In Defence of Mathematical Beauty

"Why are numbers beautiful? It's like asking why is Beethoven's Ninth Symphony beautiful. If you don't see why, someone can't tell you. I know numbers are beautiful. If they aren't beautiful, nothing is." - Paul Erdős
Linear interpolation
I can't deny it: I think maths is beautiful.

At this point a lot of people with a near-phobia of maths are probably going to look at their screen like it's about to eat them. How can you find something so scarily abstract so gorgeous? How can you love it so much you're looking at studying it for another four years at university - for fun?! Equally, a lot of people who understand maths are probably slightly confused; maths is a tool. It's functional, not beautiful. Where's the beauty in numbers scribbled on a page or in calculations you make when trying to build flats or predict the growth of an economy?

Well, you see, in part a lot of it relies on my weird head. I don't think in English - well, not strictly speaking. My thought process is actually far more akin to maths, so I find it easier to understand what the maths is telling me than people who think in words or pictures. To me maths is far more than just a collection of symbols, as writing is far more than a collection of squiggles. To me maths is far more just than a tool to work things out, as English is about so much more than dry legal documents. To me maths is nothing less than the language of nature itself.

True, its language is not like any language we speak; it seems almost otherworldly sometimes because it deals with the world around us in a completely different way. It has difficulty expressing concepts we can talk about much more with other languages, or even in the visual arts or music. Yet when it does express things, it expresses relations - and it does so with striking simplicity and power. It links concepts in a way that I find simply breathtaking.

At this point, I'm not really sure how to express myself in words properly, so I'm going to use an analogy; imagine you're happily walking along observing the world and seeing its beauty. You know that some things relate to each other, but you don't know exactly how they relate to each other, and you can't imagine how some other things might relate to each other. Nor do you even think they relate to each other at all.

All of a sudden, you hit upon something that seems to link two things together. Then on closer reflection it links another three things, then another four, then another five...Now, where once you had many disparate things, you can see that they're really part of one grand overarching scheme and suddenly you understand so much else. Feels great, doesn't it?

Euler's identity - e^(ipi)+1=0
I consider that to be one of the most beautiful feelings in the world. And it's the feeling that maths gives me, over and over again. Every time I derive or learn another equation I can learn more about how the world works and about how everything is interconnected - and so I learn more about my humble place in this universe. And yes, maths is functional, but it's also beautiful. That feeling is one of sheer beauty. To use another comparison with languages, yes, you can use Spanish or German for writing contracts or trashy bestsellers-of-the-month, but you can also use Spanish or German to write beautiful, moving poetry. It's the same with maths; yes, you can use it to design a plane or build railways (and I would argue that the use of maths in those two things is beautiful), but you can also admire the beauty of Euler's identity, which so simply and elegantly links five fundamental constants.

This is why maths is beautiful.