What's a number? 1 - naturally

In mathematics (and before that, arithmetic) we get used to dealing with numbers in various ways. But what is a number?

What sort of a question is that?

Of course, we all learn to count, and for the most part we learn to become competent. In practice, what this means is that there is an agreed-on sequences of noises that we make when we point at objects, and the number of objects is the noise we make when we get to the last one. And with practice, we find that if we're careful we always make the same noise for the last item of a collection, even if you point at them in a different order. We also find that if you have a pair of collections then they can be matched up in a one to one fashion if and only if we assign the same noise to both of them.
With a little thought, we might come to the conclusion that, for example, three is the name of a number, and any collection that we get to three on by the 'point-and-make noises' algorithm contains three items. But just what is this three? It's the thing that all collections with three items have in common, but what is this 'three-ness' that they all share?
Not knowing just what a number is doesn't stop us learning how to do arithmetic, and this arithmetic is generally very useful. It prevents us from being cheated too easily in shops, and ensures that we know how many minibuses to book for a trip to the seaside, or how with more effort to direct a spaceship to the moon. But a chosen few eventually get uncomfortable, and start asking awkward questions about just what numbers are.
One rather cunning attempt to deal with the problem is to think about sets, and then to define three to be the set of all sets with three elements. Then to say that a set has three elements is to say that it is in this set (of sets).
Alas, that doesn't work. When you try to think about sets carefully, you discover that you have to be a bit careful about just which descriptions actually describe sets. And when you do that, you find that 'has three elements' doesn't describe one, because if you try to include it as a set you get contradictions: this is a Bad Thing. This was a bit of a blow to early developers of set theory, so another solution has to be found.

Who cares what a number is?

An alternative, and even more cunning attempt to deal with the problem is to ignore it. After all, humanity has been counting successfully for millennia without ever coming to any grief caused by not knowing what a number is. Surely all that matters is that, whatever they actually are, we know how they behave. So the solution to the problem is that if we want to do mathematics rather than just indulge in unimportant activities like making sure we get the right change when we buy a bar of chocolate, then we describe how numbers behave by means of some mathematical rules, and then work with the rules.
Well, time to be a bit more specific. I'm talking about the numbers we use to count things and do elementary arithmetic with here, so I'm talking about what are sometimes called the natural numbers, and I'll include in that the special number zero. (Some people like to start the natural numbers at one, but I'm not one of them.)
It's important to realise that there is no right way of doing this. We have to sit down and think about how numbers behave, and try to capture that behaviour is a neat collection of rules.
So we play with numbers (whatever they are) or rather some kind of representatives of them, and we come to various conclusions. We can add them, we can multiply them, order doesn't matter with addition and multiplication, and so on. Observations like this can be gathered indefinitely, so we need to pick out a collection of fundamental properties that the others follow from: an axiom system for the natural numbers.
One way of doing this is due to Giuseppe Peano, who wrote a book about it in 1889 (and in Latin: this was a time when by writing in Latin you might still hope to increase your audience rather than shrinking it). Nowadays, we tend to present things rather differently, but nevertheless Peano's axioms survive in somewhat altered form. So, here is one version of the Peano axioms for the natural numbers, consisting of a set \(\mathbb{N}\) with a successor function \(S:\mathbb{N} \to \mathbb{N}\):

1. \(0 \in \mathbb{N}\)
2. If \(n \in \mathbb{N}\), then \(0 \neq S(n)\)
3. If \(m \neq n \in \mathbb{N}\), then \(S(m) \neq S(n)\)
4. If \(K \subseteq \mathbb{N}\) such that \(0 \in K\) and \(n \in K \Rightarrow S(n) \in K\), then \(K = \mathbb{N}\)

