The first part of this book deals with old ways of writing numbers down. You get shown lists of squiggles with names like Acrophonic and Alphabetic. So there I am, on the train, looking at: squiggle + squiggle = squiggle, where squiggle three has the value that we would write now as 600.
I am puzzled. I still have not freed myself properly of the way of thinking about numbers, the placeholderiness, that I imbibed as a child. Eventually I get it. squiggle plus squiggle does indeed equal squiggle. But when I look up, I've missed my stop.
So I think I can fairly say that this book is totally engrossing!
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 was more of a mathematician than a historian. As it turns out, this seems to be a fair characterisation, though he takes a very keen interest in history.
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 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 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. 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. 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.
I have a standard opening speech that I give in all of the math classes that I teach and one component of that monologue is the role of notation. I point out that one of the things that make mathematics difficult to understand is the fact that the symbols are compact representations of operations. I also point out that this compact representation is preferable over having every problem expressed as a "story problem", the problem form that is most widely feared by students. What I found most enlightening about this book is the clear conclusion that the mathematical education of the masses would not be possible if the compact and efficient notation was not being used. Trying to decipher the rhetorical forms of the problems was really hard, even after reading the explanations. Furthermore, some of the examples where a long paragraph of text is translated to a simple equality are one of the most dramatic alterations you can find in mathematics. Even if you allow for the greater understanding of the rhetorical form based on extensive practice, it is hard to see how large numbers of students could solve them. Finally, it is also hard to see how mathematics could advance as fast as it has using only rhetorical forms or non-standard notation. This book is written in a popular style, there are a few equations, but nothing that a reader with a background in basic algebra can't handle. It also has the interesting characteristic that the sections that the non-professional and professional mathematicians will have difficulty understanding coincide. In some ways modern mathematical notation has a short history and we should all be grateful for what it has done for the world. This book will make you appreciate that.
Published in Journal of Recreational Mathematics, reprinted with permission
I always find Joseph Mazur engaging, and this is no exception, It is fascinating to see how our present way of writing mathematics evolved -- and one has to admire the mathematicians who produced important results when the tools they used were still so primitive. There were one or two places where I thought proof-reading could have been better -- for example in the discussion of the general quadratic equation we have one expression and two equations, but all are described as equations (at least, that's what I see in the Kindle version). Well worth reading, though.