What are numbers?

Numbers are the central object of study in mathematics. Most humans, and certainly all humans in modern, industrial societies, are familiar with numbers and, indeed, use them regularly in their daily lives. Despite this, non-mathematicians will probably find it difficult to give a proper explanation of what numbers actually are. I don’t mean that people actually misunderstand what numbers are—in fact most people do understand what they are—it’s just that it’s difficult to put this understanding into words. A lot of the concepts that are fundamental to our understanding of the world are like this—easy to understand implicitly, hard to explain explicitly. As an example, think about the concept of ‘colour’. Five-year-old kids can understand this concept. But can you explain it in a way that a five-year-old kid could understand? I don’t think I can.

One problem with explaining basic concepts like this is that explanations necessarily make use of concepts themselves. In order for an explanation of a concept to be a proper explanation it has to not make use of the concept itself (for example, it is not very helpful to explain that 2 is the second number since the explanation of the meaning of ‘second’ relies on knowledge of the meaning of 2). So it is impossible to properly explain a concept without relying on other concepts being understood. In this sense, it is actually impossible to explain a concept completely. This is why if you repeatedly ask someone ‘Why?’, they always end up not being able to give a satisfactory answer. It’s not just that the person you’re asking isn’t smart enough: no matter how intelligent the being you ask is, they will be stumped after a given number of ‘Why?’s.

The fact that nothing can be completely explained is not quite as disturbing as it may sound: really, it’s just an indication that maybe we should define what it means for an explanation to be complete in a different way, or do away with the concept altogether. It is unhelpful to think of explanations as complete or not in an absolute sense, but an explanation can be complete given a set of concepts if it does not rely on any concepts other than those in the set. The concepts in the set can be referred to as the fundamental concepts used in the explanation. As long as the fundamental concepts are understood, the explanation is perfectly satisfactory.

Now we can state the reason why it’s hard to explain basic concepts: for complex topics, you can usually find an explanation which relies fundamentally only on basic concepts, and these explanations seem more satisfying because you can assume the basic concepts are already understood. But if you’re trying to explain a basic concept, there aren’t many concepts which are even more basic; it’s likely that some of the fundamental concepts in the explanation will not be much more basic than the one you’re trying to explain. That makes it harder not to feel like those concepts need an explanation too.

The reason I say all this is to warn you that you might have this problem with the explanation of what numbers are given in this post. This explanation relies fundamentally on the meaning of the word set. So whether you’ll be happy with it depends on whether you think your understanding of what a set is relies on your understanding of what a number is. If you think it does (and I don’t think this is a ridiculous position to hold) then this explanation might seem pretty useless!

You probably do understand what a set is, even if you haven’t come across the term in its mathematical sense. The meaning of the word ‘set’ in mathematics is basically the same as the normal English meaning of the noun ‘set’: a set is a group or collection of objects. The objects contained in a set are called its members. The important thing to realise about how mathematicians use the word ‘set’ is that mathematical sets do not have any additional structure to them beside their members. To see what I mean, consider a fairly concrete example of a set: the set of items in a shopping bag. If you wanted to describe the shopping bag, there is a lot more you can say about it than just saying what items it contains. For example, you can describe the nature of the bag itself, or you can describe the arrangement of the items within the bag. But these aspects of the bag are not aspects of the bag as a set. When we think about the bag as a set, all those details are abstracted away, and we are only concerned with which objects are in the set.

If you are familiar with formal logic, it might help if I state the following rule which gives a condition under which two rules are equal.

$(\forall A) (\forall B) ((\forall x) (x \in A \Leftrightarrow x \in B) \Rightarrow A = B).$

For every object $x$ and every set $A$, $x \in A$ is the statement that $x$ is a member of $A$. Therefore, in words, this rule states that

For every pair of sets $A$ and $B$, $A = B$ if it is the case that for every object $x$, $x$ is a member of $A$ if and only if $x$ is a member of $B$.

It might also be helpful to see that this can be expressed equivalently as

For every pair of distinct sets $A$ and $B$, there is an object $x$ which is in one of $A$ and $B$ but not both.

This is just a more precise way of saying what I was talking about above, about how sets have nothing more to them than their members.

So, how do sets help with defining what numbers are? Well, first let’s note that it is possible to speak of the number of members a set contains. This number is called the cardinality of the set. For example, there is a set with no members at all (which is called the empty set); that set has cardinality 0. And sets with just a single member have cardinality 1. Note that some sets have infinitely many members, so we cannot assign those sets a cardinality. If a set can be assigned a cardinality, its cardinality will always be a natural number (a non-negative integer such that 0, 1, 2 or 3).

