Zeno’s Paradoxes

Zeno of Elea (490-30 BC) formulated (deep) paradoxes on motion, which have haunted physicists into our time, relating to the following basic questions:

  1. Distance vs Time?        (Achilles and Tortoise)
  2. Motion appears impossible.  (Motionless Arrow)

Paradox 1 is not so difficult to resolve, so let us start with that. The question can be reduced to the following: Suppose you want to walk from A to B, a distance of 1 mile. If you think about it, you can plan the walk so that you first cover 1/2 with thus 1/2 remaining. After a breath you then walk 1/2 of what is remaning, thus a distance 1/4 into a total of 3/4 with 1/4 remaining. You take a new breath and walk 1/2 of the remaining thus 1/8 into a total 7/8.

You continue like that but you understand after a while that there will always be a little distance left to reach B: Whatever is left you only cover half of that after a breath. It seems that you will thus never reach B. But you know that without breaking down the path into little pieces, it is certainly possible for you to arrive at B.  We have thus found what seems to be a paradox, a contradiction: It seems that you cannot arrive at B, while you know that you certainly can arrive at B simply by continue walking until you are there.

So you have something to think about. Can you resolve the apparent contradiction?

Hint: Motion is described by time stepping dx = v*dt, say with v=1, thus as dx = dt or x = t or t = x. The total T time after n steps as above, is then

  • T = 1/2 + 1/4 + … +1/2^n + n*C,

where C is the time for each breath. If C > 0 then the total time T increases with n*C and so tends to infinity while there is always 1/2 of the remaining distance to cover and so you will never reach B in finite time.  However, if you do not stop to take breaths, that is C  = 0, then the total time T approaches 1 as x approaches 1 and so you will reach B after time 1. Net result, without stopping repeatedly to take a breath, you will certainly reach B in finite time. Try it!

We now turn to Paradox 2: Motion appears impossible. Zeno argued: Look at a arrow flying through the air. At each moment the arrow is still, and so is motionless.  But something which is motionless  cannot move. So the arrow cannot move, but still is seen moving. This may take your breath: The arrow you see flying is both still (at each moment) and not still = flying.

Let us see how DigiMat approaches this situation. We know the basic equation to read

  • x(t+dt) = x(t) + v*(t)*dt

x(t) the position of the arrow and v(t) the velocity, at time t. Time-stepping this equation from at starting point x(0) produces x(dt), x(2*dt), x(3*dt) and so produces x(N*dt) after N time steps. We see this as repeated jumps in position from x(t) to x(t+dt) .

So we would say that DigiMat resolves Zeno’s Paradox 3 by simply letting the arrow jump from one position to another. So the arrow is still at each moment of time t, but jumps from t to t + dt.

Is this a convincing resolution of Paradox 2? Not really, since we now have to explain how a jump is possible. We know that rabbits can jump, but do arrows actually jump from one position to the next? Taking photos of the arrow may support such a picture but if you refine the time-resolution you will find new intermediate positions all time, like a big jump consisting of a sequence of smaller jumps, but there does not seem to be a smallest jump shortest in time.

We now turn to a better resolution based on wave motion. We know that when we see a wave moving over a water surface, it is not the water itself which is moving along with the wave, but the result of water molecules moving around in little circles, basically up-and-down. The water wave is seen to moves horizontally, while the water itself moves basically only up-and-down, not horisontally.

We thus see a wave moving horizontally, but the water is not moving along with the wave, only up and down. You can see this by putting a cork on the surface to see it moving up-and-down as the wave passes but it does not move along with horisontal wave. In a sense motion thus is fictional. Zeno was right! But still motion is a reality, which we perceive when we are hit by a wave: What is moving is energy and not water particles. Motion of energy is not fictional.

You can see this phenomen in this simulation: Wave through wave carrying. medium.

Compare with Slinky1Slinky2 ): Wave carrying medium.

Compare with Shallow Water simulation (horisontal motion of water particles small compared to vertical).