'Arithmetic' Delightfully Yields Anything But Obvious Results

by Andrea Tallarita

17 August 2017

What looks like a simple topic becomes a surprising trip into unexpected worlds in Paul Lockhart's beautifully executed Arithmetic.
 
cover art

Arithmetic

Paul Lockhart

(Belknap Harvard)
US: Aug 2017

There is a certain joy in going back to the basics. You may think you learned everything you needed to know about a particular subject when you were in primary school, only to be confronted with a book like Paul Lockhart’s Arithmetic and discover that adding two and two is not nearly as plain an operation as you expected.

Arithmetic does exactly what it says on the cover. It treats the discipline behind addition, subtraction, multiplication and division, intertwining an introduction to these key mathematical notions with a sweep of their conceptual history, from the counting systems used in ancient Egypt, Rome and China, through to the modern Hindu-Arabic number system.

I can imagine how trivial a choice of topic this may appear to someone picking the book up off a library shelf, as I was guilty of the same prejudice. Yet once I started reading, the text proved mind-blowing. Some of the most ingrained and fundamental assumptions about the way we count and understand numbers are here deconstructed and shown to be arbitrary. The fact that we are used to grouping numbers by an order of magnitude based on the number 10, to give but one example, is contrasted with hypothetical arithmetical systems which are based on numbers like 4 and 7. (Can you imagine what 6+8 might look like in these systems?) Lockhart, who teaches mathematics at Saint Ann’s School in New York, shows how such alternative systems might and sometimes did work in history.

The illustration of this argument is both fascinating and easy to follow. I doubt it will be much of an eye-opener for professionals of the field, but for the mathematical layman, this book will be a very pleasant surprise.

Beyond the solid contents, Arithmetic deserves even greater praise for its pitch-perfect style. I tend to be quite sensitive to clumsy attempts at casual or humorous writing, and I am perhaps guilty of a certain prejudice against the writing skills of professors in science and mathematics. So I am delighted to say that Lockhart is a fabulously entertaining writer, and that his light-hearted approach managed to keep me cheerfully engaged even when his discussions were most abstract.

I think what makes the author’s writing work is that he always sounds candid. He refrains from grand universal claims and admits the importance of arithmetic shouldn’t be overblown. He has no interest in academic form and he happily dismisses Greek and medieval number philosophy as ‘pedantic’. He intersperses his arguments with much personal opinion, which sometimes comes across as debatable and unmeasured, but which always proves to be thought-provoking.

Reading Arithmetic less generously, there are times when Lockhart’s historical preparation seems inadequate, and in that case the opinionated style backfires. At one point he claims that our way of measuring time ‘goes back to the Babylonians’, and that the hours of the day being divided into two groups of 12 is ‘a cultural and historical [choice], based on the fact that there happen to be twelve more or less equally spaced clumps of stars observable in the Egyptian night sky’.

I’m assuming Lockhart is thinking of the 12 constellations of the zodiac here. Aside from the fact that I have no idea what the Egyptian night sky has to do with Babylonian astronomy (the two cultures developed quite separately), there is the fact that the original Mesopotamian designation counted 18 constellations on the ecliptic, not 12. Even when that number was reduced, they were never visible together in one night, nor did they follow a straight or logical line. And in any case there never were any ‘equally spaced clumps of stars’ in the sky at all, as the make-up of the various constellations was always arbitrary, and it varied wildly from one culture to another!

Although this passage is quite the howler, it is, thankfully, an isolated case. For the most part, the book’s approach is very functional to its arguments. It’s also stimulating from a pedagogical point of view, as there are short questions every few paragraphs which can be used as puzzles (both conceptual and arithmetical) to test your understanding of the points being expounded.

Arithmetic is not a perfect book. The final chapters on fractions and negative numbers are not as interesting as what came before, and I felt the explanation of how modern calculators work deserved more space and detail. These shortcomings aside, though, the book should be considered a success.

By defect of its subject matter, which has never been wildly popular, I fear the book may reach a smaller audience than it deserves. The people who investigate specific branches of pure mathematics are usually the specialists, and this book is most certainly not for them. But if you are a novice like myself, then do consider Lockhart’s Arithmetic. It’s in equal measures entertaining and educational, and a pleasant surprise on more levels than one.

Arithmetic

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