h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

>In the example, however, the time is reduced to almost nothing and since anything multiplied by almost nothing equals almost nothing, the distance, of course, becomes very small.

This does not answer the problem. The fact remains that *almost nothing* is still *something*; so how does the time (infinitely divided, but still more than nothing) ever reach the limit where the hare can pass the tortoise?

Zeno's paradox still stands. The only reason motion can happen is that in the real world things are not infinitely divisible. There is a quantum that can be leapt over. How? Nobody knows, but it's a good thing there is.

By the way, this entry conflates two separate stories: in Aesop's fable the hare has no trouble passing the tortoise, or would have if he bothered to try. In the other story it is Achilles who cannot pass the tortoise on account of the infinite series of tiny steps Zeno requires him to take.

The reason, as I understand it, that the race paradox doesn't hold is because it's the sum of an infinite geometrical series, both of times and distance. This means that in a finite amount of time, the hare/Achillies will reach the tortoise, then overtake. The problem comes when people visualise it in the stages; there is automatically a small pause inserted between each movement, and this will cause the time to become infinite. This pause does not exist in the real race, and the time sums to a finite value.

It's one of Kant's antinomies: can space (or time) be divided infinitely or not? There's no exclusive answer. It can and it can't.

That's the first time I've seen the problem tackled in terms of visualisation.

I don't follow the statement that 'there is automatically a small pause inserted between each movement, and this will cause the time to become infinite'. Each movement takes time to happen in; if the series of movements is infinite, you end up with infinite time without having to insert any more, unless you allow events to occur without taking any time -- and Zeno would probably say that that's just as counterintuitive as the Parmenidean philosophy he sought to defend. (The point of his entire book being to show that common sense produces results no less seemingly absurd than Parmenides' claims, it's arguable that the need to talk in terms of geometrical series and so forth qualifies as a vindication; either way, you end up relying on logical thought instead of trying, like Antisthenes the Cynic, to refute the argument by getting up and taking a step.)

Ah, but that's the point. The sum of an infinite series often converges to a finite limit.

For example, the old thing about the frog, trying to get across the pond, and each time he leaps half the distance that is left. He never makes it, ignoring all silly things like whether space is discrete. Therefore, the sum of the infinite series converges to 1 (assuming the total length to be traversed is 1).

Smilarly, if you take a bouncing ball which loses energy on each bounce such that it reaches half the height of the previou bounce and hence takes half the time, you end up with a finite time for the ball to lose all it's energy, via the sum as well as the logic:

Let S(n) be the total time taken by n bounces. Say that the time for the entirety of the first bounce (it's thrown from the ground, this time is to get to max height and return to ground) is 2 seconds.

S(1)=2

S(2)=2 + 2*(0.5) = 3

S(3)=2 + 2*(0.5) + 2*(0.5^2) = 3.5

S(n)=2 + 2*(0.5) + 2*(0.5^2) + ... + 2*(0.5^(n-1))

As n tends to infinity, the last term becomes smaller and smaller, and with n being infinite, there is no last term. This means we can sum the infinite series (using @ for infinity, in want of a better symbol) as such:

S(@)=2 + 2*(0.5) + 2*(0.5^2) + 2*(0.5^3)...
0.5*S(@)=2*(0.5) + 2*(0.5^2) + 2*(0.5^3)...

Subtracting the second from the first:

0.5*S(@)=2
S(@)=4

Therefore there is a finite sum. Similarly, for the tortoise/runner argument, there is a finite time reached, and a finite distance.

My comment on visualisation is about where people have a problem with this. When people describe the paradox, it involves doing the movements step by step (runner reaches old tortoise position, tortoise has moved on. Repeat) In considering this, there is effectively a pause placed in between each movement. even if this pause is a millionth of a millisecond, you can see from the sum above that it no longer works and the sum becomes infinite. That is because people visualise an infinite number of movements, which is what you describe.

The problem with that isn't the time of the movement. The problem is that you see it as an infinite number of movements rather than one continuous movement because you see a pause between each described section, which in reality doesn't exist, regardless of whether time and space are continuous or discrete.

'Ah, but that's the point.'

No, it isn't. The point is that an infinite series by definition doesn't end; having a finite sum is of no use if Achilles never reaches it.

