I think everyone is aware of the fact that mathematics faces quite a public perception challenge. This is something I’ve thought about a reasonable amount over the time that I’ve been doing maths. Part of that public perception is that a lot of things in maths just aren’t real. Once I start talking about 5 dimensional spaces or imaginary numbers, I get glazed over looks and the question “why spend so much time thinking about things that don’t really exist?”. It’s a good question! For what it’s worth, here is my answer.

Well, It’s Kinda Fun…

This may be least compelling argument ever for people who have bad memories of doing maths, but hear me out. One of my favourite professors described it to me once as “the feeling of dancing on the tip of a pin”, and that description has stuck with me since. When you see interviews with athletes, musicians, or any kind of performer you’ll likely hear the same kind of sentiment, and I think these pursuits are not as bad an analogy to mathematics as they might seem.

When you first start to learn a musical instrument you are given your scales and arpeggios, you play simple tunes like Three Blind Mice and you plod your way through practice pieces. As you try a new piece you might be able to play the first few bars fine but then a section might come up that you don’t yet have the ability to play through. Perhaps a few minutes of focussing on that bar might be enough, or it could be days, weeks or months until you are able to crack it. But slowly you assimilate more abilities and combine them until you can play quite technically complex pieces. But all the while, even during Three Blind Mice, there’s another sort of learning going on. You are learning how to express the music in your own way. This might be a conscious thought process, but in all likelihood you will be too caught up in the technicalities to notice that you are developing a musical personality. Later though, this musical expression comes to the fore, and less time is spent trying to get through the music note by note. I learned an instrument at school to a reasonable level (maths departments are full of musicians and polyglots, somehow they go together in the skill-set), and I recall finishing pieces of music having made no mistakes at all and thinking “that was bad, I still have no idea how to play this” because there was more meaning in the music that I wasn’t grasping. I’ve heard professionals say that even during playing an hour of insanely difficult piano music, they weren’t at all thinking about the notes, or that they might make a mistake, they were thinking about the interpretation, and the thrill of the music is in finding the right interpretation personal to you.

The only difference with mathematics (and a very difficult one to overcome) is that you can’t hear it. There’s no way of feeling the work of a great mathematician without learning the language they write in. But otherwise this is very much the experience I have with maths. As a school student or undergraduate, you stumble around clumsily through problems getting stuck and trying to get through tricky lines of reasoning in lecture notes. But each time you assimilate a little more knowledge, and the next time you read a proof perhaps more of the pieces will feel right and automatic to you, and you get a glimpse at what the proof really says, and feel why the thing is true rather than just being convinced by a few pages of line-by-line logic. Eventually, you can build up larger proofs, where you aren’t so distracted by the individual steps, you know how to do each bit of logic and become more concerned with the underlying meaning and shape of the proof.

Often, the best moments for me mathematically were when I was trying to figure out the correct way to describe a phenomenon that I could see happening throughout a few different papers or proofs. It’s like trying to describe your emotions to someone. At first you might be able to say some things, but you can’t really hit the nail on the head. Some analogies might work, or you might reframe and try to explain in a different way - but its a very satisfying moment when you finally discover the words for how you really feel. I always love it when an author or songwriter manages to articulate something that I’ve felt in an even better way than I could. Reading great mathematics is often like this - and its even better when you come up with the description yourself.

It Has Applications…

Another argument in favour of mathematics is through all of its applications. This is definitely one of the easier arguments to make. On a small scale, it’s like a big brain gym where you learn all sort of ideas and spend a lot of time thinking hard about difficult things, so that when you are given a problem with actual applications you might have the tools to deal with it. But for many pure mathematicians, the idea that all their pure maths training is because they might one day try and solve a problem in engineering or genetics is probably far from their minds. Instead, they might opt for the slightly bigger picture version which is that we are not ourselves going to try to solve other problems, but we’ll write our maths out in a way that people with applications in mind might be able to come and learn it in order to solve their real problems. This is possibly the more likely view of a theoretical statistics or computer science researcher. They might not be implementing the applications themselves, but they know that their theoretical research will possibly be useful to someone else.

In fact, I think a lot of pure mathematicians also have this view, but the people for whom they think their work is relevant are actually other pure mathematicians! For instance my PhD thesis was in part written with a set of mathematicians in mind for whom I thought “They don’t want to go and learn all this stuff themselves, so it’d be great if I could package it all up in a convenient way so they can just use it when they need it”. If you read a paper in Number Theory or Algebra, they might talk about the applications of the work, but rest assured those applications will often be other equally esoteric parts of pure maths.

There’s also another level of applications which are a lot less tangible but for me were a cool background bit of motivation: often, strange areas of pure maths can find their way to important technological applications hundreds of years after they were invented. Maths can lie dormant, or be the playground of a few fanatics for a while, until someday that bit of theory turns out to be exactly the right thing to solve some new problem in physics or engineering. Accurate GPS requires an understanding of general relativity that is underpinned by manifold geometry developed by Bernhard Riemann in the 1850s, who was trying to figure out how to describe the general notion of space and position. The go-to application for number theory is RSA, the cryptographic algorithm that means you can type your bank details into a computer without someone stealing them - the algorithm is based on the theory of prime numbers and group theory dating back to the 17th Century (specifically Fermat’s Little Theorem).

The World Is Often Unreal Too…

This is closely related to the applications, but I want to stress that by this I mean something like “10 dimensions isn’t actually a crazy thing to think about”, or “imaginary numbers are actually just as real as 1,2,3”. What do I mean by this? Let’s take the example of multiple dimensions. When I was a child, I had a little weather station that measured the temperature outside, and the wind speed, and wind direction, humidity, and all sorts of other things. When I looked on the data history, each minute there was a new row added with at least ten different bits of data. There it is… a 10 dimensional space. To understand that think about what a point is in two dimensions. Usually, we’d give it by coordinates, say $(27.99 N, 86.92 E)$ which is the coordinates of the summit of Mt Everest. In three dimensions, you might specify your height as well, for example the summit of Everest is $(27.99 N,86.92 E, 8849m)$ in coordinates of latitude-longitude-elevation, a 3 dimensional description of space. At the time of writing, the summit of Everest is at $(27.99 N, 86.92 E, 8849m, -19 C)$ in coordinates of latitude-longitude-elevation-temperature, a 4 dimensional space. And it’s moving in that space! It’s getting colder, so it’s sliding along the 4th dimension towards $(27.99 N, 86.92 E, 8849m, -21 C)$. Each minute, I’d get the weather data of my parents’ garden recorded as a point in 10 dimensional space. Obviously I didn’t think that at the time, and it probably wouldn’t have been helpful to, but if you want to learn how the weather forecasts work you’d very quickly find yourself roaming around 10 dimensional space in your head!

It’s the same with complex numbers, group theory, linear algebra, and a whole lot more. The world is full of things that are described by mathematics that might on the face of it seem “unreal”. Sometimes those descriptions are precise, and sometimes they are analogies that help you process and interact with the world. And that’s part of the fun - you get to see new perspectives and understand things differently. Sometimes a book or poem might stick with you and change the way you see the world, and sometimes a category theory lecture might do the same.