Zeno's Paradox

Zeno of Elea is responsible for a number of Paradoxes, however his most famous one is probably Achilles and the Tortoise (Two characters which feature prominantly in GEB, BTW) which deals with the impossibility of motion and/or change. This leads me to believe that motion is an illusion and reality is false.

See: There is no spoon.

The best known of Zeno's various paradoxes is Achilles and the tortoise. It goes thusly:

A turtle starts walking. After giving it a bit of a head start, Achilles shoots an arrow after it. The turtle moves an foot. The arrow moves twenty feet. The arrow will be impaling the turtle shortly... Or will it?

The arrow flies the last twenty inches towards its target, but while it is traversing those twenty inches, the tortoise moves one inch. As the arrow is barreling its way through the last few millimeters, the turtle is lumbering along. The arrow goes one millimeter, the turtle goes one twentieth of a millimeter. Yes, the arrow is getting closer, but as long as the turtle keeps moving the arrow will still have (in this poorly estimated example) one twentieth of that distance to go, and in the time that it takes the arrow to cross that distance, the turtle will have moved just a little farther on. The arrow can never quite catch up.

Well, obviously it can, but Zeno couldn't figure out how it logically could. Hence the paradox: we could think up this logical argument that seemed perfectly correct, but at the same time, we can see that moving things do get hit by arrows, all the time. A most ingenious paradox.

It turns out that a infinite series can add up to a finite number. Someone who knows more about the mathematics involved will have to post the formula.

Zeno wrote of a lesser known paradox known as The Stadium, meant to disprove the existence of quantized time (though, of course, Zeno didn't use the term quantum). It goes as such:

Consider three ranks of 10 soldiers, lined up as such:

     a a a a a a a a a a
     b b b b b b b b b b
     c c c c c c c c c c

Each soldier is separated by one pace. In the space of a quantum unit of time (the smallest non-divisible unit), rank B takes one pace to the right, while rank C takes one pace to the left. The resulting formation is thus:

     a a a a a a a a a a
       b b b b b b b b b b
   c c c c c c c c c c

The leftmost soldier in rank C is two paces to the left of the leftmost soldier in rank B. At some point in time, in order to get that distance away, C₁ had to be one pace to the left of B₁. However, the movement took place in the span of a quantum time unit. Therefore, there had to be a smaller possible span of time than the supposed quantum unit.

I've never seen this one disproven; certainly, the math involved would seem to be a bit more involved than with Achilles and the Tortoise. If anyone could find a way to work around this (if one exists), enlighten us. Alternatively, if it can't be explained, then time indeed cannot be quantized.

Zeno's paradox cannot be disproven with algebra or any math. Why? Because it is not a mathematical paradox.

Obviously, Zeno knew that Achilles can catch up with the tortoise. Zeno was not trying to prove that Achilles cannot catch up with the tortoise, or that a runner cannot run around a stadium.

Zeno's point was to illustrate that just because something is logical does not necessarily mean it is true. The flaw is not in logic as such. The flaw is in human understanding (or lack thereof) and applying logic to premises that appear correct.

In the case of the runner the "logic" was that the runner cannot run around the stadium because first he has to run around half of the stadium, then he has to run around half of the remaining distance, then around half of the remaining distance, and so on, ad infinitum.

It is irrelevant that our current knowledge of mathematics can easily disprove it. At Zeno's time it appeared correct. The premise was false but was perceived as true.

At our own time some other premise may appear true and still be false, simply because our knowledge and understanding are limited.

What Zeno's paradox teaches us today, as it did in Zeno's time, is that we should not be arrogant about our knowledge and understanding, that we should validate all theory by practical observation, and that we should always be ready to admit that we can be wrong and often are. In other words, we should be ready to change our theory no matter how feasible and logical it may sound, and how convinced we may be about its validity.

Zeno's paradox is not about mathematics, it is about keeping an open mind. And the need for that cannot be disproved.

Zeno's Paradox is primarily illustrative of the dangers of thinking too rigidly and/or logically whilst ignoring the "real world" (he says, laughing).

