Zeno of Elea was a Greek philosopher who lived during 490BC - 430BC and messed up everyone’s minds. He said things like:

Contrary to the evidence of one’s senses, the belief in plurality and change is mistaken, and that motion is nothing but an illusion.

Seriously, this was Parmenides’ “doctrine”, says wikipedia. Zeno tried reductio ad absurdum on motion, with ease. It is called “Dichotomy Paradox” and goes like this:

That which is in locomotion, must arrive at the half-way stage before it arrives at the goal

– recounted by Aristotle

What he is saying is that since you can divide numbers (representing distance) again and again, infinitely many times you cannot get anywhere! Because you have to get to the distance that is infinitely small. Another way to put it is that you have to perform infinitely many tasks (task of taking steps) to move.

As an aside, I am sure you will also enjoy another paradox:

If everything that exists has a place, place too will have a place, and so ad infinitum.

You have fun with that last one on your own. We are going to explore the Dichotomy Paradox from a different perspective here. Of course, many people through history have tried to attack the paradox. Wikipedia lists many, but there are two I would like to point out here:

  • Henri Bergson

    Instants in time and instantaneous magnitudes do not physically exist. An object in relative motion cannot have an instantaneous or determined relative position, and so cannot have its motion fractionally dissected.

  • Peter Lynds

    Instants in time and instantaneous magnitudes do not physically exist. An object in relative motion cannot have an instantaneous or determined relative position, and so cannot have its motion fractionally dissected as if it does, as is assumed by the paradoxes.

They are almost entirely similar. Do they sound right to you? What Henri and Peter are saying is that “dude, you cannot actually divide space in infinitely small distances!” and the same for time. Why? Because that’s not how the Universe works - “do not physically exist”! Okay, perhaps space in spacetime is not a field? Perhaps we are yet to come up with a different theory of everything that will help resolve this one. But overall they sound okay, right? I mean you don’t actually move infinite distances and therefore it is a proof by existence.

Slowdown, Kevin Brown concludes:

Given the history of “final resolutions”, from Aristotle onwards, it’s probably foolhardy to think we have reached the end. It may be that Zeno’s arguments on motion, because of their simplicity and universality, will always serve as a kind of “Rorschach Image” onto which people can project their most fundamental phenomenological concerns.

I would like to use the paradox to point out another underlying idea. First, I recommend you go through “To Infinity… And Beyond!”. Though the article is about why we need “infinitesimal numbers”, it is an excellent exposition to the paradox as well.

In the story, tortoise and Achilles are setting up a race. Races have finish lines of some sort - a white line, a ribbon, a wall to ram into. Zeno says it is impossible to reach the finish line and if you do, it is an illusion.

But let us ask “what is this finish line you are talking about”?

When we draw a while line on the ground, does it have a precise edge? Can you zoom in at the levels of subatomic particles and say “there! that exact atom and electrons define the point on the edge!”? Can you then go on to say that “that damn electron was given by moisture and therefore does not belong to the edge”? No, you cannot. Things are not so precise at that level. As far as we know, Heisenberg’s Uncertainty Principle explains how we cannot know things at infinite precision.

So what is the finish line? Actually, what is tortoise and what is Achilles? These are abstractions. They’re not physical things, they are models generated by our brains imposed upon the reality to be able to interact with it and survive. Here’s a great VSauce video asking “Do Chairs Exist”?

Sorites Paradox helps us understand that defining things in vague ways helps no one and yet we must define everything in vague terms. As VSauce put it “atoms are arranged chair-wise”! But the chair-ness is an abstraction. Donald Hoffman goes far out and argues that our brains have nothing to do with the underlying reality at all, they are just evolutionary mechanisms. For further exploration, checkout Joscha Bach with Lex Fridman.

Abstractions are all we have. All we can do is operate in terms of abstractions like walls, chairs, finish lines etc. It is impossible to “hit the wall” because (as far as we can derive), the molecules of our bodies cannot ever touch those of the wall, they repel with all their might. Hence, all we can cross, reach is the abstraction of finish line, the abstraction of the wall.

Looked at this way, there is no paradox, there are no arguments for or against it, there is no proof. Only abstractions.