What the hell is Math? What are numbers? Do they actually exist? I kind of understood what the laws of physics were, but what about math?
I am going to describe what I am calling this morning "The Great Dichotomy", and when I understood this, things became much clearer.
The laws of physics are pieces of computer code, that attempt to model how the universe works.
Maths are pieces of computer code, that attempt to model how pieces of computer code work.
That is the Great Dichotomy.
Divide the universe up into two parts:
1. Things that happen inside computers.
2. Everything else.
You can divide the universe up in all sorts of ways, but why this way? Why is this dichotomy useful? It's useful because the laws in each of those two parts are different, and universal.
On the physics side, forces and particles act here on earth the same way they do everywhere. We can study electrons here, and know how they act in other galaxies, so we can create general rules about them.
On the computer side: programs we write here, will act the same way in any Turing complete computer anywhere in the universe, so we can create general rules about them.
This dichotomy is useful because computers (iPhones, living things with DNA, our brains) have this magical property that the code is substrate independent. What happens in the code has a degree of autonomy from the underlying physics. So when you deal with math, you don't have to think about electrons, protons, nuclear forces, etc. It's kinda magical. Because of this “independence” from the underlying physics, computer code can have it's own rules, apart from the rules of physics.
Physics example: a = F/m
Take any object anywhere in the universe, and the acceleration of that object will be equal to the sum of all the forces on it, divided by it's mass.
Math example: Leibniz formula for calculating PI can be written in Python:
pi, k = 0, 0
while True:
pi += 4 * (-1)**k / (2*k + 1)
print(pi)
k += 1
What this mathematical "law" is claiming is: "If you run this code on any computer in the universe, regardless of where the computer is, and regardless of what the computer is made out of, the answer will converge to pi=3.14159..."
That is the kind of thing that a mathematical law is. It is a claim on how patterns of bits work inside every Turing complete computer in the universe, which includes our brains.
Laws of physics exist entirely inside computers, and attempt to describe how the universe outside computers work.
Laws of mathematics exist entirely inside computers, and attempt to describe how the universe inside computers work.