Zeno was most likely aware that his "paradox" was entirely moot. People (and other things) do not move exclusively in half distances. Or any other precise division of distance. As such, I submit that the real lesson to be learned from the paradox is that esoteric mathematical problems merely describe the world; they do not control it.

I fully expect flames from people crying out that mathematics is the language of physics, which does indeed "control" the world. Regardless, I stand by my interpretation.

Here is a geometrical way to prove that the sum of an infinite series can indeed be finite:

What if you would take a square, add half of it onto itself, then add half of that, plus half of that, forever? Since you would keep adding all the time, even if the pieces being added grow smaller for every time, you would get an infinite amount right?

Wrong. Here's how to prove it:

Take a square. Next to said square, add half that square. Add half of that square below the second square. Proceed ad infinitum. Illustration:

1.  -----------------
    |               |
    |               |
    |               |
    |               |
    |               |
    |               |
    |               |
    |               |
    -----------------

2.  ---------------------------------
    |               |               |
    |               |               |
    |               |               |
    |               |_______________|
    |               |
    |               |
    |               |
    |               |
    -----------------

3.  ---------------------------------
    |               |               |  (I do wonder if
    |               |               |   images on E2
    |               |               |   would be such
    |               |_______________|   a bad idea
    |               |       |       |   after all.)
    |               |       |       |
    |               |       |       |
    |               |       |       |
    -----------------       ---------

4.  ---------------------------------
    |               |               |
    |               |               |
    |               |               |
    |               |_______________|
    |               |       |       |
    |               |_______|       |
    |               |       |       |
    |               |       |       |
    -----------------       ---------

5.  ---------------------------------
    |               |               |
    |               |               |
    |               |               |
    |               |_______________|
    |               |       |       |
    |               |_______|       |
    |               |   |   |       |
    |               |   |   |       |
    -----------------   -------------

Ok, you should've gotten the point by now. Anyway, as you can see, you can keep adding pieces as much as you like, but you'll never ever exceed the size of twice the original square. Thus, the infinite series will converge.

Infinities are irrelevant to the "paradox". If you do introduce an infinity by creating an infinite sum, well we all know that the series converges, and the maths is easy. But you don't need any of it.

Look at the race in ordinary finite terms. There's a start, a finish, and the space in between. The start is Achilles' line in the sand, or the arrow leaving the bow, or whatever. The finish is the tortoise, or a ribbon stretched between poles. It doesn't matter. The "race" has three components, but you don't do three things:

1. leave the starting line;
2. run along the racetrack;
3. cross the finishing line.

These aren't three things, which you do in that order. There's only one action, the running. This has a beginning and an ending. But they're part of the action of running. Similarly, when you run the whole racetrack, you progressively run one-third, two-thirds, then all of the distance. But you don't do three different things,

1. run the first third;
2. run the middle third;
3. run the last third;

It's not an obstacle course where the three thirds are distinct phases. It's just that, whenever you run one distance, you can, if you choose, look at it and mentally divide it into a first part, a middle part, and an end part.

You can perform this subdivision before, during, or after the event: you can glance back and think "That tree was where I was half-way", or study a map and think "There's a fire hydrant at the three-quarter point, so I'll know where I am when I get to that". Or you can do it physically instead of mentally: you can measure the distance and scratch a line in the sand or plant a golf flagpole at the half-way point.

You can divide up the distance any way you like. This doesn't split the task up into multiple tasks.

The race organizers can put banners over the route saying "one third done", "two thirds done", and "FINISH". In the middle third they might put small white poles to divide that third into four parts.

Now to return to the familiar formulation of the paradox. You first run half way. Now all you have to do is run the other half.

Or you run half way then keep on going and run another quarter of the way. You've covered three-quarters of the distance. What remains to be done? This: run the final quarter.

You don't have to get out more flags, or scratch more lines. You don't have to divide the remaining quarter into two eighths. It's only a single piece of one quarter. 1/2 + 1/4 doesn't yet add up to 1, but 1/2 + 1/4 + 1/4 does.

If you happen to be fifteen-sixteenths of the way along, you're not finished yet. The sum 1/2 + 1/4 + 1/8 + 1/16 doesn't add up to the whole distance, nor can any such sum, however fine you cut it, because you haven't counted all the parts. The remaining one-sixteenth does make it add up to the whole.

