Numbers

The counting numbers are as simple as 1, 2, 3. But when it comes to numbers, mathematicians are the experts, and most mathematicians take set theory to be the foundation of mathematics; and set theory makes the simplest equations complicated:

According to the standard set theory, the first counting number, one (1), is the set containing the set containing nothing, where a set is a mathematical model of a collection of things; and adding one (+ 1) to a counting number results in the set containing the set that is that number and also everything that that set contains. Because mathematical proofs can be very formal, where a formal logic is a mathematical model of logical reasoning, mathematicians have some fairly long proofs that 1 + 1 = 2.

Can 1 + 1 = 2 be proved, though? Or is the meaning of “1 + 1 = 2” that insofar as there is one thing and another thing there are two things? Insofar as there is one thing and another thing there are two things because that is what the word “two” means. You will have learnt the meanings of words like “one,” “two” and “three” at roughly the same time as you learnt the meanings of words like “round,” “shoe” and “black,” which you will have done in ways that went something like this:

How many shoes am I wearing? Two

What colour are my shoes? Black

Colours are properties, of things of various kinds (stars, stamps, rainbows, hallucinations, to name but four). To say what those properties are requires some basic physics, biology and psychology. And as for individual colours, they are as hard to describe as the tastes of wines. But shapes are easier to describe (as are numbers), and shapes like round and square are properties, of things like stars and stamps. Now, mathematicians are the experts when it comes to shapes (as well as numbers). But while mathematicians have found some very strange shapes (as well as some very strange numbers), they do not say that round is the equation for a circle, an equation written in a programming language, or anything like that. Why, then, do they say that the number one is the set containing the empty set, where those sets are defined by the axioms of the standard set theory?

Well, mathematics is all about precision and proof, so when mathematicians say that set theory is the foundation of mathematics, they may just be saying that their proofs begin with set-theoretical axioms. After all, set-theoretical proofs that 1 + 1 = 2 could hardly be showing that 1 + 1 = 2; whereas, such proofs do show that set-theoretical models of arithmetic are not bad models of arithmetic. So, when mathematicians say that the number one is the set containing the empty set, they may just be telling us what their model of the number one is.

Why model such simple numbers, though? Why not use the actual numbers themselves? Mathematicians do use them (we all do), and they need to know what their arithmetic is in order to know how good their models of it are. Still, when it comes to questions like whether numbers are properties or not, mathematicians are not the experts. Such questions are metaphysical, not mathematical. Philosophers are the experts when it comes to metaphysical questions. Now, most of the philosophers who are interested in mathematics think that the counting numbers are not properties, of things like pairs of shoes. However, the first counting number, one, certainly seems to be a rather trivial property of things: each and every thing is one thing. And counting numbers bigger than one certainly seem to be properties of collections:

The numerical sizes of collections (such as pairs of shoes) are how many things those collections contain.

Collections have various properties (galaxies, for example, have shapes and spatial sizes, as well as being some rough number of stars), and while a pair of shoes is not just any two shoes (and a stamp collection is more than just some stamps), there are, in any given collection of things, things that are being referred to collectively, and it is how many of them there are that is the numerical size of that collection (and those numbers are precise, rough or variable depending upon the collection). The numerical sizes of small collections can be found by counting the things in them (which is why the simplest numbers are called counting numbers), but all collections have a numerical size (precise, rough or variable). And because there is no biggest counting number (the number one can added to any counting number), the counting numbers get bigger and bigger without end.

How many counting numbers are there? Infinitely many (the word “infinite” comes from the ancient Greek for unending). Most mathematicians would say that there is an infinite number of them all; and while those mathematicians might mean all sorts of things by that word “number,” the number of all the counting numbers (in the ordinary sense of how many of them there are) would be a number that was infinitely big but of the same basic kind as the counting numbers if there was a number of them all (in that sense).

