Romy asks, “What does math not explain?”
Math doesn’t explain why math works. And it never, ever will.
In the late 1800s, mathematicians were doing some soul-searching. They’d been solving problems and proving theorems since forever, but none of them really knew why these methods worked. And this was getting kind of awkward.
See, math is just a bunch of rules for turning true statements into other true statements. Assume some stuff, follow the rules, prove other stuff. And the crazy thing is: the stuff you prove is always, always true. Or at least, it’s never once been wrong.
Why?
Proving that the rules of math work was, for a while, one of the biggest unsolved questions in math. In 1900, David Hilbert made a list of the problems he wanted everyone to focus on for the next century. He had 23 problems, and this was one.
It might seem weird to prove something about math from inside math itself. It might even seem circular — if you use math to prove that math works, how do you know that proof works?
But think about it this way. Science and math are two ways to learn true things about the world. Science is when we observe things and guess what’s going on based on the patterns we see. Math is when we use these weird, magical rules to prove things.
You can say, “According to science, it’s impossible to put a ball in mid-air and have it stay there.” That means that you’ve tried it a bunch of times and it’s always been true. But hey, maybe next time…?
You can say, “According to math, it’s impossible to draw a map where five countries all border each other.” That means you’ve used the rules of math to prove that this is just flat-out true. It always has been and always will be.
For this reason, saying “According to math” is way better than saying “According to science.” Science can sometimes mislead you. The best scientists in Ancient Greece thought, based on lots of evidence, that the Sun revolved around the Earth. But math has never, ever been wrong.
So right now we can say, “According to science, math works.” That just means every time math has told us something, it’s been true — so far. We want to be able to say, “According to math, math works.” Because even though it sounds silly, it’s a more powerful claim.
Or at least, David Hilbert wanted to be able to say that. That’s why he told the greatest minds of his generation to work on it. And boy, did they!
Two great mathematicians, Alfred Whitehead and Bertrand Russell, wrote Principia Mathematica, trying to build math from the ground up. It’s three books long, almost 2000 pages, and doesn’t even prove that 1 + 1 = 2 until page 379. They were pretty sure that this was it, this was Math in its purest form, that every true mathematical statement could be proven using their rules.
They were pretty happy about it.
Until 1931.
That was the year Kurt Gödel (rhymes with turtle) ruined everything. He proved two heartbreaking theorems in a single paper. In under 20 pages, he shattered all 2000 pages Whitehead and Russell worked so hard on.
The first theorem showed that no, not every true mathematical statement can be proven using their rules. And the second theorem showed that you can’t even be sure that no false mathematical statement can be proven using their rules!
The worst part is, it’s not just Russell and Whitehead’s rules. It’s any rules. Gödel proved that no matter how well-thought-out your math is, you can’t prove every true thing, and you might even accidentally prove false things.
You can ask a mathematician to prove that there are infinitely many prime numbers and she’ll say sure, no problem. You can ask her to prove that a circle is how you trap the most area with a certain perimeter and she’ll fill a blackboard with diagrams and equations. But if you ask her to prove that tomorrow she won’t accidentally prove 0 = 1, she’ll suddenly remember she has an appointment and kindly ask you to leave.
So proving that math works isn’t just hard, it’s literally impossible. It’s not that we haven’t figured it out yet — we know for a fact that we never will. We can never prove that math actually works.
In theory, we could prove that it doesn’t work. We just haven’t yet.
More on this: George Boolos, Gödel’s Second Incompleteness Theorem
More from Romy Asks:
Originally published at milobeckman.com on March 5, 2015.