The racetrack in this case is divided not into 1/2 + 1/4 + 1/8 + 1/16 + ...; but rather into 1/2 + 1/4 + 1/8 + 1/16 + 1/16. That's five sections altogether. You only ever have finitely many pieces in the sum: the ones you've already got through, however you choose to divide them, plus the ones that are still to come, however you divide them.

This explanation was put forward by Gilbert Ryle in his book Dilemmas.

A non-mathematical solution to the problem, If I may:

The logic of Zeno's statement requires that the runner reaches the "half-way" point.

However, the conclusion states that the runner cannot get anywhere.

The paradox assumes as true that which it is trying to prove false.

This is "true implies false" logic and therefore makes Zeno's statement false.

Zeno also has another paradox, that of the two arrows:

Consider a moving arrow as it would appear at an instant in time, and consider an arrow that is standing still as it would appear at that instant. The two ‘snapshots’ are identical. Thus (so Zeno argues) there is no difference between an arrow that is moving and one that is standing still, and the whole idea of motion is an illusion. When the paradox is resolved, it is not necessarily proved that motion exists, only that Zeno’s reasoning does not prove that motion does not exist.

Zeno’s arrow: The description of the arrows that is given by Zeno does not distinguish between the still and the moving arrow. Thus, the alternative to admitting that he is right in saying that there is no difference between the two arrows, is to say that there is something wrong with his description, i.e. that giving the position of the arrow at a single time (a ‘snapshot’) is not a complete description of the arrow.

Indeed, there are two ways in which the description might be incomplete: (1) An appropriate description may be one containing information about more than one time (e.g. a collection of snapshots, rather than just one). (2) The description of an arrow at one instant of time may contain more than just information about position (i.e. you ignore relevant information if you just look at a snapshot).

Consider the intuitive notions of past, present and future. The present ‘feels’ special: the future seems indefinite, the past seems to live only through traces in the present. But what is it that distinguishes the present from the other two? There seem to be two possibilities: (a) The present is indeed real and past and future are not. But that means presumably that the past was real, the future will be real, and the present is real now. But there is a problem with what is meant by "real now" (b) Past, present and future are equally real. There is no absolute distinction between them, only a relative distinction between events that are earlier, simultaneous or later than each other. But is that enough to explain why the present feels special (or why time seems to flow)?

Pursuing option (1) above seems to commit one to option (b), in the sense that if the appropriate description of an arrow - one that makes motion real rather than illusory - includes information about the arrow at different times, then it would seem that these different times must all be equally real as well.

Something needs to be added to the instantaneous description of the arrow. This something will be the instantaneous velocity of the arrow. Intuitively, one might rather think of velocity as defined between two points (the distance between the two points divided by the time needed to travel between them). Indeed, it is also possible to talk about velocity as a property defined at one single instant. This is made possible by use of the calculus. And in fact, Newton, who was one of the inventors of the calculus, developed it precisely for the purpose of talking about instantaneous velocities (and instantaneous accelerations).

Imagine an object which is at point P at time t, and moves to Q by time s (say, moving in a straight line). Its (average) velocity v between the two points P and Q is v = |P-Q| : |t-s| where |P-Q| stands for the distance between the two points and |t-s| for the (positive) difference between the two times. Velocity in this sense is clearly a property of the object pertaining to more than one point or instant. There is one special case, however, when velocity thus defined can be said to be a property of the object at any single time; namely, when the average velocity of the object is the same between any two points along its trajectory (uniform motion).

Assume that the object goes through the points P and Q, P’ and Q’, P’’ and Q’’ etc. (not necessarily in that order), at the times t and s, t’ and s’, etc., respectively. If for any pair of such points v = |P-Q| : |t-s| = v’ = |P’-Q’| : |t’-s’| = v’’ = |P’’-Q’’| : |t’’-s’’| = ..., it can be justified in being said that not only that the object has the same velocity between any two points, but that it has a velocity at all points and that this velocity is always the same.

The more difficult case is when these average velocities are not all equal, and this is where one needs the notion of limit defined above.

