Some Glimpses of Mathematics

The intention here is a few pages just to show why I love mathematics. It has a reputation, like poetry, of being beautiful but hard. The comparison is apt; and each compliments the other in the comparison.

Maths is beautiful. Elegant, sparse, clear, it is an architecture of crystalline certainty erected using only the abstractions of mind, the ideas and logic that all minds hold in common. How could we not find it beautiful?

But it is hard, though not necessarily in quite the way you may be thinking. Athletics is hard, writing a novel, carving a statue is hard, making a new poem is hard. These things are seriously difficult; if you don't have the talent, you can't train to do it. And making new maths is hard in that sense.

But you don't have to do the hard stuff to grasp the beauty, you need to read it.

For, consider, reading a poem is not hard. And reading a poem is qualitatively different from reading, say, a history book, or a cooking recipe. Reading a poem is in some real sense doing poetry. The reader and the writer experience something together in a space defined by their shared humanity, their shared responses to the external world.

Reading mathematics is not-hard in a similar sense, once the maths has been made. Reading mathematics is in a real sense doing mathematics. The reader and the writer rebuild it together in a space defined by their shared humanity, in this case their responses in a shared internal world. Together they build, piece by piece, new simplicities from complex things.

Once the maths has been made, it really isn't hard; its common sense, applied to common sense, applied to common sense. Well, there is a snag; the common sense has been refined over and over, and getting started in any particular place is awkward. You need a guide to set your feet on an appropriate solid place to start. But after that, what is hard is to pay attention. The one qualification you really need is desire enough to keep your mind engaged.

I am not going to teach you anything here, but I would love to give you a glimpse. One of the basic methods of proof in mathematics is called induction. Could anything sound more boring? Well, come with me, and I promise you a surprise.

And there will be soon page of seriously deep mathematics, just to show you how lovely it is on the page, regardless of whatever it might mean.