Terence Tao

How are we humans supposed to deal with infinity?

Well, you can think of infinity as just an abstraction of

a finite number for which you do not have a bound for.

So nothing in real life is truly infinite.

But you can ask yourself questions like, what if I had as much money as I wanted?

Or what if I could go as fast as I wanted?

And a way in which mathematicians formalize that is, mathematics has found a formalism to idealize, instead of something being extremely large or extremely small, to actually be exactly infinite or zero.

And often the mathematics becomes a lot cleaner when you do that.

I mean, in physics, we joke about assuming spherical cows.

You know, like real-world problems have got all kinds of real-world effects, but you can idealize, send some things to infinity, send some things to zero.

And the mathematics becomes a lot simpler to work with there.

Yeah, so there's a lot of pitfalls.

We spend a lot of time in undergraduate math classes teaching analysis, and analysis is often about how to take limits.

So for example, A plus B is always B plus A. So when you have a finite number of terms and you add them, you can swap them and there's no problem.

But when you have an infinite number of terms, they're these sort of show games you can play, where you can have a series which converges to one value, but you rearrange it, and it suddenly converges to another value.

And so you can make mistakes.

You have to know what you're doing when you allow infinity.

You have to introduce these epsilons and deltas, and there's a certain type of way of reasoning that helps you avoid mistakes.

In more recent years, people have started taking results that are true in infinite limits and what's called finitizing them.

So you know that something's true eventually, but you don't know when.