Zeno’s Paradox and the Planck Length


Zeno was an ancient philosopher/cult member who had some interesting thought experiments regarding infinity.

VSauce’s video on Supertasks:




Zeno was in a cult.

Back in ancient Greece, hero cults were a thing, and kind-of a big thing. A Hero in Greek culture was a man who had died and was now revered as not quite godlike, but close.

They were somewhere between man and God.

And followers would build temples in their names and establish churches around them, and devout followers would pledge their lives to their ideals.

But Zeno was a part of the first known hero cult based around a philosopher. A philosopher named Ameinias of Elea, who created the Eleatic School of Philosophy.

The Eleatics, led by the Philosopher Parmenides, believed that the senses cannot be trusted to reach truth, that only by thinking and logic can you arrive at truth.

They also believed that change and even motion was nothing but an illusion brought about by our senses.

To prove this idea, Zeno posited a series of paradoxes, the most famous being the Arrow paradox and the Achilles and the Tortoise Paradox.

Much like my spinning example earlier, the arrow paradox says that for an arrow to get from the bow to a target, it must pass through an infinite number of halfway points to get there, and the Achilles and the Tortoise paradox takes the use of movement a bit further by saying that if Achilles were racing a tortoise and gave the tortoise a head start that every time he got halfway to the tortoise, the tortoise would have moved forward a small amount, therefore even though he’s going much faster, could he ever actually catch the tortoise?

Now logic says that of course the arrow reaches the target, there’s not some invisible field keeping it from touching, and of course Achilles beats the tortoise.

These are some of the first examples of a rhetorical device called reductio ad absurdum, where you disprove a statement by showing that it inevitably ends with an absurd result.

Other versions of this that came later include Gabriel’s cake, which says if you slice a piece of cake in half, then slice one of those halves in half, and again and so on into infinity, and then stack each of those layers on top of each other, the cake would stretch to infinity.

Now these are all space paradoxes, but there’s also one that divides time called Thompson’s Lamp.

It says you take one minute and turn a lamp on and off at every halfway division, so turn it on at 30 seconds, turn it off at 15 seconds, turn it back on at 7.5 seconds, on at 3.75, and on and on until it’s blinking so fast we wouldn’t be able to see the difference. When the minute passes, is the lamp on or off?

Now, there have been countless resolutions of Zeno’s paradox through the years, some philosophers going so far as to reasoning that space and time don’t exist… which is kind-of what Zeno was going for…

But mathematical constructs worked to prove that the sum of infinitely decreasing quantities could result in a finite number.

And the idea for a limiting value to an infinite process is central to calculus, which relies on infinitesimals in order to ascribe a finite number to an infinite number of pieces.

Thanks, Sir Isaac Newton!

But maybe the easiest answer is simply that you can’t divide time and space forever. There might be a real, physical limit to smallness. Enter Max Planck.

Max Planck was one of the most substantial physicists of the early 20th century who proposed an elemental size to the fabric of spacetime, which he called a Planck Length.

The Planck Length is insanely small. It’s 1.6×10-35 meters.

To give you an idea of how small that is, This (graphic – 15 zeroes) is the length of a proton. To get to Planck’s length, you have to add not 5 (graphics change for each), not 10, not 15, but 20 zeroes. It’s one hundred quintillionth the length of a proton.

He came to this by combining three fundamental constants, Gravity, the speed of light, and his own Planck’s constant.

And from there, he created Planck Time, the time it takes for light to travel one planck length.

So if there truly is a smallest indivisible length of time and space, Zeno’s paradox is solved. But many still aren’t so sure.


Why Is The Speed Of Light The Speed Of Light?

Today I answer another question from my 50-question lightning round video, this one on the speed of light.


So when we talk about the speed of light, the first thing to remember is that light is just a sliver of the entire electromagnetic spectrum, ranging from gamma rays to radio waves.

So we’re really talking about the speed of electromagnetism.

James Clerk Maxwell was the genius who first described the properties of electromagnetism into physics equations.

From these equations, we can calculate the speed of light. 299,792 kilometers per second.

Einstein was able to prove through his theory of special relativity that the closer you get to the speed of light, the more time slows down for a person in that relative frame. And if you were to go the speed of light, time would stop altogether.

So if you were able to travel faster than light through spacetime, time itself would actually flip. Time would go backwards. And that would break causality.

Effect would precede cause, which is impossible. The speed of light is the speed of causality.

The other prediction that supports a speed limit is the idea that inertia increases as velocity approaches the speed of light. That means mass increases.

So mass is a speed impediment. Nothing that has mass can go the speed of light.

But if you are massless, you can only go the speed of light, because you have no speed impediments. And photons are massless particles.

Particles that must travel at the speed of light and because they are traveling at the speed of light, time stands still from its point of view.

So really, that video I talked about earlier is all wrong, from the perspective of the photon, that journey would have occurred instantly.

So when you look at a star at night, that massless photon might have traveled a million light-years to reach you, but its experience was instantaneous.

Now there is another theory that’s a little controversial but starting to gain some ground.

It says that the speed of light is actually caused by quantum vacuum fluctuations.

See, quantum field theory claims that empty space is actually not empty at all but filled with quantum fluctuations and virtual particles popping in and out of existence.

And two different teams of researchers have calculated c using electromagnetic properties of the quantum vacuum, so it could be that the quantum foam of virtual particles and fluctuations may be slowing the speed of light.

But what if the speed of light wasn’t the speed of light?

What if Galileo was right and the speed is infinite?

Then nothing would exist. Because matter is made of energy, it would take infinite energy to create any mass. Time and space wouldn’t exist because all things communicate with each other instantaneously. Cause and effect wouldn’t exist.

But if the speed of light were slower, that might be even cooler. Because then we could see all the way back to the big bang.

The speed of light, of course, is just one of many constants in the universe, like gravity, the specific charges and masses of the fundamental particles, quantum effects, and the list goes on.

A whole handful of very specific constants that if they were just a little bit different, we would never exist.