The past few years have seen a welcome crop of fine mathematics titles that are intended for the general reader, but are also valuable as inspiration and sources of interesting material for those of us who teach the subject. Jordan Ellenberg¡¯s outstanding book pretty much shares its subtitle with Rob Eastaway and Jeremy Wyndham¡¯s classic Why Do Buses Come in Threes?, although the two books are very different in terms of style and content.
Ellenberg, a mathematics prodigy as a child, harboured literary ambitions and wrote a (rather good) novel, The Grasshopper King, before concluding that ¡°every day I devoted to Serious Literary Novel-writing was a day half spent moping around wishing I were working on math problems¡±. While continuing his career as a research mathematician, he writes a maths column for the web magazine Slate, and, as one might expect given this background, How Not to Be Wrong is beautifully written, holding the reader¡¯s attention throughout with well-chosen material, illuminating exposition, wit and helpful examples.
Ellenberg¡¯s theme is that mathematical thinking helps us to make better decisions. He begins by considering Abraham Wald, a theoretical mathematician who advised the US military on adding protective armour to their aircraft during the Second World War. The weight of armour makes planes less manoeuvrable, so the aim was to fit limited amounts at key positions. Analysis of bullet holes in aircraft returned from combat showed that the engines had fewer bullet holes than other parts of the plane, so Wald advised that, paradoxically, it was the engines that required armour ¨C planes whose engines took bullets often did not make it back, whereas bullets hitting the fuselage were survivable.
Ellenberg shows how this same principle applies in evaluating the returns of stock market funds ¨C ¡°survivorship bias¡± occurs in many situations. One of this book¡¯s strengths is the author¡¯s ability to produce these killer examples ¨C simple, memorable scenarios that communicate complex and often counter-intuitive technical points. Another is the vividness of his sympathetic accounts of key figures in the story he tells: Wald, Francis Galton, Harold Hotelling, Richard Hamming and many others.
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For those of us who read too many such maths books, over-familiarity with the canonical examples is an occupational hazard. But Ellenberg¡¯s examples are usually fresh, and even where he treads a familiar path, he presents new angles. For example, Buffon¡¯s needle, a clever way to estimate the value of ¦Ð (the ratio of the circumference of a circle to its diameter) by dropping needles and noting what proportion cross the lines between floorboards, is a widely known experiment in recreational mathematics. But Ellenberg¡¯s account also presents a beautiful analysis by Joseph-?mile Barbier that adds a new twist to this curious piece of mathematics. I am reminded of the great writer of recreational mathematics, Martin Gardner: Ellenberg shares Gardner¡¯s remarkable ability to write clearly and entertainingly, bringing in deep mathematical ideas without the reader registering their difficulty.
A section on the mathematics of lotteries shows how a minor technical point in the operation of one state lottery provided an opportunity for syndicates to buy large numbers of tickets with guaranteed profits. We¡¯re shown how an idea from projective geometry ¨C a highly abstract branch of mathematics with no obvious connection to the choosing of lottery numbers ¨C was exploited elegantly by one syndicate to guarantee that they would realise the maximum possible profit from their investment. This is just one of many examples Ellenberg deploys to show how the same mathematics can be applied in wildly different situations to solve problems with no obvious connection to each other.
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One of the major strands of the book is the importance of getting it right in analysing data and statistical evidence, and the traps into which the unwary can fall in complex but important real-life situations. The mathematical difficulties in finding a satisfactory electoral method where there are more than two candidates present a problem for democracy. Ellenberg lucidly explains the issues and, although his conclusion that coin-tossing might be the best way to decide close, disputed elections might seem surprising, his argument is persuasive. Similarly, issues in evaluating evidence in the context of medical statistics will concern most of us at some point in our lives. Once again, Ellenberg presents insights by analysing illuminative cases, as he does in a sympathetic discussion of the great statistician R. A. Fisher¡¯s suggestion that cancer causes cigarette smoking, rather than vice versa. We now have strong evidence that Fisher was wrong, but Ellenberg demonstrates that Fisher¡¯s logic was not as crazy as the quick summary might suggest, and thereby makes an important point.
Counter-intuitive outcomes abound in the interpretation of statistics: plausible arguments fail and misleading conclusions are easy to draw. Knowing that my doctor had read How Not to Be Wrong would greatly increase my confidence in her recommendations on, for example, whether or not to take drugs to control my blood pressure.
What I like about this book is that, here and elsewhere, it avoids simple black-and-white conclusions and doesn¡¯t hide the inherent complexity of data-based decision-making.
In my first-week class for new maths undergraduates, I present Arthur Laffer¡¯s mathematical argument about the connection between tax rates and government revenue. One would expect that the higher the taxes, the more revenue the government will gather, but Laffer produced a simple argument showing that this cannot be the case: if the tax rate is 100 per cent, so that everyone¡¯s income goes straight to the government, then no one has any incentive to work and the government¡¯s revenue will be zero. Republican policymakers in President Ronald Reagan¡¯s administration seized on this argument to claim that cutting taxes would increase government revenue. It didn¡¯t turn out that way, and Gardner wrote a devastating article demolishing the argument. I present this to my students, but then argue that Laffer¡¯s curve, despite its oversimplistic application by politicians, nevertheless gives a genuine and important insight. This is exactly Ellenberg¡¯s treatment (although, irritatingly, he does it much better than I do). Over and over again, he makes the points that I try to communicate to students when I teach mathematical modelling. In a way, I rather hope this book does not reach the wide audience it deserves, because if my students have read it before starting university, I¡¯ll have to find new examples.
