Zeno's paradoxes

I don't see these as meaningful or interesting paradoxes,
but it should be -mentioned-, as it has a good interpretation.

1:
Suppose Achilles moves at 1m/second, and the tortoise moves at p meter/second,
and suppose that Achilles starts 1m behind the tortoise.
Then there's two ways to calculate where they catch up:
  distance between 'em is 1m,
  closing speed is (1-p)m/second,
  so it'll take 1/(1-p) seconds to close;
  as we've cleverly chosen for Achilles to move at 1m/second,
  he'll have gone 1/(1-p) meters.
[Incidentally, this is a -fabulous- way of remembering the formula!
 Also, if he starts p meters behind, you recover p/(1-p) too!]

Alternatively, he needs to catch up with where the tortoise was at
first, which'll take 1 second (and he'll move 1 meter).
Now the tortoise will have moved p meters,
so now Achilles needs to run p more meters (for p more seconds),
during which the tortoise will have moved $p^2$ meters.
Repeating, we get
$$1/(1-p) = 1+p+p^2+\dots$$
ta-da!

Indeed, you can even recover the -partial- geometric sum from this:
after $n$ iterations, he'll be within $p^{n+1}$ of the tortoise,
whence:
$$(1-p^{n+1})/(1-p) = 1+p+\dots+p^n$$

This is actually pretty cool -- it gives a completely concrete and
explicit interpretation -- and further, it totally squashes complaints
like "but you'll never actually -reach- the wall
(for the EG of "take a step half-way closer to the wall")"
as it's obvious (to all but philosophers) that he'll actually -reach-
the tortoise.

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You can do more:
- the above interpretation only works for $0\leq p < 1$
- if $p>1$, it also works, but gives a negative number
  -- this is okay: it indicates that -in the past-, you were at
  the same point
  [indeed, more deeply, if you replace $p$ by $q=1/p$, then
   it does work mathematically: basically you're looking at the
   involution on the Riemann sphere that takes "outside the disc"
   to "inside the disc".]
- if $p<0$, then you can also interpret the sum, I think:
  the naive interpretation "over-shoots", so you have to backtrack
  etc.
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2:
Throwing a rock at a tree
(so here the tree is stationary).
In order to hit the tree, the rock needs to first get half-way to the
tree -- and now to get to the tree, the rock needs to again go
half-way, etc.

``Following Dr. Johnson, I refute it thus: <throw chalk at wall>''

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So what's the philosophical -problem-?
Or rather, -why is Zeno confused-?
Possible answers:
1: you need to pass through infinitely many points in finite time.
2: He's not keep track of time.
  That is, in addition to going a shorter distance at each step,
  it also takes less time -- Zeno's implicitly suggesting that each
  step takes the same amount of time.

Apparently the point of them is not to argue that motion doesn't
happen (duh -- it obviously does), but to clarify what we mean by
motion and stuff.

The question is ``how is motion actually possible?''.
And it seems to me that Zeno is just horribly confused,
and mixing up 10,000 things.
EG, if the universe is described by continuous functions,
then here's how motion is possible (draw a function).
Also, it's mixing up the question of ``how can a given (static) function
describe a universe where time happens over time?''

EG, for the arrow paradox, the problem (to me) seems to be the
statement that ``the position of the arrow at an instant in time
completely describes the system''. Which is wrong: you need both
position -and momentum-.

Alternately, you can just describe the system by the positions over
time (which is what we implicitly do in math: we take the ``god's eye view''
and don't look at an instant in isolation).
That is, just b/c a function is constant at a given instant (which is a
tautology), doesn't mean the function is globally constant.

Obviously the question of what the universe is -really- like depends
on physics, but a very naive view of the universe can resolve Zeno.
EG, maybe the universe is -discrete-.
So there's a minimum distance you can move, and a minimum time.
There's still no problem.
Obviously you'd need a physics that can deal with that.

(apparently part of Zeno's point was to point out problems
 in atomism)
