of dealing with this paradox, by setting up a
logical structure that leads one to the conclusion that the entire sequence
must be completed in finite time. Aristotle proceeds differently, by
carefully questioning what is meant here by "infinite". Ultimately he
distinguishes two kinds of infinite: infinite by addition, and "infinite by
division," which means capable of being subdivided over and over again
without end. The infinity being discussed here is infinite by division, and
since time is likewise infinite by division, there is no paradox, since we
are also using an infinite time, in the same manner as the infinite (by
division) magnitude. According to Aristotle.
Some may not like the calculus-type argument, that since no finite sum of
steps takes longer than a certain amount of time (depending on your
velocity), the whole process must be completed in a certain amount of time,
otherwise a logical contradiction ensues. This grants a special status to
time, that of proceeding at a constant rate, fixed and unchanging. I suppose
if you define completion in terms of time, then it has to work out, but the
whole thing seems a bit self-referential. Not that I don't use it often. :-)
Chuck