We will also use the convenient shorthand \(S(0)=1\), \(S(S(0))=S(1)=2\) and so on, with the familiar numerals; privately we think of \(S\) as adding \(1\).
This certainly seems to capture some of how we think the natural numbers behave. \(0\) is the first one, we get all the rest by adding \(1\), and we never get the same answer by adding \(1\) to two different numbers. The last is less obvious, but it's just a slightly disguised way of saying 'proof by induction works'.
Now comes the surprise (at least, it surprises me).
These axioms essentially specify the natural numbers uniquely.
By that I mean that if we have another set, \(\mathbb{N}'\) with an element \(0'\) and a successor function \(S'\) satisfying the same axioms, then there is a (unique!) bijection \(i:\mathbb{N} \to \mathbb{N}'\) such that \(i(0)=0'\), and \(S(i(0))=i(S(0))\). So any two sets claiming to be the natural numbers are really just the same set with two different ways of labelling the elements and successor function.
Hang on, though (you may be thinking). It these things are meant to be the natural numbers, shouldn't there be operations of addition and multiplication?
Indeed there should. And they should have all the usual properties too of associativity, commutativity and distribution. But now we know that if we can define them in terms of \(0\) and the successor function, then the natural numbers are still basically unique. (And we don't have to come check the properties of addition and multiplication work separately.) So, here's how addition and multiplication work for \(m,n \in \mathbb{N}\), using \(+\) for addition and \(\cdot\) for multiplication:
Addition

1. \(m+0 = m \)
2. \(m+S(n) = S(m+n)\)

Multiplication

1. \(m \cdot 0 = 0\)
2. \(m \cdot S(n) = m \cdot n+m\)

These definitions are both inductive: they tell us how to combine a number with \(0\), and if we have to combine a number with something other than \(0\) we reduce it to dealing with a number nearer \(0\).
I wouldn't like to claim that it is trivial, but it is straightforward to show that addition and multiplication as defined are associative and commutative and that the distributive laws hold.

That's all very well, but are there any numbers?

And here's the snag. It's all very well saying that we don't care what they are, only how they behave. But how do we know that anything behaves like that?
Remember, we have a bunch of properties that we have abstracted from experience. For the numbers we're directly familiar with (say, up to a dozen or so) we have a fairly direct perception of the truth of the various properties. But why do we believe that they keep on going for numbers which are arbitrarily large? Why do we even believe that the mathematical structure makes sense? How do we know that the axioms are consistent? It could be that we have asked for too many properties, and defined the natural numbers out of existence.
Aside: probably apocryphal stories about PhD students who have proved hundreds of interesting properties of things that don't actually exist abound. We really want to avoid that situation.
In fact, asking about consistency of axioms open an enormous can of worms, which I don't propose to get into right now. I'll settle for a less rigorous approach that is at least very (very very) plausible, and which was invented by the amazing John von Neumann.
First, we assume that set theory is consistent. We don't even need scary bits of set theory, we just assume that no problems arise with some simple sets that I'll build as we go along.
First of all, we have the empty set, \(\emptyset\). This will be \(0\) in the model I'm going to describe for Peano's axioms. (It's a model in a highly technical sense too, but I also don't propose to get into that.)
Then the first few natural numbers are given by \[ \begin{split} 0 &= \emptyset \\ 1 &= S(0)= 0 \cup \{0\} = \{0\} \\ 2 &= S(1)=1 \cup \{1\} = \{0,1\} \\ 3 &= S(2)= 2 \cup \{2\} = \{0,1,2\} \end{split} \] and in general \[ n+1 = S(n) = n\cup \{n\} = \{0,1,\ldots n\} \] with \[ \mathbb{N} = \{0,1,2,3\ldots \} \]
You might wonder, as I did when I first met this, why we don't just use \(S(n)=\{n\}\). That's simpler. Well, there's nothing to stop you doing that, but it's less elegant: for a start, in this picture the set \(4\) has 4 elements, which is handy. In fact, there are other nice properties of these integers, such as the fact that \(m \lt n\) is equivalent to \(m \in n\), which has some huge ramifications. Those are for maybe a future post. For now, we settle for having a set and successor function that look as if they work.