To define the instantaneous velocity at a time t (when the object is at point P), let us consider any infinite sequence of times t1, t2, t3, ..., converging to our given time t, and consider the corresponding points P1, P2, P3, ..., where the object is at those times. The average velocities can be calculated between P and P1, between P and P2, between P and P3 etc., thus obtaining a sequence v1, v2, v3, ... If this sequence has a limit v, this limit can be taken to be the velocity of the object at the time t. In the terminology of the calculus, taking such a limit of rates of change is called taking a derivative.

But now, there is a property of the arrow at a single time, namely the instantaneous velocity, which can distinguish between a moving arrow and an arrow that is standing still.

So we can resolve Zeno’s paradox by saying that Zeno is not giving us the complete description of the arrows at a single time; indeed, the information given on a snapshot of the arrows does not include their velocities, and is thus an insufficient basis for a distinction. Zeno is trying to convince us that there is no difference, but he has not given us all the information we are entitled to.

As whizkid says, the problem with Achilles and the tortoise is not really one of logic or maths, but one of wrong premises or assumptions.

What is the wrong assumption?

What seems to be the problem is that each time Achilles reaches the place where the tortoise started this time-step, the tortoise has moved on a smaller distance.

t0 : A---------t-----------
t1 : ----------A----t------
t2 : ---------------A--t---
t1 : ------------------A-t-

The wrong assumption is illustrated by the figure above. Each line represent a different point in time. But the points in time are not linearly distributed. In fact, we look at smaller and smaller time-steps, since both Achilles and the tortoise have constant speeds (but move shorter and shorter distances). Before Achilles reaches the tortoise, the size of the time-step goes towards 0.

This may or may not be obvious when you think about the paradox the first time. If you keep the time-steps constant, Achilles will naturally pass the tortoise (and indeed, the paradox does not occur).

Surprisingly, nobody has yet given the mathematical explanation as to why the "paradox" of Achilles and the tortoise (or the runner and the stadium) is no paradox at all. People have pointed out that with today's understanding of mathematics, we can easily show that the infinite series have a finite sum. Zeno was born somewhere around 495-480 BCE. Newton and Leibniz are credited with the discovery of The Calculus not until the 17th century of the common era.

Let's take a look at the geometric series 1/2 + 1/4 + 1/8 + ..., which describes the segments that the runner must move around the stadium.

This geometric series has ratio r=1/2. We will prove that a geometric series with a common ratio whose absolute value is less than 1 converges, and that its sum is described as a/(1-r), where a is its initial term and r its ratio. In the case of the runner, a=1/2 and r=1/2 . This series converges, and its sum is a/(1-r) = (1/2)/(1-1/2) = (1/2)/(1/2) = 1.

Each term of a geometric series is obtained by multiplying the previous term by the common ratio, r. The series can thus be described as a + ar + ar² + ar³ + ... + ar^n-1 + ..., or Σ(n=1, ∞, ar^n-1).

We can define the nth partial sum s_n = Σ(i=1, n, ar^i-1) = a + ar + ar² + ... + ar^n-1.

We know that when r=1, the series is a + a + a + ... + a. In this case, s_n = na, which goes to infinity as n grows to infinity. In this case, the series diverges.

When r≠1, s_n = a + ar + ar² + ... + a^n-1. rs_n = ar + ar² + ... + ar^n-1 + arⁿ. Subtracting these equations, s_n-rs_n = a - arⁿ = s_n(1-r). So s_n = (a(1-rⁿ))/(1-r).

when -1 < r < 1, rⁿ goes to 0 as n goes to infinity. So lim(n→∞, s_n) = lim(n→∞,(a(1-rⁿ)/(1-r)) = (a/(1-r))(1 - lim(n→∞,rⁿ)) = (a/(1-r))(1-0) = a/(1-r).

So there we have it. A geometric series with |r|<1 converges, and its sum is a/(1-r). The length of each successively smaller segment that the runner must run around the stadium is a term of the geometric series 1/2 + 1/4 + 1/8 + ..., in which r = 1/2. This series' sum is (1/2)/(1-(1/2)) = (1/2)/(1/2) = 1. So the runner does run each half of each segment, and eventually runs around the whole circumference of the stadium.