joshuazelinsky

The "numbers rant" is someting I've posted various places on the internet in the past, but it seems like I've never posted it here. Figured made sense to get it down here. So, sure. Here we go.

People who haven't seriously studied mathematics might reasonably assume that mathematicians have a unified notion of "number". There are numbers, you can add them, you can subtract them, you can multiply them, you can divide them (except by 0). Sometimes you might restrict to a particular subset of the numbers, like the integers or the real numbers.

The way people learn about numbers in elementary and high school reinforces this. First you learn about whole numbers; then fractions; then negative numbers; at some point you learn there are irrational numbers, implicitly passing from rational numbers to real numbers without learning just what that really means; and then finally you add in complex numbers. And that's all the numbers.

(If you're a bit more into mathematics, you maybe hear that there's some sort of infinities or something, like ℵ₀ and ℵ₁ or something? You don't learn that in school, but you hear about it elsewhere, and you're not sure how it fits into the above picture, but surely it does.)

This picture is wrong, and moreover, I think it encourages some bad ways of thinking. There is no unified notion of "number" in mathematics. Rather, mathematicians use different systems of numbers -- the integers, the rational numbers, the real numbers, the complex numbers -- depending on context. Some of these systems embed into larger ones; the integers embed into the real numbers, for instance. But the integers can also be considered separately from the real numbers.

OK, so what, you might stay. It still sounds like the complex numbers are all the numbers, and sometimes we just focus on subsets of them. Well, they're not.

One way this is false is that you can expand further on the complex numbers -- like, the 2-dimensional complex numbers embed in the 4-dimensional quaternions. But that's sort of a silly way this is false. If that were the only reason this were false, then it would just be the case that the quaternions were all the numbers. But they aren't! And no, not just because you can expand on them to the octonions, etc.

Rather the key point here is that there are distinct number systems that are entirely incompatible with one another. We've already mentioned an example above -- cardinal numbers. Cardinal numbers include (and are therefore compatible with) the whole numbers; but they're incompatible with negative numbers, let alone real numbers or complex numbers. How do you add ℵ₀+½? You don't! This is a nonsense sum, because you're trying to add two numbers from entirely different systems. You have to work within some specified system of numbers; there isn't some overall general thing called "numbers" where you can take any two of them and add them together.

There are many more systems of numbers; for instance, there are the p-adic numbers, which is not one system but a family of systems, one for every prime p (the 2-adic numbers, the 3-adic numbers, etc). The p-adic numbers (for some fixed p) can be thought of as base-p expansions, except that, whereas in the real numbers, base-p expansions can go finitely far to the left and infinitely far to the right, p-adic numbers can go infinitely far to the left and finitely far to the right. (Note that, as mentioned above, different values of p yield different, incompatible systems; this is in contrast to the real numbers, where, regardless of what base you write it in, it's still the real numbers.) How do you add a p-adic number to a real number? You don't! These are incompatible systems. Now, the p-adic numbers always contain the rational numbers; so if the p-adic number and the real number you're adding are both rational numbers, then sure you could perform the addition within the rational numbers; but there is no more general system that contains both the p-adics and the reals and allows you to add one to the other without special additional knowledge, they're not compatible.

So, there is no unified mathematical notion of "number". Or as I sometimes put it, there's no such thing as a number.

There's something else that bothers me about the way number systems are taught, beyond just that it's wrong, though. It's that if you present things as if complex numbers are all the numbers, it puts the focus on complex numbers as the most fundamental type of numbers. It's like, complex numbers, that's the real meaning of numbers, why would we focus on silly particular subsets like the integers? Presenting the complex numbers as "all the numbers" devalues number theory. It suggests complex numbers as the most fundamental thing, when of course, in reality, whole numbers are the most fundamental thing; other number systems are built on top of the whole numbers, and much of what they do is in a sense just reflecting on and elucidating properties of the whole numbers.

(It's also worth noting that it's important to pick the correct number system for your application. Many people use the real numbers without thinking about it because they're not aware of other possibilities. To be fair, the real numbers are indeed very useful and frequently the right answer, but it's still worth thinking about.)

Annoyingly, there's no formal mathematical definition of a "number system" -- most things we might consider "number systems" are rings or at least semirings, but some fail these criteria (ordinals with their usual arithmetic are just a near-semiring). And of course plenty of rings and semirings are of things we wouldn't normally consider "numbers". Still -- numbers come in systems; different systems are used for different purposes, so pick one that matches your purpose; and there's no such thing as "all the numbers".

That's the numbers rant!

-Harry

PS: If you want a semi-trollish take, one could argue that p-adic numbers are numbers, but complex numbers aren't, because only systems lacking nontrivial automorphisms should be deemed systems of numbers. :P (Because a number should have some sort of absolute meaning, and if there's a nontrivial automorphism, then it doesn't.)