*Charlie Ahrendts*

When we think about something that is universally true, mathematics is one thing that often comes to mind. No discussions or interpretations, we can prove everything to be either wrong or right… or can we? The bad news is, that is in fact not that simple. The good news, however, is that we can explore a super fun and interesting side of mathematics that you likely haven’t been taught in school. Let us imagine journeying across the universe, visiting different alien planets to discover the maths that they use to understand their world. But, before we start, we need to learn a little bit about the basic rules of mathematics: introducing axioms.

### What are Axioms?

Axioms are the most fundamental truths in Mathematics which are used to prove or disprove statements. The axioms we most commonly use are called the real number axioms (short explainer article here). They are a collection of powerful yet simple statements about numbers and the operations we perform on them. One example is called commutativity. It says that calculating x + y is the exact same thing as y + x. The same is true for multiplication. This might seem trivial, but without this rule there would be no mathematical way of disproving the statement that 5+7 is smaller than 7+5.

### Why do we have the ones we do?

Every Axiom exists for a reason and most of these reasons fall into one of three categories: empirical knowledge, usefulness and advancement.

The first and most intuitive one is empirical knowledge. It describes the information we gain by observing the world around us. If we look at the axiom of commutativity again, it is easy to see how it corresponds to our physical world. It makes no difference whether I take 2 apples and add one or whether I have one and add two. The fact that in the end I have three apples is so natural to us that it only makes sense to also be reflected in our mathematics.

Most of the time, we use mathematics as a tool. A very versatile tool, which nowadays we surely couldn’t live without. One of the earliest fields of mathematics is geometry. The ancient Greeks loved geometry and used it to build their massive temples and statues, but they were not the first ones to discover its usefulness. It is believed early forms of geometry are likely to have originated in the way early farmers measured and compared fields, demonstrating yet another reason for a new axiom to be included: practical use. Almost every axiom relating to multiplication can be said to fall into this category. When we talk about multiplication in everyday life, we usually mean adding something multiple times to itself. The notion of multiplication was likely introduced because it makes calculations such as these easier, as well as more intuitive.

The third and final reason for the existence of axioms is advancement. Progress in every new area of life and science provides new obstacles, and some require us to rewrite the rules of mathematics to overcome them.

This last reason might seem a little odd at first: we just decide to add some statement as a new axiom. This does not need to be rooted in any physical observation we make or any direct practical use, but it often opens the door to new areas of mathematics, some of which will later prove to be very useful for some problem which is as yet undiscovered. One such axiom is called the axiom of choice. It belongs to a branch of mathematics called set theory (read a short explainer here), which is concerned with collections of objects, and has its own system of axioms called the Zermelo-Fraenkel axioms. The axiom of choice states that if I have a collection of bins, all with objects inside them, it is possible to make a new bin with exactly one object out of every existing one. This seems obvious at first, but gets very complicated once we try to apply it to infinities. The interesting thing about the axiom of choice is that mathematicians can’t decide whether it should be added to the Zermelo-Fraenkel axioms or not. Some say that its implications are so unintuitive, that it would make no sense to add it. Others argue that it allows us to solve a number of interesting questions and advance into new areas of mathematics. I won’t go into much detail here, but the relevant thing to take away here is that adding a new axiom can be very controversial, with arguments mostly based on feelings rather than facts.

### Universality and truth

As we have already seen, not every mathematical axiom we have is based on our physical world. There is nothing inherently true about most of them, but that doesn’t mean that they aren’t valid. An axiom can be added to a system as long as it doesn’t contradict any other axiom, and as long as it is not deducible from the already existing ones. With only these constraints, we could make up many different axiomatic systems, all of them self-consistent, but still providing different answers to certain questions. And we already have done so. Our axioms have changed over time, the same way that our societies have changed too. There are even different axiomatic systems coexisting, with mathematicians sometimes having to explicitly state which one they are following when producing a piece of work. This is especially true with the previously mentioned axiom of choice because there is no consensus about whether it is a valid axiom or not. But, apart from that, almost every branch of mathematics has its own set of axioms that are tailored for the specifics of that branch. All of these agree with the basic observations we make about our universe, but that again is more a matter of practicality than anything else.

At this stage in our discussion, you may be wondering whether there might exist one set of axioms that is more complete than any other that, once reached, provides an answer to all of our mathematical problems. And you would not be the first. This belief was held for a long time, but unfortunately we now know it is not true. In the first half of the 20th century the Austro-Hungarian mathematician Kurt Gödel proved that no axiomatic system can ever be complete. There will always be problems that cannot be decided, meaning that we cannot prove them to be either true or false. This is the infamous ‘Godel’s Incompleteness Theorem’ (read a short explainer here).

Now that we know the foundations behind axioms, we can look some more interesting questions such as: would aliens develop the same mathematics? We are in no way the first mathematicians to explore this question and in fact it has been looked at a couple of times already on this very website through interviews with Professors Ian Stewart and Adrian Moore. Some people think that maths is discovered, whilst others think that it is invented. Adrian Moore explains how likely we think it is that aliens understand mathematics, depending on these two sides of the argument. Ian Steward goes into detail around the origins of mathematics on earth and how our perception of mathematics is influenced by our perception of the world around us, as well as how aliens on Jupiter might structure their mathematics.

I think we are now just about ready to begin our journey into the vastness of intergalactic space. We have a lot of civilizations to visit, all of which will surely help us to understand more about our own world. So, I invite you to let go of what you think you know and come with me on this mathematical journey of discovery…

Stop 3: Multi-dimensional World

### Conclusion

Back on earth we finally have time to reflect upon what we have learned. We set out on a journey to experience alien maths – which we undoubtedly have done – but we also came back knowing a lot more about our own mathematics. We learned that maths is not universal. There can never be a complete axiomatic system. We now understand that the axioms we choose influence the way we think about most problems which are in turn influenced by observations about the world around us. All of the aliens we visited have managed to adapt their mathematics to their environment. It allowed us to understand that taking a new perspective, thinking outside of the box, or maybe even questioning your fundamental axioms, can be a very good way to solve the great problems of our time. And most importantly, we see that maths is much more than just boring calculations and there will always be more for us to learn…