Book review: Pluses And Minuses, by Stefan Buijsman

A question that has often vexed me in the watches of the night is “Can dogs solve quadratic equations?” Let me explain. When I am out with my brother and his dogs, Loki, Daisy and Jake, I throw a tennis-ball over-arm and they hare off to roughly where the ball is heading; sometimes with Jake leaping to intercept the ball mid-air. So hypothesis one is that dogs can solve the equation for a parabolic curve, and deep in their canine brains they are woofing: when ax2+bx+c=0 then x= –b ± √(b2 -4ac) divided by 2a. Or, hypothesis two: they have scarpered after balls a fair bit and have become quite adept at it.

I am being facetious, but only mildly. This book by Stefan Buijsman, a wunderkind who got his philosophy degree aged 18, a PhD 18 months later and now is studying the philosophy of mathematics, poses two simple enough questions: can we live without mathematics, and what does mathematics do for us?

Buijsman says at the outset that the book was written for his 15-year-old self, who was frustrated by learning equations and formulae and proofs that would be of scant use in later life. He even says that, as a mathematician, he rarely needs to use them now given a computer does the calculations more efficiently. The book is written in a breezy, unpretentious tone befitting the target audience, with very few equations (the one quoted above is longer than any in the book). In fact, I can see it being quite a success. It would be an ideal gift for a 14 or 15-year-old who may be interested in mathematics. Indeed, I can envisage a great many parents getting a lot from it in our era of home schooling, especially as many of them will not have had to use “taught” mathematics for many years.

If you are a reader of popular science, particularly maths – writers like Marcus du Sautoy or Ian Stewart or even the classic Gödel, Escher, Bach by Douglas Hofstadter – you might find it a little on the thin side. But what it sets out to do it does well.

The book is subtitled “How Maths Makes The World More Manageable”. The crux is that we would never have developed the discipline at all were it not for some inherent utility. This reminds me of how my Dad – a maths teacher – would encourage reluctant Foundation Level classes by asking them what they wanted to be when they left school. Someone said, for example, a decorator; which meant you could teach about calculating surface areas, and the number of pots of paint or rolls of wallpaper the job would require. Maths was about what you could do with it.

Buijsman concentrates on three main areas; calculus (the name itself seems daunting), neatly explained as how quickly things change and how much things change; probability and statistics; and graph theory. A few big names crop up – Pythagoras, Archimedes, Newton, Leibnitz, Bernoulli, Bayes, Euler – usually with a toothsome anecdote. The best chapters are the early ones on human cultures which manage well without any form of mathematics and the transformation to “mathematical societies”. The anthropological sections are fascinating, though again readers interested in tribes that have no or rudimentary counting schemes will probably know this material already. (There is at least a good snicker for the schoolboy – I use the gendered term deliberately – about the Yupno, who can count to 33 using fingers, toes, ears, nostrils and testicles and penis, which might mean women can’t count to 33). The chapter on Mesopotamia, Egypt, Greece and China has the key pitch that bigger urban populations required a new form and thought-world to deal with crop yields, land allocations, taxation, large scale building works and all the other things we think of as markers of a less primitive society.

The opening chapter is perhaps a tad too much “cool schoolteacher”. Google Maps! Coffee machines! Facebook! Netflix! Did you know they all rely on mathematics! Well, I never. Here was me thinking the internet ran on fairy dust and unicorn dung. It is salutary to see some scepticism about the idea that a piece of software “thinks” or “learns”. I can’t imagine the algorithm that would guess that the last things I watched were The Bishop’s Wife (1947) and Star Trek: Discovery Season 1 (2017). The headline parts of this are reiterated in the conclusion, which seems to pad out a modest 180 pages. I would have traded the breathless enthusiasm for 20 pages on trigonometry or topology. If you want a book that deals with the more outré shores of set theory, imaginary numbers, multiple infinities and so on, such books exist. But if you are looking for one that might inspire the younger generation into taking mathematics seriously, this is an assured first step.

At least Buijsman is very honest about the central paradox of the entire subject. To what extent are numbers “real”? Is there an “idea of 2” untethered from any specific things that come in pairs? Is mathematics just a set of approximations that get refined and refined over time, or are there some eternal verities? I have long argued that in all the sciences, rote-learning and memorising are necessary, but to make it go further one needs wonder. If you want a supple brain, you might need some mind-bending first.

Pluses And Minuses, by Stefan Buijsman, Weidenfeld and Nicolson, £14.99

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