To me, there is a mystery at the heart of mathematics. Why does it work? Why does it tell us useful things about the world?

One philosophy is that mathematical truth is like Plato’s Forms, an abstract, perfect truth that exists outside of physical reality. You can think of mathematics as a type of language; it’s language with a convention that any possibility of error is completely forbidden. As Wittgenstein suggests, things we cannot speak of we must pass over in silence. Mathematical language cannot say anything about politics, about current events, about the precise nature of a real-world cannonball lofted through the air, because it’s possible that our observations about these things are mistaken. Burn all this away, and what is left is the mathematics.

This way of looking at mathematics is powerful, but it’s missing something. Think of the greatest drama of historical mathematics, the rise and fall of Euclidean geometry. For about two thousand years, Euclid’s Elements was the main textbook for teaching mathematics. Euclid started with a few axioms for geometry, and proved vast swaths of geometry from these axioms. The one aesthetically displeasing part was the fifth axiom - if you have points ABCD, and ABC and BCD are acute angles, then the rays BA and CD must intersect. In other words, the “parallel postulate” - that for every line and a point not on that line, you can only draw one parallel line through that point.

People thought this axiom was obvious for a long time, and it was only considered aesthetically displeasing that it was so much more complicated than the previous axioms. Mathematicians tried to prove it from the other axioms for a long time, but it turned out this was impossible.

In fact, this axiom is worse than aesthetically displeasing. This axiom is false. With the discovery of special relativity, scientists realized that space is not flat. It is curved. You can draw multiple parallel lines through a point. They just have to be really, really close to each other.

It’s worse than that. “Real numbers” are also not real. Quantum mechanics now tells us that the precision of a measurement is limited. But the rigorous definition of “real numbers” requires an infinite sequence of numbers getting closer and closer. The mathematics doesn’t work if small numbers are prevented from going smaller than the Planck length. A circle of radius 1 doesn’t really have an area of pi - it’s a little bit off.

Euclidean geometry is also no longer considered to be the highest standard of rigor. It’s interesting to look at a modern mathematical computer formalization of geometry, like Lean’s geometry library. Euclid took all sorts of theorems for granted, that the computer insists we prove. How do we know that the angle ABC is equal to the angle CBA? How is the angle ABB defined? If you say it can’t be defined, then every single time you use the angle ABC, you need to prove that A does not equal B, and B does not equal C. Euclid basically ignored this issue, and got lucky in that it never led to a bug in his proofs. Lean decides that the angle ABB should be considered to be a 90 degree angle. But from a modern point of view, geometry is hopelessly messy to make rigorous. It is far cleaner to build up integers, sets, rationals, real numbers, real analysis, and then tack on geometry at the end.

Euclidean geometry ends up being far from abstract mathematical truth. With modern progress in mathematics, we see that Euclidean geometry is not a great approximation of the world, and it isn’t the most rigorous area of mathematics either. We keep it around for historical and teaching purposes, basically.

So how can we explain the fact that for thousands of years Euclidean geometry was the pinnacle of mathematics? We could have another mathematical revolution, where we discover that our current mathematics methods are not great approximations to the world, and they aren’t sufficiently rigorous, either. The value of mathematics cannot lie in the fact that it is an abstract truth. It is the quest for this abstract truth. And the quest for this truth is a human activity. This quest would not exist without people to go questing.

Personally, I suspect that mathematics is on the verge of another revolution like the non-Euclidean revolution. Software will eat mathematics. The standard for mathematical rigor will rise again - any proof that cannot be checked by a computer will be considered insufficiently rigorous. It will be obvious there is a mathematical revolution once the computers knock off a few of the longstanding open problems. Just like nowadays many subareas of physics basically require that you write a decent amount of Python code and become a decent software engineer, in the future mathematics will also grow much more intertwined software engineering.

That’s my prediction, anyway.