Well, the other day I printed out the Web version of that recent article regarding “Zenos Paradox” and the nature of time by Peter Lynds, the college dropout who’s got a lot of physicists and philosophers all riled up these days. (Sorry, I forget the site, you can find it by doing a Google on Peter Lynds). Some people call Lynds a modern Einstein, and some say he’s just restating the obvious. My old mind can’t quite keep up with either physicists or the philosophers any more, but they still interest me. So I tried to read the “lite version” of Lynds paper (you have to buy a copy of some physics journal to see the long version).

I was trained in basic calculus and physics long ago, so at first I couldn’t really see what the problem behind Zenos Paradox is. Zeno himself obviously didn’t know calculus; they didn’t have it when he came up with the problem. But if he did know calculus and the concept behind it, i.e. that time and space can be cut into infinitesimally tiny segments, usually called “dx” and “dt”, his problem would drop away.

(Zeno’s paradox involves the logic of motion and how it relates to time. It’s a thought experiment about a tortoise and a fast guy who decide to run a race. The tortoise is promised a head start, and thus accepts the challenge. This is a thinking man’s tortoise, but he still doesn’t know calculus. He figures that the runner will never catch him. Why not? Because by the time the runner reaches where Mr. Turtle was when the starting gun went off, Mr. Turtle will have moved forward an inch or so. By the time the runner closes that inch, Mr. Turtle will have moved another eighth of an inch. By the time the runner closes that eighth of an inch, … and so forth. In sum, the runner will get closer and closer, but will never pass the struggling terrapin.) (This reminds me of those brain-puzzler things, which usually start off like this: Three guys are looking for a hotel room one night, and finally find a place. The guy at the desk says $31 for the night. One guy puts in $11, the second guy ….)

In the lite article, Lynds explains Zeno and gives the calculus answer to his paradox. But then Lynds says that there’s still a problem. And that’s when he loses me. Actually, it seems as though he knows that he will lose me, because after that he keeps on repeating his conclusion. (When you hear the punch line repeated, you know the joke wasn’t all that good). The Internet article was supposedly meant to be a “Lynds’ Time for Dummies”, but it actually doesn’t do a very good job of explaining things. In one of the footnotes on the article, Lynds in effect tells you to shell out a few bucks and buy the main article if you don’t get it … another 21st Century huckster.

Just what is Lynds’ conclusion? That there does not exist an exact instant in time, e.g. precisely 10:00 PM. In reality, Lynd sez, all that exists are little intervals. Just how little are the intervals? Well, that depends on what you are using to measure time with. With a good watch, maybe you know time down to the hundredth of a second. With an atomic clock, maybe it get it down to a billionth or something. But you never get the uncertainty down to zero. Why? Well, Lynds doesn’t say this, and at one point even seems to deny it, but the ultimate limitation appears to be related to quantum mechanics. At some incredibly tiny point, you can’t split a particle any further. There is a fundamental quantum length, below which you can’t go. Since measuring anything is a function of seeing it (whether by regular light or by some other electromagnetic force), at some point you can’t get any more accurate than to say that there was an event that triggered a photon to be shot out, that the photon reached you, and you saw a little flash from it.

Let’s say that your eyes let you know exactly where this photon hit you, but because of Heisenberg and his darn Uncertainty Principle, you can’t perfectly know both the exact position and exact momentum of the photon. In our case, you know position, and thus you are a bit unsure about momentum; momentum is proportionate to wavelength, and wavelength determines what color you see. Thus you aren’t 100% sure about the photon’s wavelength. You think you know, but any good quantum physicist would tell you that you can’t be 100% sure that the color isn’t really off a little. Thus you don’t know exactly how long the waves are.

Therefore you are a little bit unsure as to exactly where that photon was sent from and thus how far it traveled. (I believe that you can tell where the photon was along the wave when it hit your eye based on how bright it seemed – thus the “wave phase” of the photon is not the problem; the problem is that you can’t know the exact wavelength, which you need given that the distance to the starting point is some multiple of it, with a slight adjustment for the “wave phase” of the photon when when it hit your eye). You can take measurements and narrow this down, but there is still a range of possible places, however tiny that range is, where the event took place that launched the photon (remember, you need other photons to make those measurements – they also have uncertainty). You know the speed of light for sure, but you don’t know exactly how far the photon went before it hit your eye. Using statistical techniques, you can get a 99% range, or even 99.999% range of distances. But that means that you will get an earliest time and a latest time when the event took place, relative to your seeing the photon that it launched. You’re stuck with an interval, in other words. You can’t do any better than that. Just no way.

Another approach to this is to remember that a photon acts according to a probability wave (under the wave-particle duality theories), which means you could have seen the photon in a different place, even given identical starting conditions; and conversely, that different starting conditions could have sent a photon with the same wavelength to the same spot where it hit your eye. You could never know for sure. The event could actually have happened a little bit earlier or a bit later, given the differences in distances that would have been covered, and you wouldn’t have known the difference. Thus, you’re a bit uncertain as to when or where the event actually took place. And you can never do any better, no matter how hard you try.

Therefore, you don’t and can’t exactly know anything. If you fire a cannon, your physics professor could give you some equations that will tell you how far away the cannonball will be at, say, 5 seconds after you shoot. But Lynds would say THERE IS NO “FIVE SECOND POINT”. Depending on how good your watch is, you DO know something about a little time interval that surrounds that hypothetical 5 second point. Therefore, all you can know about the cannonball is that it is somewhere between the two points where the physics equations say that the cannonball should be at the lower and upper limits of the 5 second mark (e.g., 4.999 and 5.001 seconds from shooting, if your watch is that accurate). If you can somehow come up with a quantum watch, your interval is a whole lot smaller — but there’s still an interval, not a perfect instant in time.

In sum, motion is really a blur, as any photographer suspects. Take a picture of anything moving, and the thing is a little blurry. Use a 1/30th shutter time and you really notice the blur. Set the shutter to 1/1000th and it looks a lot better. But there is still a little bit of blur. That’s just the way things are, if I read Lynds correctly.

Or maybe it’s just that Lynds is himself a bit blurry (it ain’t easy to think in his terms; but philosophers and physicists sometimes demand that we think in ways that hurt). Well, this all doesn’t mean that you can throw out your calculus and differentiation and integration lessons. You still gotta do your homework. Calculus is still a darn good and useful approximation of how things work. But at some point, reality is lumpy and bumpy. And I think that this is what Lynds is ultimately getting at. Now, as to whether that’s a better way to deal with Zeno and all that, I’m not sure. Calculus ultimately agrees with Zeno’s presumptions that you can keep dividing up a space into an infinite number of pieces (but then says that Zeno got the rules of calculus wrong). Lynds seems to say that you can’t keep on dividing time or space into smaller and smaller bits; at some point you can’t go any smaller, you have to use the smallest “mosaic piece”, and that piece puts the runner out in front of the turtle. And the race is won!

But of course, I could be wrong here.