Ancient Greeks
One of my favorite characters from history is this guy from ancient Greece called Zeno of Elea.
He came up with all these crazy paradoxes.
One of them was the idea that motion is impossible.
There was this Austrian guy named Godel who studied infinity. He came with all these different theories on different kinds of infinity.
One of them involved two lines. One flat, and one at an angle. But the start points and end points were lined up along same points on the X axis.
One Line Longer Than The Other
Since two lines had their end points lined up vertically, and one was flat and one was tilted upward, the one that was tilted upward was a little bit longer.
But he figured you could take each point on the shorter line, and connect it to a point on the upper line.
And each point on the upper, longer tilted line could be connected down to the shorter flat line.
Which means they could have the same number of points, which of course would be infinite.
Crowded Number Line
On the number line you’ll find three types of numbers. Whole numbers, rational numbers and irrational numbers.
Whole numbers are zero, one, two, three, etc.
Rational numbers are the ones that can be expressed as ratios, or fractions.
1/2, 1/3, 1/4, etc.
But the number of fractions between zero and one are infinite.
You can essentially keep counting up forever, and always add a number one larger on the denominator.
Math Doesn’t Exist
There’s an idea that math doesn’t exist.
Plato and all those old school philosopher guys had the idea that nature is imperfect, and only philosophy, which is a function of the human brain, can discern the perfected nature, which can only exist in the abstract.
But if those ideas can only exist as thought of by humans, and nature is imperfect, what was nature like before humans came long with our brains?
What was the state of nature when there was only monkeys throwing poo at each other?
The Life of Pi
What about Pi?
Pi is the most famous irrational number. It’s irrational not because it’s crazy, but because it is not a ratio, it cannot be expressed as one whole number divided by another whole number.
And it goes on forever.
If you could look in the infinity of pi on the right side of the decimal, you’d find every number that ever existed.
You’d find your phone number, your social security number, and every other number that has ever been created by humans.
At the same time, pi doesn’t exist except in our brains.
Nearly All Pi Is Useless
You can calculate both the size of a hydrogen atom, and the size of the universe, with only a couple dozen digits of pi.
You can land people on the moon within a couple of inches with only a couple dozen decimals of pi.
Pi, as an abstract symbol, shows up in all kinds of crazy places.
But what if it only exists in our brains?
What if Plato is right?
Only In Your Mind
Perfection is only a hallucination, and nature is imperfect.
What does nature think about nature?
What was pi before humans showed up?
What does it mean that we humans can hallucinate things that don’t exist and then use those hallucinations to build things that do exist?
What if nothing exists, and everything is hallucination?
I Think Therefore I Am
What if Descartes was right all along?
What if all we really know is that we are, and we are thinking.
What if I, and all the words I’ve said so far, and all the words I haven’t said yet are a hallucination in YOUR mind?
The amount of rational numbers between zero and one are infinite.
The number of irrational numbers between zero and one are infinite.
The rational numbers represent a countable infinity.
Two Different Infinities
And the irrational numbers represent an uncountable infinity.
That means between zero and one are two kinds of infinity!
What Godel realized was that since both lines contained the SAME number of infinite points, but one was longer than the other, then some infinities are denser than other infinities.
Zeno’s paradox went like this. Before you go from point A to point B, you have to get halfway there.
You’re Always Stuck
And before you get to THAT point, you have to get to another halfway there.
Eventually, you have to cross an infinite number of “halfway there” points.
And since crossing an infinite number of points would take forever, the only conclusion would be that motion is impossible.
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