Consider the following version of one of Zeno's Paradoxes. You are in the middle
of a room. To leave the room you must first cover half the distance between you
and the exit. Then you must cover half the remaining distance. Then you must
cover half the remaining distance again... Considering this infinite process,
how do you manage to finally get out of the room?
I have heard it said that this paradox, and perhaps others of Zeno, are
"explained" in the calculus. Assuming you have heard this too, and agree, I'd be
interested in how it is explained. Or a reference to an appropriate source.
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I enjoyed the responses I've seen. Thank you to those who did respond.
I've always doubted that calculus does this, but I have seen reference to that
effect. Indeed I can find that statement in Newmann's wonderful "The World of
Mathematics," i.e. that infinite series somehow accomplish this.
I've done some thinking on the subject, but didn't want to slant the responses I
might get when I posed the question. Since this idea is used in math classrooms,
I thought it reasonably appropriate for the list.
The first pass explanation is that infinite sums do the trick. Since 1/2 + 1/4 +
1/8 + ... equals 1, in the calculus, then the sum of the distances which Zeno
requires us to traverse adds up to the distance required to get out of the room.
(or the time required).
My problem with this is that 1/2 + 1/4 + 1/8 + ... does not equal 1 until we
make a suitable definition. It seems to me that we create a system of symbols
which includes infinite series, operations on infinite series, and which is
isomorphic to the real number system. We do this with the definition of limits,
then simply say that 1/2 + 1/4 + 1/8 + ... corresponds to 1.
Then, if we apply the correct algorithms to infinite series the results are
isomorphic to performing the corresponding operations on real numbers. So that
2(1/2 + 1/4 + 1/8 + ...) = 1 + 1/2 + 1/4 + ... = 1 + (1/2 + 1/4 + 1/8 + ...)
which corresponds to 1 + 1 = 2.
So, to define 1/2 + 1/4 + 1/8 + ... as 1 is no more true than saying that you
can get out of the room.
I do like the interpretation that calculus has an answer by "by setting up a
logical structure that leads one to the conclusion that the entire sequence must
be completed in finite time." Assuming that it takes half the required time to
cross half the room, and 1/4 of the time to traverse half the remaining
distance, etc. then the idea of boundedness seems to have some validity. And the
whole thing seems reasonable. Of course, as that writer noted, we must make
assumptions about the physics in any case. It's "possible" that the relevant
series is the harmonic one - in which case you won't get out of the room.
I also liked the following.
"I don't think that Zeno's paradox IS "explained" by the calculus. ...
Zeno's paradox is only a paradox under the assumption that we cannot perform a
process an infinite number of times. What Leibniz and Newton did was ask the
question "What if we CAN perform a process an infinite number of times?" Under
this assumption there is no paradox."
However, I don't believe that one can do something an infinite number of times.
(Call me Bishop Berkeley I guess.)
Anyway, I guess my feeling that calculus does not "explain" Zeno's paradox is
not off base, and that I'm not missing something. Of course I will also continue
to model the paradox with infinite sums - I just won't (I never did) claim that
the one explains the other.