Abstraction: numbers and structures

The first part of Chapter 2 of Gowers' book, "Mathematics: a very short introduction" and the previous web page introduced us to the idea of natural numbers via what they do, not what they are. What they do is recorded as rules called axioms true for all objects in the set of natural numbers with certain operations (in this case, addition and multiplication). We duck the question of what natural numbers are by agreeing that whatever it is they are we are only really interested in the list of things that they actually do.

This is the first real example of abstraction. A collection of numbers is defined through a list of axioms. The thorny problems of what numbers actually are are no longer important, and we have an important new view on maths. There is a additional benefit to this view, which is that by explicitly denying the relevance of what natural numbers are we are ensuring that no misguided mathematician will in the future ever attempt to use this irrelevant information.

Gowers, in the rest of Chapter 2 takes us on a brief tour of the number-kingdom.

For the record, there are two things that Gower does not say in these respects.

Mathematicians quickly get used to a great number of subtly different sets of axioms for different systems. Gowers is trying to simplify the situation somewhat, at the risk of being possibly slightly misleading.

Natural numbers are not adequate for all purposes. For example natural numbers do not work well with subtraction. In terms of models, they are not very good numbers to use to model our bank balance when we go in the red, or the temperature in celsius on a cold day. The solution is to consider a larger set of numbers called the integers. The integers have a slightly different set of axioms, but interestingly enough the familiar commutativity, associativity and distributivity axioms remain OK. Of course, when we look at the integers, we need to decide things like what happens when you multiply two negative quantities. Why do we choose the particular convention that a negative times a negative is positive?

But if one wants to divide, not even the integers will do. The rational numbers are an extension of the integers with division. They have slightly different axioms too, but still commutativity, associativity and distributivity remain OK.

If one wants to take square roots of positive numbers, or take limits (chapter 4) not even the rationals will do. Now we need the real numbers. These (at last!) are the numbers that you will be most familiar with at GCSE, A-level, or from using a calculator. However there are a great many tricky details with real numbers and mathematics with real numbers is possibly the first place that even very good mathematicians make mistakes and accidentally prove ridiculous things like $0=1$. (Needless to say all such proofs do indeed have errors in them. Writing proofs out properly is the only way such errors can be found. This is the topic of Chapter 3.)

If one wants to take square roots of negative numbers as well, it is necessary to pass to complex numbers. These include numbers such as $i$, the square root of $-1$. Of course the complex numbers have yet another set of axioms, but (interestingly) commutativity, associativity and distributivity axioms still remain OK.

Gowers doesn't take the number-theoretical zoo any further, but we could if we wanted to. After the complex numbers (which is like a two-dimensional version of the reals, see chapter 5 on dimension) there is a system of numbers called the quaternions which is a four-dimensional version of the reals. Again, the quaternions have a set of axioms of their own. In this case associativity and distributivity remain OK but commutativity breaks down.

So what is the use of four-dimensional numbers? Well, quite a lot really. A quaternion can naturally be thought of a point in Einstein's four dimensional space-time (one value for time, a triple of values for coordinates in space), or as a value in electromagnetism (a value for charge and a triple of values for the quantity and direction of magnetism). So numbers and models continue to go hand-in-hand, where numbers are used to model real world situations.

(I should point out that it is not necessary for this course to find out details of the quaternions, though they are not very difficult. But it is interesting to know that they are there and have been studied.)

So what is a number? Is there a small set of axioms that will be true for all numbers? I don't know of any such set of axioms. The basic ones we have seen, commutativity, associativity and distributivity, would seem to be a suitable suggestion, but we have seen that commutativity breaks down for the quaternions (and also for other systems I do not have time to mention). Also, there are other objects that behave rather like numbers which might or might not be reasonably called "numbers". It's simply a matter of taste! An example is "sets", where operations of intersection and union behave in many ways just like multiplication and addition. Are sets numbers?

The end point of this discussion is that the concept of "number" is a spectrum of possibilities, with different lists of axioms at different points on the spectrum, different properties for the different structures, and different uses for each. In other words we have an abstract idea of "number" in all its possible guises, all with different possibilities and uses.

