## persuading the world that mathematics can be fun

As we begin the year 2017, I like many others have spent some time looking back on the year just come to a close, and perhaps years farther back, and I find pieces of unfinished business. One big piece involves something that I’ve tried my hand at a bit, not as much as I want to, and this piece is explaining hard bits of mathematics, interesting bits of mathematics, to people who are not mathematicians. If you wish to read the current state of my efforts here, you can find them in the numerology of unrelated constants and the 1numerology of unrelated constants 2.

A couple of years ago, one of my non-mathematician colleagues challenged me to do exactly this, to take my 10 favourite mathematical facts and find a way of describing them in a way that he would understand. I’ve not yet risen to the challenge, but I do think about it most days. What are my 10 favourite mathematical facts? How can I winnow down my list of favourite facts down to 10? And more interestingly, how to describe them, explain them, in a way that a non-mathematician would be find intelligible and interesting.

Looking back at those two earlier posts on the numerology of unrelated constants, I can see that I am still trying to explain mathematics as I would for my students, rather than aiming at the broader audience of people who might find the concepts interesting. And to be frank, I’m not entirely sure how to explain some of my favourite things, such as the many different sizes of infinity, to non-mathematicians. It’s a challenge, and not one that I’ve entirely yet come to terms with. But I do like a good challenge.

I think that I fallen afoul of one of the things I’ve written about in the language of mastery versus the understanding of the student, which is the difficulty inherent in explaining something that I’ve spent a lot of time thinking about, to someone who has never encountered it before. And the more I think about it, the more interesting and difficult a challenge I think this is going to be.

For instance, consider the idea I mentioned above, which will be one of my 10, the many different sizes of infinity. One of the great achievements of 19th century mathematics, which arose from Cantor’s development of set theory, is the idea that sizes of non-finite sets can be made precise, and in the realisation of this precision, we get an infinitude of different infinities.

Where the challenge arises is that when I mention this idea to non-mathematicians I know, they give me the strangest look. Infinity is infinity, isn’t it. How can one infinity be different from another infinity. They sometimes use choicer language, but what comes across is the perfectly reasonable lack of knowledge and understanding of a part of 19th century mathematics, that we don’t teach to pre-university students and increasingly don’t teach to university students. And so, the question is, where to start unpacking this strange and beautiful idea and how to pull out the essential non-technical bits.

I will admit that there’s something I’m not doing as part of this whole process. Perhaps I’m being a bit too reductionist, in wanting to figure out how to construct such explanations on my own, without reading about how others have done it, but for me that’s part of the challenge. There are some excellent expository of mathematics for the general audience in the world, but I am taken back to a story I read as a much younger man.

In a future society, those with exceptional talents are noticed and cultivated from a young age. As part of this cultivation, they are prevented from knowing what others have done, so that their talents are not contaminated. The story revolved around a musician who had heard none of the classic and classical composers but rather took his inspiration directly from rain and streams and the wind through the trees. But someone sneaks in and plays some Bach, and this changes the young man’s music. Elements of fugue disappear from his works, and it becomes obvious that his music has been fundamentally changed by this external influence. And so I’m going to give myself to see what I can do on my own, knowing that I can always learn should I desire to do so.