What’s a number?

🕒 5 min

Three is a number, right? So are pi, the square root of two or i. But mathematics is all about abstraction, so how do we abstract the concept of a number? How do we properly define numbers?

Starting off

In school, you’ve probably heard the “intuitive” definition of natural numbers, as “numbers for things you can count”. They start off at one, then increase by one and go up to infinity. It’s usually written like this: ℕ = {1, 2, 3, …}. Then you introduce the concept of whole numbers as a union of ℕ with {0} and -ℕ, writing it like ℤ = {…, -3, -2, -1, 0, 1, 2, 3, …}.  Rationals get a better definition in high school, where they’re defined as fractions of whole numbers and natural numbers, like this: ℚ = {m/n: m ∈ ℤ, n ∈ ℕ}, while we really don’t dive into how real numbers are defined (usually as “anything that can’t be written as a fraction”, but that isn’t even correct!), and complex numbers are just “things you can make from the square root of -1”.

But those definitions aren’t particularly useful, and they sure aren’t abstract enough to study on their own.

Why do we even need to define a number?

Sure, we can just say that a number is “the mathematical abstraction of the concept of the size of a set of objects”, as dictionaries do, but a nice definition would allow us to study the properties of the concept, not numbers themselves. Furthermore, as mathematics is a highly formal field of study, just saying something is… isn’t really acceptable. 

But numbers are a very complex concept. Like, think about it for a second. Three, without any added context, has no meaning in day-to-day speech. “Three apples” makes sense, but just “three” doesn’t. There isn’t a “three” anywhere in the world, but there could be “three of something”. 

Sets first

So, to start defining what a number is, let’s look at the structure surrounding it: in the same way that we don’t really define a vector by itself, but only a vector space – and then define a vector as a “element of a vector space”. (A vector space is similarly defined using axioms, if you’re not familiar.) Similarly, numbers also don’t make a ton of sense to define on their own, so we start off by looking at the set of naturals. We notice that there’s a pattern: every number has a successor (there’s always a bigger fish number), but not every element is a successor: we start with 1 and there’s nothing smaller than 1: nothing preceding it.

But that isn’t enough on its own: these two properties of the set of natural numbers don’t define just the set of natural numbers – they also hold true in some other sets that don’t have the same properties as naturals: for instance, they hold if we define a set like {0, 0.5, 1, 1.5, 2, 2.5, …} and define that the successor of x is x+1. However, this isn’t the set of natural numbers – and it’s not really alike (formally, they’re not isomorfic: we can’t construct an order-preserving bijection whose inverse is order-preserving as well). So, we need something more.

Luckily, the principle of mathematical induction holds true in the naturals. If you’re not familiar with induction, it’s the idea that if you can prove that something holds for n=1, and using only the assumption that it holds true for a general n, that it holds for n+1, then you’ve proven that that statement is true for all natural numbers n. 

These three things are enough to define the natural numbers and were first formalized by Italian mathematician Giuseppe Peano and are called “Peano’s axioms”. He published them in a somewhat different form, but they’re usually stated along these lines, with just a touch of mathematical formality to hold our ideas together in a more sensible notation 🙂

If you’re interested in how that’s written down:

1. For all natural numbers n, there exists a unique successor n⁺ in natural numbers and if  n⁺ =  m⁺, then n = m. 
2. There is a first element in the set of naturals, i.e. 1 ∈ ℕ. It is the only* element of the set of naturals that is not a successor of any natural number.
3. The principle of mathematical induction holds:

Of course, these things hold true for some other sets as well. If we define 1 = ∅ and n⁺ = {n}, we get the set {∅, {∅}, {{∅}}, …} – and these axioms hold true there as well.

Note: the word “only” in * prevents the {0, 0.5, 1, 1.5, …} example from working, but is usually left out from the Peano axioms. This depends on how Peano axioms are stated. See https://math.stackexchange.com/questions/132855/why-do-we-take-the-axiom-of-induction-for-natural-numbers-peano-arithmetic for more info.

How’s that any better?

It is: because that means that the same conclusions are valid whether we think of 1 as an empty set (which lets us define operations and study the sets more closely in an easier to grasp manner (which I know sounds very counterintuitive)) or of 1 as “just one thing” intuitively. This leads us towards the conclusion that it isn’t really about the number itself – it’s more about the structures those numbers build together. Proving a property of 3 isn’t really interesting (it’s easy to prove 3 is a prime: you’ve done it at some point in your life), but proving properties that hold for sets of numbers is more useful (as it’s a more general statement) and more fun, to be honest 🙂

We can go further!

How high can you count? If you’re a child (congratulations for reading this blog in that case! :D), maybe up to a million? If you’re an adult, you’ll probably say… well, up to infinity. But, as anyone who’s read The Fault in Our Stars knows, not all infinities are the same. So, up to which infinity could you count? 

Could we abstract natural numbers even further? Would this give a somehow better definition? How would we define reals? Are these actually nontrivial mathematical questions? 

The answer to the first, second and last question is yes, and the answers to them (and the ways we got there) are quite interesting. Along with the question “what’s actually a set?”, which is something we haven’t touched upon here but took for granted as something that we just know (and it’s a really good question to ask!), these questions have built entire fields of study in mathematics and gave some shocking discoveries along the way. 

Perhaps it’s a story for a different post, perhaps this post will get a sequel if you like it, but to leave you pondering, I’ll end on a riddle.

There’s a village in which a barber shaves every person that doesn’t shave themselves. Who shaves the barber?

Let us know what you think in the comments below.