This post is part of the What is Maths series.

I'm sure that asking what maths is would seem strange to most people. We have studied maths at school, we know what numbers are, and we know our times tables. On top of these fundamentals, maths education in England includes many sophisticated topics such as calculus, simultaneous equations, and more.

To make my meaning clearer, let me ask another question. What is a car? If you asked this to the average person, they would, after giving you some funny looks, tell you about how it's a machine that can be used to get from one place to another quickly. After further interrogation, they may tell you about how it's controlled by turning a wheel and pushing a series of pedals. If you asked a mechanic what a car is, you would still get a rather funny look, but afterwards you may well get a very different response, and learn about internal combustion engines, suspension geometry, and fuel mixes.

While neither of these people are wrong, they describe wholly different aspects of what it means to be a car. To be clear, I don't intend to suggest that either point of view is better or more correct; it takes many brilliant engineers to build a Formula 1 car, but they need Lewis Hamilton to get it across the finish line. Equally, Lewis Hamilton is unlikely to be able to build a world class race car.

To start to see this new perspective on maths, let's use the decimal system as an example. We all use this system to write numbers, but how do we know it works?

The Decimal System

For simplicity, we shall only think about how to write non-negative, whole numbers (also called natural numbers) for now. To recap, what do we mean when we write a number such as 123? When first learning this system, we often would have labelled the digits, from right to left, as ones, tens, and hundreds. This would mean that we can read 123 as 3 ones, 2 tens, and 1 hundred (or, slightly more symbolically \( ( 3 \times 1 ) + ( 2 \times 10 ) + ( 1 \times 100 ) \)).

This system begs the question, can every non-negative whole number be written this way, or do we miss some out? If we start with a number such as 123, we can first divide by 10 to get 12 with remainder 3, so our ones column is 3. Then, 12 divided by 10 is 1 with remainder 2, so our tens column is 2. Next, 1 divided by 10 is 0 with remainder 1, so our hundreds column is 1. We have now reached 0 so we can stop, and see that our answer is, somewhat unsurprisingly, 123. This example seems somewhat contrived as we started with a number already in decimal form. However, we can see that the method does not rely on this; we could still use it if our starting number were instead written in a different form such as a tally.

To make this argument slightly more formal, we should provide it in a general form. This means that rather than starting with a specific number such as 123, we should start with an unknown number and give it a name. In this argument, we shall call our number \( n \). This style of argument makes it easier to ensure that we don't accidentally use some property that is unique to a specific number and only use universal properties of numbers. It also has the added bonus of making us sound a bit more fancy.

We can now start by dividing \( n \) by ten—calling the result \( n^\prime \) and putting the remainder into the ones column. Now, let's divide \( n^\prime \) by ten—calling the result \( n^{\prime\prime} \) and putting the remainder into the tens column. Repeat until you reach 0 and you've successfully converted \( n \) into a decimal number!

The Flaw in all Arguments

I would now like to pose another question: is this proof correct? While I certainly believe it is, what might a deeply pedantic cynic say? They may say one of several things:

• how do you know that you can divide any number by 10;
• how do you know that the remainder is always a single digit (i.e. between 0 and 9);
• how do you know that the division process will always reach 0?

Upon hearing these complaints, there are a few things we could do. We could provide an argument for each of those criticisms. Alternatively, we could simply claim that it is obvious any number can be divided by 10, so no elaboration is needed. It turns out that each of these options are fraught with danger!

If we choose to provide further arguments, they will rely on various other facts being true. Each of those other facts must also have arguments justifying their use, or we claim those facts are obvious. However, whenever we claim something to be obvious, we risk the possibility of being wrong.

To demonstrate that this is a real concern, we can consider a real example. I won't go too deeply into it, but during the development of set theory, it was believed that any collection of sets could themselves be contained within a set. For example, there is a set containing all sets. This seems like a totally reasonable belief until you consider Russell's paradox. He noticed that some sets, such as the set of all sets, contain themselves. Equally, some sets, such as the empty set, do not. We can now consider a set, \( R \), that is the set of all sets not containing themselves. If you ask whether \( R \) contains itself, you will find a troubling contradiction. The only way to resolve this is to reconsider the very fundamentals of set theory!

The consequence of these concerns is that we can never finish our argument; whatever we produce lacks the detail needed to be truly watertight. Descartes was a mathematician and philosopher and, in order to come up with his most well known quote, he took these concerns to their logical extreme. He asked whether we can know anything at all, even questioning whether we can trust our own senses. After all, what if everything we experience is just an illusion, as in ‘The Matrix’ perhaps? Eventually, Descartes realised there is something we can know with absolute confidence: “I think, therefore I am”. In other words, even if we are not willing to trust our own senses, we at least know that we exist.

Resolving Descartes’ Dilemma

I shall not discuss the philosophical resolutions to this problem here, but I shall discuss the mathematical resolution. It is surprisingly simple: take a small number of facts (which we shall call axioms) and believe them on faith alone! From these axioms, we can use logical arguments to prove whatever other facts we need, so long as those other facts really do follow from the axioms.

It should be immediately clear, however, that the choice of axioms is rather important. First of all, if one of our axioms is actually wrong then everything we claim to have proven is likely also nonsense. This means that all of our axioms should be simple enough that they are not in any doubt. Meeting this requirement is also rather useful if you want to convince someone else that you're argument is sound. Another requirement of our axioms, is that they should be ‘powerful’ enough that we can prove whatever we need; it would be frustrating to be unable to complete a proof because we were missing some crucial axiom.

In the following part, we will begin to explore a set of axioms that aim to describe Number Theory, the study of non-negative whole numbers (i.e. 0, 1, 2, 3, …).