The argument that infinity cannot be added as a number with some "reasonable" properties such A1-3,M1-3,D and $0\times \infty =1$ seems quite convincing.

On the other hand, axioms A1-3,M1-3,D and $\infty +1=\infty$ are consistent when taken together, even though there is no "normal" number $x$ such that $x+1=x$. You can see this by making a "model" of the axioms A1-3,M1-3,D and $\infty +1=\infty$ by letting the objects in this model be the usual non-negative integers and a single special new object called $\infty$ which is not an integer. You then need to define $+,\times$ on this set of new objects. We do this by taking addition and multiplication to be the "normal" one on numbers and for all other cases, $\infty +x=\infty \times x=x+\infty =x\times \infty =\infty$ on all objects $x$ (including $\infty$). By careful checking one can show that axioms A1-3,M1-3,D and $\infty +1=\infty$ are all true for this model. On the other hand, axiom A4 is not true in this model. If you think A4 is important you may be interested to know that $\infty +1=\infty$ is not possible in a system with A1-4 and the rather obvious looking axiom $1\ne 1+1$.

This is why a lot of creative mathematical work goes by trying to invent new rules or axioms and then trying to show the new rules are consistent. The problem is you don't know in advance if your set of rules or axioms won't have some hidden "issue" like the problem with infinity on page 32. This is pretty bad, because it is not sufficient to say "I had a look for problems with my axioms and couldn't find any, so they must be OK." You do need to prove your axioms are OK. Making a model for your axioms is one possibile way you might try to write such a proof.

Suffice it to say here that arguments like this are usually possible but more much complicated than anything we will do in this course. Arguments building significant mathematical models for new sets of axioms are typically treated first at 2nd and 3rd year level in mathematics degree programmes. "Consistency" is a central topic in the work in foundations of mathematics (by Hilbert and others) in the beginning of the 20th century.

Gowers gives rules L1,L2,L3 for log. You need to be slightly careful with these rules because it is difficult to interpret $\log x$ when $x$ is zero or negative. Until you know quite a lot about complex numbers you should take with L1,L2,L3 an extra rule that says $\log x$ is only defined when $x$ is a positive real number. (Even when you do know about complex numbers, logs are sometimes problematic. In the complex numbers, when $x\ne 0$ there are infinitely many possible values for $\log x$. Logs are a bit like square roots, but worse!)

Just like the other rules or axioms for numbers, the rules for logs don't just define the way logs behave and what they do but they give you just enough information so that you can calculate logs. (Gowers shows you how to calculate $\log 1000$.)

Is there a set of axioms that will be true for all numbers?

Preliminaries:

Explain the term "axiom" in terms of an abstract mathematical model of a real-world situation.

In this case, the modelling process we saw ealier (chapter 1) for different areas of science is actually being applied to mathematics itself. Different parts of mathematics are used for the "real world" and the "abstraction". It pays to keep these separate in your mind.

Finish this section with the idea that axioms are like rules in an abstract game. They say what the objects under consideration "do" to each other, in an ideal and abstract sense.

Discussion:

Explain that what numbers "do" most obviously is that they add, multiply, etc.

Give examples of axioms for addition, multiplication.

Mention that experience shows that some are highly important, e.g. commutativity of $+$, associativity of $\times$, others possibly used less frequently. In other words not all of them are always needed.

Optional: could mention other models that don't look like numbers but still obey (some of) these axioms. E.g. sets with union and intersection; vectors with addition (and possibly multiplication) matrices of a particular shape and size with addition and multiplication.

Invent some silly model, like a set of jelly beans that behave like numbers.

Conclusion:

Axioms do not tell you anything about what the objects they describe really are. Jelly beans are not numbers, at least not usually; but sometimes they behave like numbers.

For every sensible axiom you might consider there is probably a model where the axiom is false but the system behaves in some way like "numbers".

Although axioms seem the best option for helping us recognise numbers, there is a wide range of possible sets of axioms and number-like behaviour. No single set of axioms seems to capture the concept "number" exactly. The concept "number" is too vague and has too many variations.