Hit with insomnia the other night, I began thinking about Zeno's Paradox and what made it a paradox. Zeno, in case you don't know, was a Greek philosopher who believed that existence was static and that motion and change were rather impossible. (I got this out of an encyclopaedeia, okay? This is the "official" line on what he was talking about.)
The classic argument that Zeno hit upon to demonstrate this (based on a technique called reductio ad absurdum) involves a runner on a race course. Zeno argued that the runner could never finish the course because first he must run the first half of the course, and then he must run half of what is left, and so on an infinite number of times...
The end result of this infinite halving is that the racer must never reach the finish because he always has some distance left, of which he must always first travel half. Applying this to any distance shows that it is impossible for an object to move at all!
A couple of things occured to me while thinking about this. The most obvious thing was that this is a good argument for space (or time) being quantized. If either time or space is quantized, then that cuts off the infinite recursion and allows the runner to bridge the last quanta. -- Still, the fact that the paradox doesn't happen doesn't actually imply that space is quantized. (Note: an issue of Science News from 1998 has a cover article on the latest physics theory "space-time foam" which implies that space is quantized; the smallest length you can have is 10^-35 meters.)
Another flaw in the paradox, which allows space to be continuous, is that humans typically have the mistaken intuition that the sum of an infinite number of things must be infinite. In fact, if you add 1/2 + 1/4 + 1/8 + ..., the sum you get is 1: the exact length of the race. The statement of the paradox attempts to take advantage of this misconception in both the adding of the distances, and in calculating the amount of time it would take to run those infinite number of distances.
My friend Steve Colwell had this to say about it:
"It's not really a paradox, the Greeks just didn't know about all-nighters. The way it really works is you work harder and harder and get closer and closer to the goal, then when you're close enough you pull an all-nighter and finish the whole thing. That's how you break through the infinite regression paradox thing. Remember, you heard it here first."
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Date: Wed, 10 Jun 1998 06:23:26 +0800 To: thumper From: William Subject: Zeno's Paradox
I was searching for Zeno's paradoxes on the Net, and my search brought me to your we page. Apparently Zeno was not trying to prove something; he was trying to disprove the Pythagorean idea that space and time are made up of discrete units (that they are pluralistic). His paradoxes are meant to show only the absurdity of the Pythagorean idea. Zeno said that if space and time were indeed made up of discrete units, then through recursive division of either, we should eventually hit a unit that cannot be further divided. As this is clearly not the case with either space or time (any two lengths of space or time can always be further divided into two), it is absurd to claim that space and time are pluralistic. I believe Zeno wanted to show that if we accept space and time are pluralistic, then we should not run into that infinite division problem that we do in his paradoxes. Since our experiences consistently show that some people do finish races, then there must be something wrong with Pythagorean conception of space and time.
Date: Tue, 10 Nov 1998 15:25:45 -0500 From: Kenton Green To: thumper Subject: Re: zeno's paradox [Excerpted, copiously.]
> > The conclusion drawn for that Zeno's argument is apparently false, as > > you say... therefore someone might say that this implies that space (and > > time) CAN'T be infinitely divisible. > > Right, that was my argument, in my web page. But William says that > it was supposed to be the other way... to show that space CAN be infinitely > divisible. I'm not 100% sure I follow that logic -- I always feel like > I can follow it for a few seconds when I read William's email, but then > it slips away, like one of those optical illusions where it just depends > on your point of view. > Heh heh, yes I *definitely* know what you mean, I don't think I have been able to (partially?) comprehend more than one of Zeno's arguments at a time. Here is another wording of that conundrum we are discussing, that I think explains why I feel an appeal to calculus isn't satisfactory: To get to some arbitrary point A, you would first have to pass some point between you and point A (call it point B). But to get to point B you would have to pass point C, and to get to pt. C you would have to pass pt. D, etc. In other words, we have (presumably) an infinite number of obstacles in our way. But, we know that motion IS possible, so it must be that there are not an infinite number of obstacles: i.e. there is a finite number of check-points between me and my destination, at some sufficiently small level. Therefore, according to this argument of Zeno's, for motion to be possible space must be discrete/quantized. > > Ah, but is your explanation, that the geometric series (1/2+1/4+1/8+...) > > while consisting of an infinite amount of numbers is actually finite, > > really sufficient to explain away the problem? > > I guess I still think that the answer is yes, although I suppose it > depends on what "the problem" is. My explanation is trying to address > why it "feels" like the runner should never finish the race; and the > answer is still that it plays on the instinctual misconception that > the sum of an infinite number of things must be infinite. > As I (tried to) show above, regardless of the understanding that an infinite sum of numbers isn't necessarily infinite, the conclusion must still be reached that for motion to be allowed, I cannot be forced to pass an infinite number of check-points. Why doesn't calculus or limits apply? because I'm not (er, Zeno isn't) talking about a (converging) sum of numbers, like the geometric series, that sums to a finite number. The scenario is about an infinite number of separate entities (checkpoints), each as significant (each must be passed) as the last. So, in a sense the sum is 1+1+1+1+1+... which is NOT finite. One could try and resurrect calculus and limits by saying that each checkpoint, as it gets infinitesmally closer, is reached infinitesmally sooner, and so the rate ( = distance divided by time) approaches a finite number, using for example L'Hospitals rule. But this proves motion is possible by assuming motion exists! > > This is I feel the true nature of the paradox-- not that each argument > > taken alone is a paradox, but that together they imply that our > > conception of space and time is inconsistent. And if space and time are > > logically inconsistent, then perhaps Parmenides is right and these are > > just illusions (which is the original thesis most people feel Zeno was > > supporting anyways). > > AH! Now there's some real meat. This is the part that I'm really > ignorant of. I've heard of the arrow paradox, but my memory of it > is that it's about the same as the runner paradox (just a rephrasing). > Clearly, I need to go back and read what it actually says, now that > I'm old enough to appreciate the subtleties of language. > There are four well-known scenarios that Zeno creates (out of about 40 that are credited to him). They are the 'motion is impossible' one you were considering, the 'achilles and the tortoise' one that is related to the first (both of which strive to force one to conclude that space and time are quantized), the 'arrow' one, and the 'stadium' one (both of which conclude that time and space cannot be quantized). Sincerely, Kenton -- University of Rochester/Center for Optoelectronics www.ece.rochester.edu/~kgreen/
Date: Tue, 23 Apr 2002 16:04:10 -0600 (MDT) From: jesse k johnson To: thumper Subject: Re: Zeno Hi Bo, I'm still not sure if I totally understand Parmenides' idea on oneness, as it is so strange. He claims that the universe is unchanging, and that all change is actually an illusion. I suppose that from the standpoint that most philosopher's views were opposite from Parmenides', Zeno was trying to disprove everybody else's ideas (including the Pythagorians'). in disproving other ideas, he would have been adding weight to Parmenides'. What I recall from class is that Zeno thought up his paradoxes to prove Parmenides' ideas, which would actually not be in conflict with the view that he was trying to disprove the Pythagorians. I'm pretty sure that their ideas were just about opposite Parmenides'.