The argument being that the arrow has to do an infinite number of things
before even getting started.
This is related to the paradox of Achilles and the tortoise: Achilles gives
the tortoise a headstart (d1) in the race, then runs after it. By the time
Achilles reaches d1, the tortoise has moved ahead some additional distance
d2. Now Achilles must cover the distance d2 but during this time the
tortoise has moved ahead a distance d3, etc. Obviously the distances are
getting shorter, but equally obviously there are an infinite number of them!
How can Achilles ever catch up with the tortoise, if he has to "cover an
infinite number of distances" first?
Of course, from calculus we now know that the sum of an infinite series can
quite possibly be finite, so here:
d1 + d2 + d3 + ... = x (not infinity)
x can also be found by solving x = v_a * t and x - d1 = v_t * t, where v_a
and v_t are the speeds of Achilles and the tortoise, respectively. The
second unknown in the system of equations is t, the time it takes Achilles
to catch up with the tortoise (again, not infinite).
The arrow paradox doesn't translate so conveniently into an infinite series,
but it also ceases to be paradoxical if we remember that we can do an
infinite number of (sufficiently small) things in finite time.
KC
-----Original Message-----
From: Forum for Physics Educators [mailto:PHYS-L@list1.ucc.nau.edu] On
Behalf Of Fakhruddin, Hasan
Sent: Monday, March 28, 2005 3:09 PM
To: PHYS-L@LISTS.NAU.EDU
Subject: Re: Zeno's Paradoxes
What's the solution for the first Paradox: An arrow travles half the
distance to the apple and again half of the half ad infintum. How is it
possible for the arrow to ever reach the apple?
Hasan Fakhruddin
Instructor of Physics
The Indiana Academy for Science, Mathematics, and Humanities
BSU
Muncie, IN 47306
E-mail: hfakhrud@bsu.edu
-----Original Message-----
From: Forum for Physics Educators [mailto:PHYS-L@list1.ucc.nau.edu] On
Behalf Of Dan Schroeder
Sent: Saturday, March 26, 2005 8:36 PM
To: PHYS-L@LISTS.NAU.EDU
Subject: Re: Zeno's Paradoxes
There's a pretty large body of literature on Zeno's Paradoxes, and not
all philosophers would agree that they have been solved. Do a google
search or a library search to learn more. One interpretation of the
paradoxes is that they challenge us to think about whether space and
time are continuous or discontinuous. The answer, according to modern
physics, seems to be something more subtle than either. Some
philosophers might go so far as to claim that Zeno's paradoxes told us
this all along, though I wouldn't put it that way myself.