# Does Zeno's paradox prove the continuity of spacetime?

Zeno's paradox seems to umply that motion is not possible. The paradox is easily resolved though by pointing to time intervals that get smaller if smaller space intervals are chosen in the formulation of the paradox. Like that there is no ground to make motion impossible.

If space or time are discrete though then there will be intervals from which a particle in motion can't travel to the next interval. If the basis unit of time is a Planck-time (10exp-43(s)) then how can time be measured between these intervals? In every interval everything is motionless so what determines the transition to the next interval in which things are slightly different? How can a particle move from space interval (a Planck interval is about 10exp-33(m)) to a from this interval spatially disconconnected next interval?

Now there are physical theories that include exactly discrete spacetime structures to account for a quantized version of gravity. One of these theories is loop quantum gravity. Such a discrete spacetime shouldn't be considered though as ordinary spacetime litterally built up from almost tangent chuncks of spacetime. But discreteness is involved. This doesn't automatically mean that continuity is gone though. Maybe there is a spacetime process going on between the intervals that we can't perceive. It looks like discreteness but behind the scenes there is continuity. But spacetime is no longer diffeomorph, which is a prerequisite in general relativity.

Does Zeno's paradox prove that this has to be the case? I mean, does the fact that things can move trough spacetime prove that there is continuity on every level? Can there be processes outside 4D spacetime that determine how each new interval must look like?

• Comments are not for extended discussion; this conversation has been moved to chat. Commented Jul 9, 2021 at 8:19
• Zeno's paradox at most proves that ancient greeks had not discovered calculus Commented Jun 13 at 0:15

No.

Zeno's paradox assumes that motion is fundamentally continuous. If spacetime is fundamentally discrete, then motion is fundamentally discrete. An argument never proves its own premises.

In every interval everything is motionless so what determines the transition to the next interval in which things are slightly different?

Some rule other than continuous motion, of course! Plenty of things (like temperature) are emergent features of more fundamental rules; perhaps motion is one of these.

• Good answer. The question assumes that time and space HAVE to be continuous. But that is just a human assumption, one which we seem to be programmed for. All that would be needed for discrete physics is that APPARENT continuity show up at macro scales. Commented Jul 23, 2022 at 15:56

Zeno's paradox shows that there is something fundamentally crucial about motion that is not captured by modelling it as the real line. It remains mysterious.

This has been spectacularly verified by the advent of QM in the 20C which showed that motion isn't at all lik we thought it was.

I think it proves the opposite.

To run 200m one must first run 100m, then 50m, then 25m, and so on.

Sensors are placed after 100m, 150m, 175m, and so on. These sensors approach the 200m finish line but importantly no sensor is placed on the 200m finish line.

When each sensor is passed the distance is displayed on a screen, as well as the time that the sensor was passed. This display remains until the next sensor is passed. Nothing else controls the behaviour of the screen.

Say we run at a constant speed. We pass the 100m sensor at 12:00:10, the 150m sensor at 12:00:15, the 175m sensor at 12:00:17.5, and so on.

When we reach the 200m finish line at 12:00:20, what distance and what time is displayed on the screen?

The sequences may approach 200m and 12:00:20, but because there is no sensor on the 200m finish line neither 200m nor 12:00:20 will display on the screen.

No answer consistent with the premises is possible.

Spacetime being continuous seems to entail a contradiction and so it would seem that spacetime being discrete is proven.