Perhaps, then, we could define the natural numbers in terms of cardinality. We would have to find some way of distinguishing sets that have different cardinalities that does not rely on the notion of number already. This would allow us to classify the sets according to their cardinalities. Then each natural number can be defined as a symbol representing one of the classes. For example, 0 would represent the class of all empty sets, 1 would represent the class of all sets with a single member, and so on.

There is, in fact, quite a simple way of distinguishing sets with different cardinalities. But before I can articulate this I’m going to have to introduce a couple more concepts. The first concept is that of a correspondance from one set to another. A correspondance from a set $A$ to a set $B$ is simply a way of distinguishing certain pairs of objects $a$ and $b$ where $a$ is a member of $A$ and $b$ is a member of $B$. The objects in the distinguished pairs are thought to correspond to each other. Correspondances between finite sets can easily be represented as diagrams: just write down the members of each set in two separate vertical columns and draw lines linking the distinguished pairs. Here are some examples of correspondances, represented by diagrams, between the sets $\{1, 2, 3\}$ and $\{A, B, C\}$ (i.e. the set of the first three positive integers and the set of the first three capital letters; the curly-bracket notation used here is a standard one for writing sets in terms of their members). I’ll call these correspondances $R$, $S$ and $T$, going from left to right.

The last correspondance, $T$, is the kind we’re interested in. By $T$, every member of $\{1, 2, 3\}$ corresponds to exactly one letter, and every member $\{A, B, C\}$ corresponds to exactly one number. In terms of the diagram, each object is linked to exactly one object in the other set by a line. Correspondances like $T$ are called bijections. $R$ and $S$ are not bijections, because 3 corresponds to no letter by $R$ and 1 corresponds to both A and B by $S$.

It’s no coincidence that $\{1, 2, 3\}$ and $\{A, B, C\}$ both have three members. In fact, there is a bijection between two sets if and only if they have the same number of members. This may be pretty clear to you from the definition of “bijection”. If it isn’t, consider this: when you count the members of a set like $\{A, B, C\}$, you assign 1 to A, 2 to B and 3 to C, and so you are actually showing that the correspondance $T$ exists. Of course you could also count the other way and assign 3 to A, 2 to B and 1 to C; that would still be a perfectly valid way of counting, because the correspondance thus established is still a bijection. But if you were to “count” by establishing the correspondance $S$, i.e. by assigning 1 to A, 1 to B and 2 to C, and then conclude that $\{A, B, C\}$ has only 2 members because you only got up to 2, this would obviously be wrong. The ways of counting which work are exactly those which establish a bijection between $\{1, 2, 3\}$ and $\{A, B, C\}$.

This, then, is how we can distinguish sets with different cardinalities. So then we can define each natural number, as explained above, as a symbol representing the class of all sets with a given cardinality. Of course, to be totally sure that this definition works we have to prove that all the things we expect to be true about natural numbers are true when they are defined in this way. Now, in the 19th century, Giuseppe Peano gave an axiomatisation of the natural numbers: a set of self-evidently true statements (called axioms) which could be used to prove all true statements about the natural numbers1. These axioms are as follows.

1. For every natural number $n$, $n + 1$ is not 0.
2. For every pair of natural numbers $m$ and $n$ such that $m + 1 = n + 1$, $m = n$.
3. For every set $S$, if $S$ contains 0, and $S$ has the property that for every member $n$ of $S$, $n + 1$ is a member of $S$, then for every natural number $n$, $S$ contains $n$.

In order to check that the axioms are satisfied we’ll have to define 0 and 1 and explain how natural numbers can be added using this definition. Clearly, 0 should represent the class of all sets with no members and 1 should represent the class of all sets with exactly one member. Now, suppose $m$ and $n$ are natural numbers and let $A$ and $B$ be sets from the respective classes which they represent. $A$ and $B$ can be chosen so that they have no members in common, and since $A$ has $m$ members and $B$ has $n$ members, the union of these two sets, $A \cup B$—the set which contains the members of both $A$ and $B$—has $m + n$ members. So we define $m + n$ as the class of all sets equal in cardinality to $A \cup B$.

Now you have the tools you need to prove that the axioms are satisfied. I’m going to end this post here, although perhaps I’ll post the proofs some other time.

Footnotes

1. However, this is not actually true: there are true statements about the natural numbers which cannot be proven by Peano’s axioms. One example is Goodstein’s theorem. Goodstein’s theorem is known to be true because it can be proven if you develop the theory of the ordinal numbers, a larger class of numbers which contains the natural numbers. There are, however, still true statements about the ordinal numbers which cannot be proven within that theory. In fact, any consistent theory stronger than that given by Peano’s axioms is incomplete in this way—this is the famous incompleteness theorem of Kurt Gödel.