Diversions in Mathematics #0

Introduction to the series

I have always been interested in maths, and not only in the subject itself but also the ways in which maths is explained and taught. In general, a crucial part of studying and researching is to be able to communicate one’s findings to other people, who may or may not be knowledgeable in your field. For this reason, I’m all for popular science books and magazines, which (provided that it is done well) serve to explain scientific research in an accessible way, and to promote scientific awareness and appreciation amongst the general public. However, in my opinion, popular science books too often simplify, and even completely skip the mathematics behind the science. There is certainly a cultural aversion to mathematics — at least from my perspective as an Australian, and from my awareness of similar attitudes in the US and UK — which may be part of the reason for the lack of ‘real’ mathematics in popular science writing. Here is an anecdote: apparently Stephen Hawking’s publisher advised the great scientist that every equation he included in his A Brief History of Time would result in reduced sales. (There is one equation though: Einstein’s ). Of course, this is one of the bestselling science books ever, and sits atop many a coffeetable, but I wonder how many people have seriously read it…

So what about popular mathematics? Undoubtedly there are many who enjoy recreational mathematics, but my feeling is that the average person will not suddenly decide, “gee, I really feel like reading something about abstract algebra today!” In fact, I wonder if there are any books on abstract algebra intended for the general public! Sciences such as astronomy and astrophysics, biology and psychology tend to capture the public imagination better. For lack of a better word, they are simply more “relatable”. Maths demands more abstract thinking, and understanding of mathematical concepts is inseparable from understanding mathematical notation. We write equations not because we want to be deliberately obscure — quite the opposite is true! Equations are extremely precise, and represent relationships between mathematical objects that would be tedious, cumbersome, perhaps even impossible to describe in words. By way of illustration, take a look at this beautiful equation:

In fact, this presentation is already quite verbose, in some texts you’ll see it written simply as:

If you’re curious, this is Laplace’s equation, and some mathematicians and physicists have devoted signficant parts of their career studying the properties and applications of this equation. Clearly, there’s a lot more than meets the eye (for example, what do those triangles mean?). It is often the case in mathematics that simple things turn out to have profound consequences.

Hopefully, Diversions will become a recurring feature of my blog. It is my intention to write about mathematics in an informal way and with the general public in mind, but I will not attempt to hide the ‘real’ mathematics. I will present equations, guide the reader through calculations, and discuss complex concepts. But if I succeed, hopefully I will not intimidate the reader either, and ideally there should be an element of fun. I use the word ‘diversion’ in this sense too, similar to the French musical term divertissement (or divertimento in Italian). At the time of writing, I am studying an introductory abstract algebra course at the University of Sydney. It is a challenging course, the material is (not surprisingly) very abstract, there is a lot of new (mathematical) language to learn. However, our lecturer Dr. Stephan Tillmann has emphasised the idea of having fun in maths. Frequently, we are asked to “play around” with a new concept, his assignment questions are geared towards exploration, and we are encouraged to ponder “what happens if…?” It is in this spirit that I write these blogposts. If you are interested, then watch this space!

Updated: March 2019