Resolution of Zeno’s Paradox of Particle Motion

resolution by wave motion



We suggest a resolution of Zeno’s 2500 year old paradox about the impossibility of material particle motion, based on viewing particle motion as ultimately resulting from immaterial wave motion, which is possible.

If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless. (Aristotle, Physics VI:9, 239b5)

But maybe that is our mistake: maybe there are no particle positions and velocities, but only waves. It is just that we try to fit the waves to our preconceived ideas of positions and velocities. (Stephen Hawking 1988)

                                                                   Arrows in motion.

Zeno’s Arrow Paradox

Zeno of Elea (490-430 BC), member of the pre-Socratic Eliatic School founded by Parmenides, questioned the concept of change and motion in his famous arrow paradox:

Parmenides and Zeno were followed by Plato seeking a timeless reality without a flow of time towards death.
Parmenides argued that if there was Becoming, a thing must grow from nothing into something, which he regarded to be logically impossible. Heraclitus took a different more constructive attitude stating that (cf. The Direction of Time):

  • All things are in flux.
  • Becoming is the very essence of life.
  • You cannot step twice into the same river, for fresh waters are ever flowing in upon you.
  • The sun is new every day.
  • For it is death to souls to become water, and death to water to become earth. But water comes from earth; and, from water, soul.
  • All things are exchanged for Fire, and Fire for all things as wares are exchanged for gold, and gold for wares.It is cold things that become warm, and what is warm that cools; what is wet dries, and the parched is moistened.

Is Zeno’s Paradox Resolved Today?

What answer would you get if you would ask a physics/mathematics teacher or physicist/mathematician how to resolve Zeno’s paradox, today? Try! 

It is likely that you would get either an indication that the question is silly, and anyway since it is 2500 years old has lost relevance, or you would get a lengthy lecture about infinitesimal calculus with infinitely many infinitely small changes summing up to motion.  

A mathematician would probably shift to instead the question of Achilles and the Tortoise and claim that the resolution is hidden in the convergence to 1 of the the geometric series 

                                         1/2 + 1/4 + 1/8 + 1/16 + 1/32 +  …. 

which however has nothing to do with the arrow, see Math Academy.

A physicist would probably say that quantum mechanics can offer a resolution because by the probabilistic interpretation of quantum mechanics particles are jumping around and thus are moving, but then tell you that quantum mechanics cannot really be understood by anybody. In neither case, there is a high risk that you would not feel fully satisfied. The net result is that Zeno’s paradox seems to be unresolved, still today. Amazing, isn’t it?

                                  Artistic representations of motion by Duchamp and Boccioni.

Resolution of Zeno’s Paradox

We now suggest a resolution of the paradox based on wave mechanics: We are familiar with the phenomeon of waves moving across the surface of a lake or ocean. We know that the wave motion is real in the sense that we can feel the impact of the waves on a boat or when we are swimming. No doubt that the waves are moving, but what is in fact moving? See video of surfer. For a further discussion see also the knol Slinky as Resolution of Zeno’s Paradox .
Is it chunks of water particles that get displaced like a swarm of flying arrows? No, it is not! The water particles are all moving in little circles back-and-forth up-and-down and do not get displaced with the wave, as illustrated in the following figure and the video1 video2 video3 (see also waves in a pool and soccer public making the wave):

The wave is moving from left to right but the water particles are not moving along with the waves. If we agree to view the wave motion as macroscopic and the particle motion as microscopic, then we can say that we see macroscopic motion without macroscopic motion of material particles. 

This suggests a way out of the paradox, because the paradox is about motion of material particles. We have seen that macroscopic motion is possible without macroscopic particle motion. The wave moves and the wave carries energy, and so macroscopic energy is moving, but the energy is not carried by macroscopic particle motion. 
This phenomenon is even more strikingly illustrated by the propagation of  electromagnetics waves through vacuum which is not carried by any particle motion because the vacuum is empty. There is no ether medium
carrying the electromagntic waves, like water carries water waves, but still there are immaterial propagating electromagnetic waves carrying energy.
As concerns water waves, we can repeat the argument and consider the circulatory water particle motion as the macroscopic motion carried by some microscopic wave mechanics all the way down to an ultimate immaterial quantum mechanical wave motion like immaterial electromagnetic wave motion.
We can thus resolve Zeno’s paradox of particle motion, by like Hawking denying the existence of particles and only speaking about wave motion, ultimately immaterial wave motion. If there are no particles, then we do not have to explain how they can move. And we can explain wave motion.

Walking Mushrooms

Another illustration is offered by advancing rings of mushrooms:

The ring widens every year, as if the mushrooms are walking on the lawn, while in fact the old disappear and new get born. Similarly you can, when giving the impression that you are moving ahead by walking, claim that what in fact happens is that your old self disappears at each step and gets reborn into a new self at each next step ahead… Right? See also Slinky as Resolution of Zeno’s Arrow Paradox.