If Achilles attempts to throw a javelin at a target,
before it reaches the target it has to go half the distance (that makes
sense), but before it can go half the distance it has to go half of that
distance (once again it makes sense), but before it goes half that
distance it has to go half that distance. What this means is that the
javelin has to go one half the distance, one fourth the distance, one
eighth the distance, one sixteenth the distance, ad infinitum. Now the
idea has become absurd because it concludes that the javelin will never
leave Achilles' hand. And thus, Zeno is showing that motion is an
illusion.
After Zeno presented his ideas, and after realizing the absurdity
of them, an ancient Greek philosopher by the name of Diogenes, made
no attempt to refute Zeno's new theories; instead, he got up and
walked across the room, as if proving that motion exists. But this,
of course, did not solve any of Zeno's paradoxes.
Comparing these two paradoxes they basically have the
same conclusion: no motion exists because that would involve
overcoming the repeated splitting of the distance to be traveled.
According to Zeno's logic this proves that an object never starts in
motion, and thus concludes that all motion is an illusion.
Convergence on an Infinite Series  Using Calculus
Before solving the two paradoxes in
question it will be shown that convergence on an infinite series,
using Calculus and algebra, will not solve these arguments.
Recently a news reporter critiqued my essay on Zeno's
Paradox and stated, "It was solved long ago [meaning the paradox]
when mathematicians showed that an infinite series can converge on a
finite solution. (The wikipedia article with the great title 1/2 +
1/4 + 1/8 + 1/16 talks about this.)"^{1}
But for this concept to be true it has to be proven
that calculus and algebra can eliminate the infinite aspect of
Zeno's paradoxes and create a measurable distance between the
javelin and the target.
Because of this news article and because of the fallacy
of this idea I have been motivated to rewrite my essay in order to
address this issue.
Calculus and Paradoxes
