What’s a number?

In math, we could answer that question a few different ways. We could answer it semantically by looking at what numbers mean. Or we could answer that question axiomatically by looking at how they behave. Or we could answer the question constructively by looking at how we could build the numbers from some other kind of simpler object.

We’ll start with semantics. What do numbers mean? We all think we know the answer to this, but in fact, we’re wrong most of the time! People think that a number is just one thing, the thing that you use to count, and that’s it. But that’s not really true. Numbers can have two different meanings, depending on how they’re used.

There are two kinds of numbers. When you see the ...

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