Today I discuss how some infinities are bigger than other infinities with Zeno's Paradox.
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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 1.6x10-35 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.