Math for understanding reality
I’m going to argue why I think math is the best tool we have to understand reality. And how I’ve come to believe its less perfect than I thought.
With a few things you are willing to believe are true (your axioms), you can combine them in various ways you believe are logically valid, to get new ideas that must also be true. You repeat and eventually you’ve got proofs for why the earth is a sphere and the speed of light is c.
If your axioms are actually correct, and you didn’t make mistakes, your conclusion must be correct. You must believe it, or abandon your axioms. That guarantee is incredible, and it took me a few months to really grasp the weight of it.
That’s all great, but the first sentence in my last paragraph already revealed a flaw.
Your axioms are just assumptions. And so are your rules of logic themselves. You’re just taking these on faith. And if they are wrong, your house of card topples.
This realization was triggered when I listened to Donald Hoffman talk for three hours about how nothing is real. At one point he criticized a colleague for assuming math itself is fundamental. I was a bit offended at first. How could math not be fundamental? But after chewing on it, I understand his point. You still have to make assumptions, and they could be wrong.
1+1=2 feels obvious. Modus ponens feels obvious. But “feels obvious” isn’t a proof. You can’t step outside logic to prove logic, because you’d need logic to do the proving. This feels adjacent to Gödel’s incompleteness theorem, although I’ve learned about that only in passing, I might be misconstruing.
So math isn’t the silver bullet I thought. It doesn’t give you truths about reality. It gives you a conditional: if you accept these axioms, and if you accept these rules of logic, then everything downstream follows.
But I still think it’s the best tool we have.
That’s because the assumptions math asks you to swallow are the ones basically every human already agrees on. Nobody’s really out there disputing that 1+1 is not 2, or that a thing can be both true and false at the same time. The foundations are are assumptions, but it seems they’re as good as assumptions get. Math is what you get when you push your assumptions down as far as they’ll go, and you land on a layer almost nobody disagrees with.
That’s pretty good, I think.