So the natural numbers are these weird sets?

Fortunately not. These von Neumann numbers (often called the von Neumann integers, even though they don't include the negative ones) are just to show that the axioms make sense. The point of them is to say that the axiomatization makes sense, so it is in fact safe not to worry about what numbers are, just how they behave.
It's worth noting that there are models of the Peano axioms that look completely different. The fact that there is a bijection that preserves the structure doesn't mean that the objects have to bear any resemblance to each other.
So here's an entirely different approach. We consider a world of sets and functions, where the domain and codomain of each function is the same. I mean here if I look at any function, it maps some function to itself: different functions can do this with different sets. Then my integers are operators on the collection of functions that are built up like this:

1. \(0(f)\) is the identity function.
2. \(S(n)(f)\) is \(f \circ n(f)\)

This is the idea of Alonzo Church, and can be expressed neatly using his lambda calculus . (In fact the lambda calculus gives a nice way of expressing addition and multiplication in this model of the integers.)
The point is that these integers then are nothing like the von Neumann integers. They are instructions that tell you how often to compose a function with itself. But they have exactly the same structure as the von Neuman integers.
We could even make a model using strings of characters using the alphabet \(\{0,1,2,3,4,5,6,7,8,9\}\) where everything is defined using the usual rules for decimal arithmetic. (Yuck.)
And none of these is really the natural numbers. Or they all are, because they're all really the same, we're just expressing one ideas in several different way. The point is that the axiomatic approach tells us that it doesn't matter how we think about the natural numbers, we can reason about them just by using the axioms, secure in the knowledge that whatever we deduce will be true of whatever concrete realization we prefer.

Are you glossing over anything?

Lots. One is the the subtlety of the inductive axiom. I gave it as involving an arbitrary subset of \(\mathbb{N}\). But if we do logic (and sets) carefully, this is not part of first-order logic. We'd have to say something like

If \(P\) is any predicates such that \(P(0)\) is true and \(P(k) \Rightarrow P(k+1)\) for al \(k \geq 0\) then \(P(n)\) is true for all \(n \in \mathbb{N}\).

This doesn't look like such a big deal. And, after all, it's how we use it.
But it's an enormous deal. If you use this as your axiom, then the axioms are not strong enough to tie down the natural numbers to essentially one version. You can have non-standard models, which included numbers larger than all the normal integers. This is also tied up with non-standard analysis.
I've also glossed over the whole of (axiomatic) set theory, the attempt to put the idea of a set on a reliable footing so that it can't be broken by something like Russell's paradox. It's a fascinating topic in its own right, and a study of it explains why you get into trouble by trying to have a set of all sets with three elements. Interesting as it is, though, we don't need to go to all that effort to appreciate why the von Neumann numbers give a nice model.

Hang on, \(0\) isn't a natural number! This is all wrong!

It is true, not everybody includes \(0\) as a natural number: some only consider the strictly positive integers to be natural numbers. It's one of mathematics's little ironies that there is no universally agreed definition of a natural number. Neither definition is clearly better than the other: each has its minor context dependent advantages and disadvantages.
If I had to draw up approximate lines, I'd probably say that in general terms logicians and set theorists tend to include \(0\), number theorists tend not to. And I will admit now (to save anybody the effort of pointing it out later) that Peano started at \(1\), not \(0\).
I've included \(0\) because it's what I'm used to, and because the von Neumann integers are such a compelling model, and it has a natural place there. I also like there being an additive identity. If you really can't bear the thought if \(0\) being a natural number (and if you can't, I can't imagine what compelled you to get this far) then just consider this whole thing as am discussion of what the non-negative integers are.

Why does the title have a 1 in it?

Now that we know what the natural numbers are (or aren't) we can build on this to make other number systems. I intend to explore a couple of them (eventually) in subsequent blog posts.