At the other extreme, zero is another precise answer to a how many question (“how many sheep are there in a field that has no sheep in it?” for example), but is zero a property? Maybe not. But then, it might seem wrong to say that zero is a number. After all, we do not use it when counting. And it is not the size of a collection. Nor is the number one, of course. And if you have only one sheep, then you do not have a number of sheep. Nevertheless, the number one is definitely a counting number. And these doubts about zero and one being numbers just go to show how similar numbers and colours are, because there is a sense in which black and white are not colours; in that sense, the colours are made out of red, yellow and blue. Orange, for example, is red plus yellow, and adding a bit of blue makes brown. There are three primary colours because there are three types of cone cells in the human eye; and counting numbers are made out of ones because collections of things are made out of things:

One thing and another thing makes two things, and adding another makes three, and writing that in symbols gives us 1 + 1 = 2 and 1 + 1 + 1 = 3.

There being three primary colours shows that colours can be counted, even though colours merge into each other in the rainbow. Clouds too could be counted, if they were distinct enough for long enough. Still, it is more obvious that you and I are two people. And even if we had never existed, we would still have been two possible people. Possible people can be counted if they can be referred to and they are distinct from each other. What about impossible people? Well, Superman is not a realistic fictional character (he flies in the face of the laws of physics), but Superman and Batman are two superheroes. I suppose that while Superman is physically impossible, he is a logically possible person, in the sense of there being no contradictory information about him in the stories about him; or, if there is contradictory information, then we could say that there are two or more versions of Superman, or different characters called Superman if the stories are sufficiently different.

Are the most basic numbers one and the sizes of logically possible collections?

Well, suppose that they are not. Suppose, to begin with, that while some name did seem to be the name of a number bigger than one, no collection of that many things was a logically possible collection. In what possible sense would that name have been the name of a number, in the sense of a precise answer to a how many question? Going the other way, suppose that while some description certainly seemed to be the description of a logically possible collection, there seemed to be no answer to the question of how many things were in that collection. Could we not say that the answer to that question was the number of things in that collection? Could we not give the number of things in that collection a name, such as nu (and a symbol, v), to make it more like two (2)? Any problems with introducing such a number would be prima facie problems with the description in question being the description of a logically possible collection.

Not too dissimilarly, my stamp collection exists, so it certainly seems to be a logically possible collection. Let us call the number of things in it v. Yesterday v was 100, but today I got a new stamp. Today, v is 101, not 100. Does that apparent contradiction mean that some logically possible collections have no numerical sizes? No, it just means that v is a variable. Insofar as “collection” means distinct things being referred to collectively, the description “my stamp collection” is obviously naming different collections (collections of different things) at different times.

If we take the word “numbers” to mean precise answers to how many questions, then numbers bigger than one do seem to be the sizes of logically possible collections. And it makes sense for such numbers to be composed of ones. So, the counting numbers, together with any infinite numbers that are similarly composed of ones, would seem to be a rather natural kind of number (although the question of the existence of such infinite numbers is even more complicated than the question of the existence of xenobiologically possible colours).

Note that when mathematicians and philosophers say that numbers are something else, they are usually thinking of wider ranges of numbers. They might, for example, be trying to say what numbers like zero, a half and pi are at the same time as they say what one, two and three are. Those six numbers are precise answers to how much questions, and mathematicians call them “real numbers” (the real numbers also include negative numbers), and there are many other kinds of numbers that mathematicians study (imaginary numbers, transfinite ordinal numbers, and surreal infinite and infinitesimal numbers, to name but four).

Why do philosophers think that numbers (numbers like the counting numbers) are not properties (of things like pairs of shoes)? Well, some philosophers take the mathematicians at their word and are thinking about sets. And some regard logically possible collections as problematic, usually for such reasons as Russell's paradox (see Freedom for why they are wrong), even though the idea that numbers are not properties predates Russell slightly (see Brown is the Brightest Colour). And some philosophers think that numbers do not really exist, that numbers are more like the stories that we make up than things like pairs of shoes and the atoms of which they are made. Would they say the same about colors and shapes, though? Those philosophers are usually writing about mathematical models of such things, rather than the things themselves (in the interests of precision, they say), so it can be quite hard to tell sometimes.