If the universe is quantized, it would indeed solve the paradox, because then there wouldn’t be an infinite number of small distances.
I’m not saying the infinitessimals issue doesn’t also solve it, but the paradox simply does not exist in a quantized universe.
<sigh> Yes, yes, I know that…it was a joke post. I’m sorry…I’ll never do it again.
BTW, this statement is a lie. 
<laughs evilly, and goes to find the turtle, which is somewhere on Crete>
My apologies about the misspelling of Zeno. I also apologize, in advance, for the misspelling of other words in this thread, including “mispelling.”
So the problems with Zeno’s paradox are:
…and, possibly…
Thanks, everybody!
I think I’ll go punch the fellow who mentioned it and stuff a tortoise shell down his throat now.
IANAGenius or IANAMathguy, but here’s my thoughts…
Zeno’s paradox only deals with REQUIREMENTS. The fact is that you’re moving.
You say: “You have to go through this infinite number of points, so you’ll never get there.”
I say: “Fine” and take a step. I’ve just passed through your infinite number of points. Requirements met.
I think the problem is that our number system does not relate perfectly to the real world. Like the whole .999~ = 1 thing.
Basically what I’m trying to say: You’re just passing through an infinite number of points. Points are an imaginary thing, not something real.
Right. You don’t have to stand there with pencil and paper and ADD the numbers up…you just have to walk. If you had to do the former, then it would take you forever.
The thread below explores that issue in some depth.
“Stupid math question”
http://boards.straightdope.com/sdmb/showthread.php?threadid=71810
Let’s settle this once and for all:
THE UNIVERSE IS NOT QUANTIZED. (or is that quantatized?
Thank you.
Now, that said, we can look at the paradox from another angle.
Assume you DID slow down and DID actually have a measurable amount of time that you halved the distance.
Well, then eventually, you would come to a point where you were so close, we couldn’t MEASURE the difference between your position and the turtle’s.
And since we couldn’t MEASURE it, for all intents and purposes we say it’s not there. That’s one way to look at the quantum mechanical answer… but the assumptions are so absurd as to be ludicruous. Frankly, nothing in the world can move that way because we just have too much energy.
Pah! Maybe in your little nancy-boy eastern colleges…
Well, I seem to recall an article in Scientific American several years ago that claimed it was. I certainly could be misremembering, though. What is your source?
Here’s an article that seems to support the idea that the universe is quantized at the Planck Length:
http://faculty.washington.edu/smcohen/320/GrainySpace.html
Not 100 percent sure where it comes from, it’s on a Philosophy professor’s web page UW, but at the bottom it says “Copyright 1999 The New York Times Company” so it may have originally come from the NY Times.
And, from the prestigous science journal “Nature”:
“General relativity remains a purely classical theory: it describes the geometry of space and time as smooth and continuous, whereas quantum mechanics divides everything into discrete chunks.”
douglips wrote:
So what was the deal with those Greek mathematicians? They did some things that seem amazing to me, such as prove that the square root of two is an irrational number. But when it comes to the sum of the series 1/2 + 1/4 + 1/8 + 1/16…, isn’t it obvious to any non-idiot who thinks about it, that the longer you go, the closer you get to one? You didn’t need Newton or Leibniz (sp?) to figure that out!
There’s a parallel post about whether .999… is in fact equivalent to 1 that sounds suspiciously Zeno-esque…
By the way, you spelled Leibniz correctly. Trivia moment - I was at a rare book exhibit, and they had a first edition of Newton’s Principia - and it was Leibniz’ personal copy with notes in the marginalia! Can you imagine what he might have said?: “Damnit - I thought of that first!” or: “sure, you could state it that way, but my way is more elegant!”. Apparently the collector who bought it in the 1920’s picked it up from a university (Leiden?) who sold it because it was a dupe and the other copy was not written on!!!
WordMan
It has always seemed to me that the easiest-to-grasp way to address this paradox is this:
Given your particular “racecourse,” you are free to speak of it as being constituted by however many subdivisions as you want–even an infinite number. Just remember that the amount of time it take to traverse each subdivision will shrink correspondingly with its length. Divide the racecourse in two–each length is traversed in half the time. Divide it infinitely–each length is traversed in an infinitesimal time. The total time will always be the same. If Achilles could beat the tortoise on a 4-part course, he could beat the tortoise on an infinite-part course.
Note: philosopher Gilbert Ryle wrote an essay on this paradox.
Okay, remember reading that one in NYTimes. Really got my goat at the time. Yeah, I say, Planck length, shlanck length. I don’t understand the superstring theorists anymore than the next guy. Even if the universe is quantized (and that’s a big if)… we’d have problems measuring the tortoise and Achilles far before we got to that grainy character. Your Nature article didn’t work for me… Try again, maybe?
I know, I know, the philosopher in me too doth protest… but I think the article actually makes one good point… we’ve been arguing about lumpiness/smoothness of the universe for millenia, inductively we need to accept a continuum… and especially as far as the comments on this post are concerned. For good ol’ Zeno, quantization of the universe (oooh, I cringe!) does not have the slightest effect.
(Or it has all the effect in the world, seeing as at Planck length, that’s the way superstring theory tends to veiw reality. Again, something that loses me every time.)
Maybe the best way to get to the Nature artice is to go to http://www.nature.com and quick search for “quantum mechanics divides everything into discrete chunks”. You will have register to view the article, but it’s free.
To be honest, I really fail to see why people say a quantized universe wouldn’t make a difference in the paradox. If there isn’t an infinite sequence, then how is there still a problem? At some point, you cannot take a half-step towards your goal, because there is only a Planck-length left. You must, at that point, step to the end. End of paradox.
Revtim, I don’t necessarily agree with you about the quantization of the universe, but I do agree that if it were a quantized universe, Xeno’s paradox would be irrelevant. The basis for Xeno’s four paradoxes (someone mentioned there were eight, I’d like to know when these other four turned up) is whether or not time and space are discrete. That said, I have a problem. The proponents of quantum mechanics purport that space is quantized. They do not purport that time is discrete, rather, it is continuous. Time and space are interchangeable. Interesting dilemma, eh? Maybe I should take this one to GD.
Nen, I think quantum physicists do in fact purport time to be discrete and non-continuous. Those guys think everything is quantized!
Yeah, that dawned on me right after I posted. My damn almost degree in physics ain’t worth the pulp it might be printed on. Anyway, here we go so I don’t hijack this thread.