Math is hard, but it makes hard things easier. In fact, since math is a hard thing, math also makes math easier.

Many people are afraid of math, or baffled by it, or both. Or they were completely turned off it by their classes in high school. I understand this—I was completely turned off sports in high school and have never really recovered. I was so bad at sports in high school, my teachers were incredulous that anybody so bad at sports could exist. And yet I’m quite fit now and have even run the New York City Marathon. At least I now appreciate physical exercise, but I still have a horror of any kind of team sports.

math this is how we get things like polynomials

they mate, how they look after their young,

The wonderful thing about abstraction is that when you get very used to an abstract idea, it starts to feel like an actual object instead of just being a made-up idea.

Part of the process of abstraction is like using your imagination. Mathematical abstraction takes us into an imaginary world where anything is possible as long as it’s not contradictory.

Then you add a circle of radius 1/3 attached at the same point, and then a circle of radius 1/4, and then 1/5, and you keep going “forever.” Remember, this is math, so you don’t actually have to sit there drawing forever: you just have to imagine that you did.

Now, I’ll be the first to admit that a lot of it didn’t really mean anything to me. In fact, I was quite proud that I found a way of doing extremely well in physics without really having to understand any physics

. I just read the question, wrote down all the letters corresponding to the quantities given in the question, and then scanned the formula sheet for a formula containing all the correct letters.

More generally, this sort of jigsaw puzzle in category theory is called “making diagrams commute,” and is something I find fun and satisfying.

More generally, this sort of jigsaw puzzle in category theory is called “making diagrams commute,” and is something I find fun and satisfying.

The key in math is that things exist as soon as you imagine them, as long as they don’t cause a contradiction.

The key in math is that things exist as soon as you imagine them, as long as they don’t cause a contradiction.

Do you think it’s cheating to solve a problem by inventing a whole new concept and declaring it to be the answer? For me this is one of the most exciting aspects of math.

Your first objection is probably “But there is no such number!” However, what’s more true is that there was no such number, but we’ve now invented one.

I think this is why toddlers get so excited about learning to climb stairs, because they realize that all they have to do is learn to climb up one step, and then if they do that repeatedly they can go higher and higher and higher, perhaps all the way to the sky. (Except, usually, some killjoy adult comes along

I think this is why toddlers get so excited about learning to climb stairs, because they realize that all they have to do is learn to climb up one step, and then if they do that repeatedly they can go higher and higher and higher, perhaps all the way to the sky. (Except, usually, some killjoy adult comes along and removes them from the stairs.)

The 100 m sprint fascinates me for the same reason. It’s not because I’m glad Usain Bolt will be

The 100 m sprint fascinates me for the same reason. It’s not because I’m glad Usain Bolt will be able to catch a bus better than the rest of us.

When a small child keeps asking “Why?” repeatedly, you might wonder if it’s ever going to end. The answer is: no, it isn’t.

I find mathematics the most exciting of all subjects, because it only relies on brainpower.

math is easy in the same sense that custard is difficult.

This is the crucial point: we make things easy by ignoring the things that are hard.

If Math Is Easy, Why Is It Hard?

place. It’s like building bridges: it’s hard to build a bridge across a river, but easy to cross the bridge once someone else has built it. And while you’re trying to build the bridge, it’s helpful to be able to fly. Logic Is Too Inflexible Logic is too inflexible in the face of a flexible and often rather random world.

It’s like building bridges: it’s hard to build a bridge across a river, but easy to cross the bridge once someone else has built it. And while you’re trying to build the bridge, it’s helpful to be able

It’s like building bridges: it’s hard to build a bridge across a river, but easy to cross the bridge once someone else has built it. And while you’re trying to build the bridge, it’s helpful to be able to fly. Logic Is Too Inflexible Logic is too inflexible in the face of a flexible and often rather random world.

It’s like building bridges: it’s hard to build a bridge across a river, but easy to cross the bridge once someone else has built it. And while you’re trying to build the bridge, it’s helpful to be able to fly.

Apart from the fact that logic is too weak, logic is also too powerful.

a couple of years later, Japanese interest rates really did go negative,

I was once in a hotel bar in London when Arsenal walked in—the whole soccer team and the entire entourage. I was sitting there doing some math, as I sometimes do in bars because I like being surrounded by people and especially people having fun.

(This is the kind of example that can make people think math is useless. But some examples aren’t there to be “useful” as much as “illustrative.”)