How Zeno’s Paradox was resolved: by physics, not math alone
Zeno’s Paradox: before you can ever reach your destination, you must travel halfway there, always leaving another half. Despite Zeno’s Paradox, you always arrive right on time. Travel half the distance to your destination and there’s always another half to go.
Zeno’s paradox: motion is impossible
The fastest human in the world, according to legend, was Atalanta
- She must first travel half of the total distance, then travel the remaining distance, and so on, ad infinitum, until she reaches her destination
- As long as you can demonstrate that the total sum of every jump you need to take adds up to a finite value, it doesn’t matter how many chunks you divide it into
- Simple, straightforward, and compelling
Zeno’s paradox at the quantum level
Certain physical phenomena only happen due to the quantum properties of matter and energy, like quantum tunneling through a barrier or radioactive decays.
- In order to go from one quantum state to another, your quantum system needs to act like a wave: its wavefunction spreads out over time. Eventually, there will be a non-zero probability of winding up in a lower-energy quantum state.
How to inhibit this: observe/measure the system before the wavefunction can sufficiently spread out
If you make a measurement too close in time to the prior measurement, there is an infinitesimal probability of the system tunneling into the desired state. If you keep the system interacting with the environment, you suppress the inherently quantum effects, leaving you with only the classical outcomes as possibilities.
Zeno’s Paradox
This mathematical line of reasoning is only good enough to show that the total distance you must travel converges to a finite value.
- It doesn’t tell you anything about how long it takes you to reach your destination, and that’s the tricky part of the paradox.
The takeaway is this: motion from one place to another is possible, and because of the explicit physical relationship between distance, velocity and time, we can learn exactly how motion occurs in a quantitative sense.
Motion over a finite distance always takes a finite amount of time. Thanks to physics, we at last understand how.
The Zen Paradox
There’s no guarantee that each of the infinite number of jumps you need to take – even to cover a finite distance – occurs in a finite amount of time.
- It’s possible that the time it takes to finish each step will still go down: half the original time, a third of the original, a fourth, a fifth, etc., but that the total journey will take an infinite amount of Time.
- Figuring out the relationship between distance and time quantitatively and quantitatively did not happen because the reason for this is that distance and velocity do not always move in the same direction.