What is mathematics?

I was recently asked by a friend what mathematics actually is. And what mathematicians actually do all day long.
In this post I will try to answer these questions.

But first, let me rant a little.

Imagine you want to learn how to do art. You're fortunate because the regular school curriculum has a subject called "Art". Once you enter it, all you'll ever do is transcribe scores. You get books upon books of scores and are supposed to write down the individual notes as letters. Nothing else, ever. No painting, no reading, no looking at paintings, no listening to music, and certainly no playing music. Only transcribing, for twelve years or more.
Do you think you'd be interested in Art after that treatment? No? Yet that's exactly what happens with "Mathematics" in school. Instead of showing the wonderful beauty that is mathematics, students are supposed to calculate, of all things. I'm sure that in the early 1900's, when computers weren't available and people had to to accounting and bookkeeping and calculating all by themselves, this was probably a great idea. After all, the "modern" school system is mainly set up to prepare young people to enter the industry as usable workers.

In any case, that's not mathematics. That's computation and since the 1960s probably best left to, you know, computers.In fact, before we had computing machines, the word "computer" referred to a person doing computations.

Mathematics is something completely different from this drudgery and torture.
Mathematics is a game.

The game is very simple: You start out with a handful of simple statements about something. The goal of the game is to find out as much as you can about the things the statements are about.
That, more or less, is all there is. There are a few rules what you can do with the statements and the things they are about and the way of finding stuff out, but they're not complicated, so you could learn them in a few hours.

The amazing thing, however, is that starting from very few little facts, you can find out a vast number of things. You'll find out lots of interesting facts about the things you're thinking about, and you'll get much better at the game. How much you find out that's your score. If you find out more things your score is higher. Of course, your score is given out by fellow mathematicians so the better you can describe what you've found out concisely and clearly and short the higher your score will be. Some things you will feel are right, but they're not and you'll prove it.Naturally, disproving is also finding something out, so you'll increase your score. Some things you will feel are right but no one can prove it -- that's when mathematicians get edgy and start challenging the world to prove them or even hand out million-dollar prizes to anyone who can prove it.

After a while of this, however, you will come to a wall. You will have exhausted everything there is to find out about the things you started with. Now starts the second part of the game. The second part is even better, because now you get to define new parts of the game. There's nothing more to find out, so you just think of something new, and then start finding out about these new things. The less you need to say about the new things and the more you and others can find out about them, the higher your score. Some old mathematicians had such high scores that we attached their names to the things they defined. Not many people play on this level, so all of those are pretty well known.In fact, for the top player, Leonhard Euler, we started naming things after the first person to find out things after him, just because his list would be so large. Even so, it is a very long list.

But, as you'll probably think by now, that gets old after a while, too. So then starts the third part of the game. Because now that you have a decent overview over mathematics and the things it's about, you'll start to see the structure behind it. Wildly different fields of mathematics will start so seem similar and you'll start to see the universe behind it. You'll start to see that, no matter where you start, people will think along the same lines and things will be more similar than they seem at first. And if you can then connect two seemingly different things, that's when the really high scores come out. Almost no one plays on this level, so everyone here is revered as a hero of mathematics. You might even have heard some of those names: Leonhard Euler, Isaac Newton, Evariste Galois, Erdős Pál, Andrew Wiles, ... Or, probably you haven't.Each of theses is worth looking up, by the way.

At that point it isn't even about the game any more. It's about much more, as you've glimpsed the structure of the human mind and of the world, because you've seen the ties that hold together all those different mathematical descriptions of the world. You've seen that mathematics is the canvas on which the universe is painted.
And it's beautiful, it's glorious. Everything is connected, no matter where you start.

That is mathematics.

You probably think you know what mathematics is, because you had that subject called "Mathematics" in school. Well, you don't, because what's taught there is just a tiny subset of mathematics called arithmetic, which today is usually better left to machines. The reason you had to learn this is because when the modern schooling system was created in the early 1900's, those machines didn't exist yet, so people had to learn to do the computations themselves in order to be accountants and bookkeepers. But, of course, other people have pointed out this century-old flaw in our education a lot more eloquently than I can. In any case, people learn computing and think they know what mathematics is.

Most of those people will miss out on the beauty that is mathematics.

And that is very sad.