How to Write About Mathematics

Occasionally I rant about the general awfulness of mathematics textbooks. If I were to express my major objection in the most charitable possible way, it is that most textbooks are written like reference books. They are usually very good at recording the basic facts of a subject and proving them with admirable rigor. If you just need to look up some elementary theorem or formal definition, then by all means consult a textbook. The trouble, though, is that textbooks are seldom written from the perspective of a student encountering the material for the first time.

If I were to express things more melodramatically, I would say that the problem is that textbooks seldom tell a story. They have no plot, no characters. And they should! Good mathematical writing is marked by a progression from the initial statement of a problem, through the rising action of our initial, fumbling attempts to solve it, reaching its climax with the solution itself, and then proceeding to its denouement in the form of a proper, rigorous proof. The best part is that the story never really ends, since the solution to one problem leads inevitably to new problems.

Over time, however, I have come to the sad realization that many of my colleagues do not share my view. A piece of mathematical writing with two consecutive sentences of exposition will be derided by many as too wordy. Books that are nothing but a sequence of dense, unmotivated definitions followed by equally unmotivated theorems are praised for being concise. It has happened many times that I have heard people gush about textbooks I would be embarrassed to use to level off a table.

Occasionally, though, I grow uncertain. Maybe I'm the crazy one, and those deathly-dull, jargon-filled, notation-fests are actually on the right track. That is, after all, how most mathematicians write. Who am I to argue?

Every once in a while, though, I come across a book that really does it right! I have just finished reading one such book. It is called Measurement, by Paul Lockhart. It is quite simply the best math book I have read in quite some time. Anyone thinking about writing a math textbook should be required to read it.

It is possible that I am a little biased. Lockhart was a post-doc at Brown when I was an undergraduate there in the early nineties. I had two courses with him, one in linear algebra, the other in complex analysis. He had a big influence on me, especially during my periodical crises of confidence about my future in mathematics. I often tell my students that math is easy, it's math classes that are hard. Sometimes I would feel beaten down by the drudgery of my courses and start wondering if maybe I should pursue something else. Professor Lockhart played a big role in getting me over that. I mostly remember him as the funniest, but also the most lucid, math teacher I have ever had.

I am happy to report that he writes the way he teaches. The book has two main sections. The first deals with certain topics from Euclidean geometry, while the second discusses calculus. I found myself learning many new things, and looking at some old things in a new way. More than that, though, what impressed me was the naturalness of the discussion. Take the section on calculus, for example. At no point does he say anything like, “We shall now define the notion of the limit of a function at a point...” Not at all. Instead he describes a certain problem: How should we think about the motion of a particle through space? Then, with some clear thinking and a sequence of natural questions, he develops all of the standard ideas presented in the calculus sequence. In around 200 pages, he starts from scratch and gets to some fairly sophisticated ideas in multi-variable calculus. It all seems so natural and inevitable and interesting, and not at all like the tedious, symbol-laden, thousand-pagers we inflict on freshman math majors.

Here's an example of something I had not really thought about before. Lockhart is contrasting the approach toward calculating areas taken by the ancient Greeks with the more modern approach we employ in calculus. The Greeks tended to use the “method of exhaustion,” in which irregular areas are broken up into a large number of more regularly shaped regions. The modern approach is to perturb the area a bit, and then to solve the differential equation that results from measuring how it changes. Lockhart writes:

I love the contrast between the ancient and modern approaches to geometric measurement. The classical Greek idea is to hold your measurement down and chop it into pieces; the seventeenth-century method is to let it run free and watch how it changes. There is something slightly perverse (or at least ironic) about how much easier it is to deal with an infinite family of varying measurements than with a single static one.

I know quite a bit about Greek geometry, and also quite a bit about calculus, but I had never thought about it that way. Very cool.

Anyway, if you have any interest in mathematics at all you should procure a copy of this book immediately. It is brilliant. Be warned, though, that I don't mean to say the book always makes for light reading. Math, even when explained with perfect lucidity, is still challenging.

Here's a short video of Lockhart discussing his book. I hope he writes a dozen sequels. Enjoy!