Why is mathematics interesting?

Recently I started reading What is Mathematics, Really? by Reuben Hersh, an American mathematician. It's a fascinating first look at the philosophy and practice of mathematics - what mathematicians do and our ways of explaining how we should be doing it. (These often don't match up.)

I value this book for three things: firstly, its clarity. While I have a background in mathematics (well, physics really, but I like pure maths and would study it more in my spare time), my background in philosophy is rudimentary at best. This puts me at a massive disadvantage when thinking about the philosophy of mathematics.

Secondly, it advocates something new and something I've been thinking about for a long time. In mathematics, three philosophies dominate: mathematical Platonism (all mathematical objects exist in an idealised, abstract form independent of time and space), intuitionism/constructivism (mathematics is an activity invented by humans rather than the discovery of fundamental truths about objective reality), and formalism (a meaningless game in which symbol strings are transformed according to certain fixed rules). Platonism and formalism are the big ones, which is weird because Platonism has a mysticism about it which I feel is inappropriate and, as Godel showed, no mathematical formalism is consistent or complete. So I'm closest to intuitionism, which is not really something acceptable due to it being weird and ugly (a classical proof of the Fundamental Theorem of Algebra is half a page; an intuitionist proof is ten pages long).

In his book, Hersh offers a "humanist" philosophy of mathematics existing only as a part of human culture, which is closest to my own position. I guess you could call it a bit of a cop-out, akin to everything trivially being a social construct, but the alternative seems to be an objective mathematics existing out there, independent of time and space. I don't believe such a thing exists.

Thirdly, the book raises more questions than it answers. I value this because it makes me think. Thinking is good. I should do it more often.

One of the first questions it asks is "why is mathematics interesting?". This may seem like a trivial question; we do maths because it's important. But if mathematics is so important to us that we have multiple professions dedicated to applying or studying maths, it must hold some interest.

I'm going to cop out of a proper answer hugely on this one: because I'm not a sociologist or anthropologist, and because I'm too much of a recluse to talk to other people, I'm not going to attempt to answer why mathematics is interesting to societies. I'm going to attempt to answer why mathematics interests me personally and hope that the results have some applicability to other people.

I like mathematics chiefly because it's weird and beautiful.

Mathematics doesn't seem to exist physically in the same way that, say, chairs or tables do. In a naive way one might use mathematics as a descriptor; for example, when I say "there are two apples" I am using "two" to describe qualities of the apples, in the same way as I might describe them as "red" or "green". When I perform mathematical manipulations using numbers, it's as if I were using the redness or the greenness of the apples to do something. Then I can abstract further and concentrate solely on the structures (but diagrams are still useful - I feel like most of my work is just drawing squiggles sometimes). In addition, I have synaesthesia, so I see numbers and letters as colours. This is very...interesting, shall we say, when working with complex numbers. Because it doesn't exist physically, I also think of it as something we invent (like nations or systems of government) rather than something we discover (like planets or stars).

At the same time, mathematics is breathtakingly powerful. We have invented mathematical structures which very accurately describe the universe around us. In fact, I usually think of mathematics as a language describing nature, but one which describes it in a compact and elegant way. Forget pissing around with English or French; everyone on earth has a shared idea of mathematics and how to use it to describe things.

Mathematics transformed my life by transforming my understanding of the world around us and I am forever in debt to this - a language like no other, describing things that do not exist physically, abstruse and powerful and forever asking more questions than it can answer. That is why I find it so interesting.