# Using Zeno's Paradox to prove that spacetime is discrete

After recently watching a couple of videos on supertasks and Achilles-turtle paradox, I'm curious - could Zeno's paradox be considered a "proof" that physically there has to be a smallest unit of length? Because at least physically, the paradox is resolved if we make that assumption.

I'm sure the above is nothing novel, but I want to know why exactly an argument like above may not work.

EDIT: In order to clarify this further, suppose I'm going towards a point 2 m away at 1 m/s. It'll take me 1 s to cover half the distance, or 1 m. Let's define a "step" as covering half of the remaining distance. So the 2nd step will mean covering 0.5 m in 0.5 s, and so on. The entire process of reaching my final destination is thus broken into infinitely many steps of 1 m, 1/2 m, 1/4 m, and so on. Of course, $\sum_{1}^{\infty} \frac{1}{n} = 2$, but the number of steps is still infinity - there is no "last step". So how can a process without a last step be completed?

Apparently, one way to resolve this could be to say that the process actually has a finite number of steps. This means there is a physical limit to how many subdivisions of length one can make en route to the final destination. Hopefully this clears things up a bit.

• Zeno's paradox is not a paradox at all, and thus does not have to be resolved. Jan 13, 2017 at 12:11
• @Noiralef: It would be very helpful to provide and explain in an answer the solution of the 'paradox'
– user98038
Jan 13, 2017 at 12:27
• See Zeno’s Paradox of the Arrow for the resolution to the Zeno paradox. It does not need spacetime to be discrete. Jan 13, 2017 at 12:34
• There are an infinite number of steps because the problem is defined in a way that generates an infinite number of steps. The problem can be solved without doing that. And as already stated, similar infinite sums can be handled in mathematics. Jan 13, 2017 at 12:52
• Let's put it another way: it is possible to encompass an infinite (uncountable) number of points of space in a finite elapse of time. So there is no paradox at all. Jan 14, 2017 at 10:13

## 3 Answers

Just like Noiralef said in his comment. The root of Zeno's paradox lies not in some fundamental contradiction in the idea of continuous movement, but in the limited ancient Greek understanding of the infinite sums. The sum of 100 + 10 + 1 + 1/10 + 1/100 + 1/1000... (the time needed for Achilles to catch the turtle.) is not infinite (meaning that he can't reach it.) but a simple rational number, 1000/900 = 111.111111...

Edit: To make the infinite number of steps into a problem, you have to proof that: 'Any motion is composed from a finite number of small steps.' It seems to me, that, it would require space to be discretized, thus making the reasoning petitio principii.

• That kind of misses the point. Like you said, it's easy to see that the sum of the infinite series is well-behaved and converges - the problem lies in the fact that the number of terms is infinity. Here's a nice video by numberphile explaining the problem from a mathematical point of view. Jan 13, 2017 at 13:34
• I don't understand. Why should I care about the number of terms, if the sum is real? Why should have the number of terms have physical meaning? Jan 13, 2017 at 13:39
• I've edited the question the explain it further. Jan 13, 2017 at 14:01
• @ShirishKulhari The problem lies in not appreciating that infinite processes can have finite results: in this case the sum of the time taken for an infinite number of steps is finite. This is something that the ancient Greeks struggled with but which was sorted out a long time ago in mathematics.
– user107153
Jan 13, 2017 at 15:20

It sounds like the conceptual difficulty is that a process can be broken into an infinite number of steps, and an infinite number of steps cannot be done. Each step must be done one after the other, and you can't get to the end of infinity.

An arrow in flight can be thought of as one step. There is no problem with that step, as watching the arrow shows.

The problem comes when you impose steps on it. If you cannot complete an infinite number of steps, you might ask yourself if you can break a process into an infinite number of steps? Would that ever be finished?

If not, you have broken the process into a finite number of steps, and you can finish those.

If so, it is no harder to finish an infinite number of steps than to break up the process into those steps.

You might consider that the typical way to break the arrow's path up is 1/2, 1/4, ... Completing the first step is 1/2 as difficult as completing the whole flight. The seconds step is 1/4 as difficult. The total difficulty of the steps is the same as the whole flight.

Zeno's paradoxes don't constitute a 'proof' that spacetime is discrete but that common-sense notions of how motions occurs aren't simply teneable.

Historically speaking, it pushed the perpatetic school - the school of Aristotle - to a closer examination of the concepts of space, time and motion. This was of some importance to the Western revival of physics beginning with Kepler and Galileo as this had been preceded by an intense study of this school over several centuries. In fact, it's notable that a student of Aristotle, commonly called pseudo-Aristotle, and thought to be Archtas, had the parallelogram rule of forces/velocities in the work called Mechanica.

It also makes the continuum structure of space, where space is made up of points, implausible. For example, Chris Isham has written in The New Physics:

The construction of the real numbers from the integets amd fractions is a very abstract procedure and there is no a priori reason Why it should be reflected in the empirical world ... it is clear, that quantum gravity, with its natural Planck's length, raises the possibility that the continuum nature of spacetime may not hold beneath this length.