Monthly Archives: December 2014

Book Review: Enlightening Symbols by Joseph Mazur

Subtitled ‘A Short History of Mathematical Notation and Its Hidden Powers’ I was first made aware of this book on GrrlScientist’s blog on the Guardian website. I added it to a wishlist and was given it as a present for my birthday earlier in the autumn. Having been educated in maths to a degree further than most, I have used rather a library of symbols in my time and had cursory historical overviews of their development but I have not previously read a thorough history.

The book is split into three parts, the first focusing on the development of numerals, the second on algebra and the third on the power of symbols. All three parts are quite distinct and ought to be looked at one by one.

I must confess, I didn’t find the first part particularly coherent. That is partly a feature of the fact that the history of the representation of numbers is itself quite muddled. In reading this, I got the impression that Mazur, who I don’t ever recall coming across before, was more of a mathematician than a historian. As it turns out, this seems to be a fair characterisation, though, like me, he takes a very keen interest in history and (we’ll come to it below) into other areas as well.

The history of numerals is summed up on page 78 as follows: “There have been many scrupulous studies on the origins of our system, but even after a hundred years of scholarly wide-ranging research, we are left with only sketchy guesses of its beginning and evolution.” Perhaps this should have been an executive summary at the start of the section rather than a conclusion reached after having gone round the houses a few times. It’s not that the history is uninteresting, it is really quite captivating. It’s just that Mazur’s take on it didn’t allow this reader to get a grasp on it, so it was quite bewildering. So I must confess that I wasn’t overly enamoured with Mazur’s writing and as I finished the first part, I feared that the last 2/3rds of the book would be a bit of an unenjoyable trudge.

How glad I was to be proved wrong. For in moving from numeral to algebra, fresh life was breathed into the text and I was treated to the book that I had hoped to read.

As with the first section, the story is not straightforward, but we do get to see some of the significant historical developments in fresh light rather than the fairly dim gloom we had beforehand. The first major figure we encounter is Diophantus. His name should be familiar to most maths students, though if you haven’t come across him then this would be a good place to gain an introduction. The basic story is that problems that we think of as algebraic did not begin with symbolic representations.

If you had a good maths teacher (and I’ve been blessed by having a few) then you will have been presented with “word problems” where some question or other is asked which involves numbers and where the answer is required in the form of a number. The student is then asked to convert the word problem into a symbolic form and then manipulate that symbolic form using the methods taught to arrive at an answer. What Mazur gives us is an unpacking of this, showing that most early algebra consisted of such word problems.

We get to meet al-Khwarizmi and see some of the problems he posed in his seminal work Al-Kitab al-mukhtasar fi hi sab al-gabr wa’l-muqabala (yes, I did have to copy that carefully). We see the development of symbolic representation such as those for multiplication, powers and division. Without trying to summarise it here (I confess, this part of the review was written a couple of weeks after I finished reading the book), I would heartily recommend it to you. For those who dropped maths after their GCSEs, I will say that it might not be particularly applicable. For those who are university educated or who can still recall their A-levels then the final step will be very familiar, but it’s a fascinating story as to how we got here.

The final third of the book carried on in the same vein as the second part had, with less of a major change in tone that there was between the first and the second. As I read through the first two parts, I was struck by a quite sobering (or maybe dispiriting might be a better term) thought that in spite of having studied maths to a greater level than most people in the world, was my understanding of it merely the understanding of manipulation of symbols?

There is reassurance at the end, though. Mazur’s view is that our ability to shorthand things in symbolic frees up the mind to truly understand what is going on. This seems to coincide with how I view the abstraction in maths in general, as well as some specific aspects like Fourier transforms; here we phrase a question in a specific way, abstractify to the general case, solve the general case and then you have a template for answering the specific case. By working with symbols we may temporarily lose sight of exactly what it is we are calculating, but that lack of sight allows us to avoid getting bogged down in unnecessary detail. By all means, if we wish to come to back to an intermediate stage in the calculation and convert into word problems, we can – that is the power of symbolic maths.

The final section also deals with some other matters peripheral to our understanding of mathematics, such as the psychology and philosophy of maths. So it was little surprise to see Wittgenstein referenced at this point. Though Mazur was readily more accessible than Wittgenstein was. The breadth of this final view reveals the author to be more than just a mathematician, he is a bit of a polymath. So while the book was not hugely coherent to begin with, the last two-thirds are very creditable and I would recommend it to anyone interested in maths and the history thereof.