Chapter 2 of Gowers' book, "Mathematics: a very short introduction" starts with an argument that (abstract) chess pieces exist, independent of (real world) chess sets. It is designed to ridicule an argument in a TLS review that numbers exist.
Suppose, in chess, the black king can move to any neighbouring square. Then there is a piece in chess that can move to any neighbouring square. So there is a piece in chess. So chess pieces exist.
Of course the supposition that a black king exists obviously implies that chess pieces exist. Gowers' point seems to be that this is a obvious and trivial argument. And I agree. More importantly, he also goes on to say that the question of what these objects that exist really are is unimportant. It's much more what these objects do that matters. For of course, many designs of chess sets have been used for what is the same game. On the other hand, the abstract design of the board (an 8 by 8 square) and how pieces move on that is essential for chess. In this respect chess pieces are just like mathematical objects.
You could think of this in terms of models: we may be playing a game of chess, with pieces made out of wood following the classic Staunton design. If someone proposes a game using the Lewis chessmen it would still be chess. When we play chess we hold a model of the position in our head when we think about it. That model is inexact and abstract: it does not contain information about the height of the pieces or the fact that some of the pieces are painted badly, or one of the kings has a chip on its crown.
The reason that absolutely no information is lost at all in this model actually suggests that it might be better to think of the game "chess" as being the abstract model itself, and thinking of the board and pieces as just a convenient physical representation of the abstract game. In this view it might even be possible that we could play chess even if every last chess set in the world was destroyed.
I can't resist raising a classical problem in logic that is very close to this - a problem that was not solved satisfactorally until about the beginning of the 20th Century and is still widely misunderstood today - since the solution is essential for careful and rigorous mathematics.
Consider two versions of the above argument.
One of these arguments is correct, in the logic of mathematics. The other isn't. Which?
Both arguments would have been regarded as correct by all, or the vast majority of, logicians up to about 1900. From the way the question is phrased, it is not difficult to guess that the second is correct, and therefore I am saying that the other is not correct. But what is wrong with the first?
The problem is that the first only tells you what to do if you have a king. It does not tell you that you do have a king. The supposition is that
From this you cannot assume you do have a black king! (You cannot even deduce there is a game called chess.)
The "logic in mathematics" is often subtly different to "logic in the street" where no one would think that you'd be told what to do with a king unless such things actually exist. But "existence" in maths is more subtle and it pays to be precise in these matters.
Numbers are like abstract pieces in chess. We specify numbers by saying what they do, not what they are. (So numbers are not baskets of apples, or strokes on a page, or any other kind of notation.) As Gowers says, this is not a rule you must accept as a piece of mathematical faith or dogma. It's just the best attitude to take to get things done.
For example, I write numbers in one particular script, and then I meet an alien who uses a different script. E.T. and I cannot agree on how to write numbers (and I secretly think my notation is better) but we agree in what they do. So I can have mathematical conversations with E.T. perfectly well. Doing mathematics is frequently like that.
In this sense, all mathematicians actually do is prove theorems of the following kind:
So mathematicians don't even need to know that numbers exist! (Check with the last section why the statement above does not imply numbers exist.)
Actually, it is probably a good idea in mathematics (just like in real life) to have a reasonably good idea why you think numbers exist. That's because if they don't exist, all the work we do is a waste of time. But the point is that you don't have to have a definite proof that numbers exist and know what they are. It suffices to explore the consequences of "what if numbers exist".
Natural numbers are often called "counting numbers" and are intended to be the numbers that count how many discrete objects in a set or collection.
There seems to be some dispute whether to include zero (0) as a natural number or not. Gowers seems to follow the small majority (perhaps 51.9%) of mathematicians that choose not to allow 0 as a natural number. I am in the other camp, with 48.1% of mathematicians and 100% of theoretical computer scientists (who do a subject very closely related to pure mathematics) who think 0 is a natural number. One argument is that if natural numbers are counting numbers then it should be a major embarrassment to all of us that 51.9% of mathematicians cannot count the number of leprachauns in the room.
Irrespective of what you think of that particular argument, you may make up your mind for yourself, and decide one way or the other. In written work, if it matters whether 0 is natural or not you had better say what your convention is.
Gowers lists six rules on page 23 that are essential for understanding of the natural numbers, and indeed all six rules are correct for the natural numbers whether or not one accepts that 0 is natural or not.
So these six rules are amongst the things that natural numbers do. This is the slogan "it's not what you are but what you do that matters" and fits with what was said earlier. Since the point of these notes is to add a little more information to what Gowers writes, I will add a couple of things at this point. Firstly,
Secondly, Gowers does not really spell out the form that the properties of a natural number, or list of things that a natural number "does", really takes. Instead, he asks us to look at rules that apply to all natural numbers. But if you were to ask for a complete list of what it is that the natural number 5 actually does the answer would be slightly complicated and look like this.
(I won't give the Dedekind-Peano rules here. Note that they are slightly different depending on whether you believe 0 is a natural number or not. If you are in the camp that says that 0 is not a natural number then 1 is the only natural number such that , and is not needed.)
The point here is that "what it is that 5 does" is a list of things of the form "5 is a particular object in a set with operations for addition and multiplication obeying certain rules". As stated before, exactly what "the thing that is 5" really is becomes unimportant, but in fact the "5" itself is also of secondary importance, and what is really important is the set of natural numbers, the operations of addition and multiplication, and the Dedekind-Peano rules.
This is the starting point for a more sophisticated view of what numbers are, related to something often called "structuralism" by philosophers. That is, the structure satisfying the Dedekind-Peano rules (or axioms as they are more usually called) takes centre stage. In fact there is even a theorem (due to Dedekind) that says that any two structures satisfying these axioms must behave in all respects in exactly the same way. So if I met E.T. and we started talking about natural numbers, then provided we agreed on the Dedekind-Peano axioms we would know we were talking about the same objects, or if they weren't exactly the same object it wouldn't matter in the slightest.
title: What is a natural number?
Start with the intuitive concept of natural number as counting number, and say this is what we are trying to capture more precisely.
Mention that there are two camps, and I have decided to accept that zero (0) is a natural number here, because we should surely be able to count no objects, and because the Dedekind-Peano rules are simpler if we include 0.
Say that "primitive" ideas of number (such as apples in a basket or strokes on paper) do not work for numbers we wish to consider such as which is too big to be represented in any of these ways.
Start to list properties of numbers. Commutativity, associativity and distributivity are good examples. Recognise numbers through these operations rather than as objects in their own right.
Advanced: explain the axioms for subtraction of one number by another that is no larger (this requires zero of course) and the rest of the Dedekind-Peano axioms, including the rule of mathematical induction. If you can't do this at least say that the Dedekind-Peano axioms have been written down and work for these numbers.
Say that by a theorem of Dedekind, two structures satisfying the Dedekind-Peano axioms behave in exactly the same way. This consitutes the beginnings of structuralism, in which different systems satisfying exactly the same properties may be regarded as "the same" and objects in one such system can be "identified" with objects in the other system.
Illustrate this with the E.T. scenario where E.T. and I disagree on what natural numbers actually are (I might think they are strokes on paper but E.T. thinks they are hairs on the back of a hiffogrump) but we agree on the Dedekind-Peano axioms. So we can talk about something together perfectly well. We are therefore talking about abstract objects independent of our particular way of perceiving them.
Finish with the definition, "A natural number is an abstract object that is a member of a structure satisfying the Dedekind-Peano axioms." Point out that it doesn't really say what a natural number really is but it no longer matters.