For example, one can ilustrate the Stadium version of the paradox using a disc: divide it into halves, then halve one of the resulatant semicircles, and so on ad infinitum. It's obvious that you have a finite sum, since you begin with one and divide it up. Since you've dividing into halves, if you were to move some imaginary counter from the semicircle to the surviving quarter-circle to the eighth-of-a-circle, and so on, at what point would you reach the end?

Well, since you've been dividing in two ad infinitum, there'll always be another segment in front of you. The division was infinite, lit. 'without end'. Of course, the area of the part of the disc no longer being divided has converged on the total area of the disc, but it'll never actually reach it.

In the case of Achilles, you can have a finite distance at which he would draw level with the tortoise, but first he has an infinite number of steps to get through, hence infinite time. So you make time infinitely divisible too, as Aristotle did -- but if each step in an infinite series takes _some_ time, then the series... takes infinite time.

The fact that a finite sum for time is expected _reinforces_ the paradox; now it's not just your senses it contradicts, but your maths as well. To solve the paradox you have to show that Zeno's reasoning itself is invalid and/or has one or more false premises (or deny that space & time are infinitely divisible); by circumventing his reasoning all you've actually shown is that the paradox is paradoxical. Never mind the sum; how does Achilles reach it?

For those who like visualisations, here's another.

Achilles racing the tortoise can be represented on a graph. Assuming they run at constant speeds, their time-against-distance lines will be straight, and will cross at an easily determined point.

Zeno manages to turn them into asymptotes. How?

Think of a fractal, which you can zoom in on indefinitely. He zooms in as the lines are about to cross, and if he can zoom faster than the crossing approaches (which imagination can do) he can put off the crossing event indefinitely. Which he does.

I probably ought to clarify one point, since my statement that 'if each step in an infinite series takes _some_ time, then the series... takes infinite time' is in retrospect very poorly phrased. I should proofread more carefully...

What I should have written was that although the infinite series has a finite sum, an infinite series by definition still has no _end_; and consequently it isn't necessary to get to infinite time by adding finite times together. Instead one can rule out a finite duration: if a series of events can occur within a finite time, then the series has an end, since if it didn't have an end it would continue after the end of the finite period. (If Achilles catches up with the tortoise, his attempt to catch up comes to an end.) If the series has an end, it isn't infinite. ('Infinite'='endless'.) Therefore, an infinite series of events cannot occur in a finite time period. Therefore, if an infinite serise of events can occur, it can do so only in an infinite time period.

(Of course this assumes, as we naturally do with Achilles, that we have a series of events taking place consecutively, and that the series takes time; I think trying to recognise that was what produced the misleading exposition.)

"You can't add up an infinite number of things and end up with a finite total".

"You can if they're small enough."

Apparently Berkeley's objections to Newton's infinitesimals (e.g. that he made them out as both something and nothing at the same time) were only formally resolved in the 20th century. http://en.wikipedia.org/wiki/George_Berkeley says, under "The Analyst Controversy"

It was not until 1966, with the publication of Abraham Robinson's book Non-standard Analysis, that the concept of the infinitesimal was made rigorous, thus giving an alternative way of overcoming the difficulties which Berkeley discovered in Newton's original approach.

I share a surname with the bold Abraham, but cannot claim kinship.

Unfortunately, one way of reading Zeno's 'Stadium' paradox (in which things pass other things moving in the opposite direction, and each of these things passes further, stationary things - it gets a bit complicated) is as an argument that motion is impossible if space *is* quantized, though Zeno didn't frame it in those terms. The paradoxes throw you back and forth between themselves. You try to pass the tortoise by positing quantized space and you run into the stadium, so to speak.

Parmenides argued, in a slightly peculiar way, that all was one and unchanging. Many people said that sort of talk was just nonsense, and Zeno said well, if you stop and think about it, assuming that motion and plurality are possible gives you just as much bother; here, watch. He then demonstrated that you can't walk to the door, and that just as you think you've worked out how, you're prevented from passing anyone moving in the opposite direction.

Sure, it looks as though all these things are possible, but reason tells you otherwise, so why trust your senses?

Russell famously said that every generation solves Zeno's paradoxes, and then every subsequent generation feels the need to solve them again, or something like that. So he claimed they were finally solved, and now people argue that he was missing the point. Heigh-ho.

Isn't life wonderful! The Hindus said centuries ago that all appearance is illusion; since then people have been saying "that's just a cop-out!"

I totally agree. There is no paradox.

I meant to say I totally agree with LQ above. Is there no way to edit a post on this forum?

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