though I imagine, somehow, we'll be able to turn this into a discussion on george bush or something.

ANYWAY, http://en.wikipedia.org/wiki/Zeno's_paradoxes

Zeno's Paradoxes:

Quote :

"Achilles and the tortoise "You can never catch up." "In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead." (Aristotle Physics VI:9, 239b15) In the paradox of Achilles and the tortoise, we imagine the Greek hero Achilles in a footrace with the plodding reptile. Because he is so fast a runner, Achilles graciously allows the tortoise a head start of a hundred feet. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run a hundred feet, bringing him to the tortoise's starting point; during this time, the tortoise has "run" a (much shorter) distance, say one foot. It will then take Achilles some further period of time to run that distance, during which the tortoise will advance farther; and then another period of time to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, Zeno says, swift Achilles can never overtake the tortoise. Thus, while common sense and common experience would hold that one runner can catch another, according to the above argument, he cannot; this is the paradox."

Quote :

""You cannot even start." "That which is in locomotion must arrive at the half-way stage before it arrives at the goal." (Aristotle Physics VI:9, 239b10) Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a fourth, he must travel one-eighth; before an eighth, one-sixteenth; and so on."

Quote :

"The arrow paradox "You cannot even move." "If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless." (Aristotle Physics VI:9, 239b5) Finally, in the arrow paradox, we imagine an arrow in flight. At every moment in time, the arrow is located at a specific position. If the moment is just a single instant, then the arrow does not have time to move and is at rest during that instant. Now, during the following instances, it then must also be at rest for the same reason. The arrow is always at rest and cannot move: motion is impossible. Whereas the first two paradoxes presented divide space, this paradox starts by dividing time - and not into segments, but into points."

A primary weirdness is that though these have sort of been mathematically disproven, they haven't really been disproven overall. They provide some serious fodder for discussions of topics involving how strongly our perception influences our reality or, beyond that, the universe itself.

What do you think?

1/19/2006 11:06:54 AM

I'm having a hard time relating this thread to cheese.

BAN DIRTYGREEK

1/19/2006 11:14:22 AM

ban everyone who doesn't support the cheese threads

1/19/2006 11:17:11 AM

OK, I didn't even understand the third one. As for the former two, the "paradox" is that the reader is led to believe that an infinite number of time intervals implies infinite time, which is not true. A series, which is a "sum of infinite number of terms" may nonetheless converge to a finite number, as I hope most of us here learned in Calculus.

So yes, they have been disproven.

1/19/2006 11:27:53 AM

So before I take a step I need to take half a step?


Oh god no, now I'm stuck in my chair because of this crap.

1/19/2006 1:09:20 PM

well, that's sort of the point - that your idea of the universe is far different from the reality of it

1/19/2006 1:20:03 PM

i don't get the first one

i can understand the second and third one, but at the same time i think they're stupid

1/19/2006 6:59:05 PM

These are dumb.

1/19/2006 7:20:17 PM

i remember talking about that first one in high school calculus. series and sequences and whatnot

[Edited on January 19, 2006 at 7:23 PM. Reason : .]

1/19/2006 7:23:29 PM

As my "doppleganger" has noted an infinite number of time intervals need not imply an eternity, we need only require that the time increments become suitably infinitesimal. More basic than this we should ask ourselves is it true that this is what happens just because we can write down sequences and series that describe continuous motion through the calculus.

As far as we can measure time viewed as continuous has sucessfully modeled physical phenomenon, but is it possible that there exists some fundamental timescale underwhich we cannot measure? On general grounds through the lense of quantum mechanics this seems reasonable. In you junior course on QM you should learn that systems that are bounded have discrete spectra. In contrast, the momentum of a free particle is not discrete rather it takes a continuous spectrum as it has no boundary conditions placed on it (it's free).

Now consider the universe as a whole, if it is bounded then it ought to follow that the spectrum of
the position operator is discrete. In view of relativity discreteness in time and space are linked.
Clearly such discreteness is at such a fine scale that we cannot detect it at the present, but if we
could get small enough then we might have to take quantum leaps ( very tiny leaps) in space and time. And if this is the case then we avoid all such paradoxes physically even in principal, simply because this view says that the continuum is an illusion. Instead the universe is made of many many
many many... many space time points, but not infinitely many. So all the sums we thought were infinite were actually finite. This probably solves alot of standing conceptual dilemmas in physics that
follow from trying to overextend an idealization.

Anyway, all mathematical models of "nature" are likely wrong. We see but a shadow of things unseen.
Any physical model extended to far is likely to do one of two things,

1. fail experimentally
2. fail calculationally ( it would be incalculcable )

Newtonian physics fails in the sense of 1. at high speeds. QM fails in the sense of 2 in describing
macroscopic physics, it's just inpossible to really even set-up the whole wavefunction for systems
that Newtonian mechanics very successfully describes through simple mathematics.

1/19/2006 7:50:14 PM

the first and to some extent the second one, is more clearly explained by this thought experiment:

suppose you are standing 8 feet from the wall of a room. you walk halfway to the wall, so now you are 4 feet from the wall. then you walk halfway again, so you are 2 feet from the wall. you keep doing this, and your distance from the wall decreases each time you walk halfway to the wall:

1 ft
1/2 ft
1/4 ft
1/8 ft
1/16 ft
1/32 ft

and so on.

but, you are always some distance from the wall (because the serious above goes on forever). so, you will never reach the wall.

1/19/2006 10:25:58 PM

or i could just walk 8 feet

1/19/2006 10:34:22 PM

^^ Your series fails due to rounding error.
You cannot walk 1/1048576 ft

And even if you could, it would take you an infinitesimal amount of time.

Of course, your excercise does mean one thing: As you approach, but never reach the wall, the current time is approaching the time of your arrival, but never reaching it.

1/19/2006 11:12:19 PM

the easiest way to circumvent paradoxes one and two is to aim further than the distance you are headed. instead of aiming for a specific point which is where your destination is, aim a certain distance further and then in progressing halfway there you will have achieved the destination.

the third paradox is met by the heisenberg uncertainty principle, which applies to particle physics. (i think) you cannot know both the speed and location of a particle at any one time.

1/20/2006 2:14:51 AM

Quote :

"the heisenberg uncertainty principle, which applies to particle physics. (i think)"

not that anything you said is wrong, it's just funny to see "heisenberg uncertainty principle" and "i think" in the same sentence

good point about aiming past the target.

1/20/2006 9:41:28 AM