That aside, this is a wonderful book. I have some very small quibbles: naturally, many of the examples are drawn from the other side of the Atlantic (the ¡°maths¡± rather than ¡°math¡± of the book¡¯s subtitle is not carried through into the book itself); as a Scot, I am irritated by the description of Edmond Halley as ¡°Astronomer Royal of England¡±; and although the hand-drawn diagrams are charming and make the important point that anyone thinking about maths should be drawing diagrams for themselves, some readers might prefer more polished printed diagrams. But these are very minor points. This book will undoubtedly be read by mathematics practitioners and students, but it also has much to teach all readers about avoiding mistakes, and lawyers, doctors and politicians in particular will gain insights into issues affecting their professions.
I will be sorry if Ellenberg has indeed given up novel-writing, but fiction¡¯s loss is certainly mathematics writing¡¯s gain.
The author
The son of two statisticians, Jordan Ellenberg grew up in the state of Maryland, not far from Washington DC. ¡°A lot of famous mathematicians - Charlie Fefferman, Jacob Lurie - are from my county (well, they¡¯re famous to other mathematicians). But I can¡¯t point to any feature of the landscape that should naturally breed them,¡± muses the professor of mathematics at the University of Wisconsin-Madison. ¡°The dominant habits I have that stem from my upbringing are that I talk very fast, and I¡¯m a diehard Baltimore Orioles fan.¡±
Ellenberg lives in Madison with his wife, Tanya Schlam, a research psychologist, and their two children. It is, he says, ¡°a lovely place; almost too small, but just not. But it¡¯s very, very cold in winter, like freeze-the-inside-of-your-nose cold.¡±
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He was ¡°profoundly interested in mathematics¡± early on, and ¡°had a sort of conversion experience as a small child, which I write about in the book, where I was looking at a six-by-eight rectangular array of holes in a speaker cabinet and suddenly realised that THIS was why six times eight had to be eight times six - because both products were counting the 48 holes. That was when I understood that math offers you knowledge you can perceive directly,?and not just be told about.¡±
In 1987, 1988 and 1989, Ellenberg represented the US at the International Mathematical Olympiad, and took home two gold medals and a silver. Was he nervous, and did he expect to win?
¡°In the IMO you get four and a half hours to do three questions; with so little time pressure there¡¯s little opportunity for nervousness. As for winning, one doesn¡¯t really ¡®win¡¯ the IMO as such - a fixed proportion, usually a little under 10 per cent, win gold medals.¡±
Of the years he competed, he recalls: ¡°The atmosphere was much more cooperative than competitive. This was the late 1980s, the wind-down of the Cold War, and the IMO was dominated by the Eastern Bloc countries. There was a real spirit of East-West detente - although at the 1987 IMO in Havana, they did make us all go to a museum about US imperialism.?I wore a T-shirt that said, in English and Ukrainian, ¡°Math Brings Friends Together¡±, and it was not meant ironically.¡±
What sort of undergraduate was he - dreamy, driven, swotty? ¡°Since I¡¯m American, I don¡¯t know what ¡®swotty¡¯ means, so naturally I¡¯m tempted to do the daring thing and say ¡®swotty¡¯, or even ¡®right swotty¡¯. But the odds that that¡¯s correct are pretty poor. So let¡¯s say I was very ambitious and often insecure/confident (which combination is more or less the Harvard University ²ú°ù²¹²Ô»å).¡±
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¡°I was interested in a lot of things,¡± he adds, ¡°and took the minimal number of math courses you could take and still graduate with a math major. But I was not in any meaningful way rebellious; on the contrary, at that age, I cared a lot about grades and doing a good job at what I was told to do.¡±?
After attending Harvard University as an undergraduate and doctoral student, he worked at Princeton University as a postdoc. Canvassed for his views on the differences between the two, he observes: ¡°Each institution devotes a lot of marketing to differentiating itself from the other, but they are not in fact very different.¡±
Author of a published novel, The Grasshopper King, and a regular contributor to the online magazine Slate, Ellenberg considered pursuing a career in words rather than numbers. ¡°I thought I might want to be a writer, and even went so far as to spend a year in a creative writing programme [at Johns Hopkins University in Baltimore] after college. But every day I spent trying to be a novelist was a day I spent missing math. In the end I was very grateful for that year, though - every time my mathematical studies hit a difficult patch, I remembered that not struggling with math was way worse than struggling with math!¡±
What, with the benefit of hindsight, he might have done differently with The Grasshopper King? ¡°The answers are mostly technical and would probably bore non-writers. The epistolary section is probably too slow and not well-integrated into the main text and there are some words I would change. In all, though, I¡¯m very happy with it. I recently picked it up to look at it and see if I felt embarrassed - nope, still not embarrassed. I wrote it when I was 22 and it¡¯s very clearly the work of a person to whom many aspects of human life are a hazy mystery. But under the right circumstances that can be a good look. I think many of the jokes are still good.¡±
And he insists, despite previous statements to the contrary, ¡°I may very well write another work of fiction! I even know more or less what my next novel will be about, though I certainly don¡¯t know whether I¡¯ll ever write it. It¡¯ll be a long time from now, if ever. But when I write about mathematics I write about it the way a novelist would, even though I¡¯m not a professional novelist. That¡¯s how I was trained as a writer.¡±
Ellenberg is the child of two statisticians. Reflecting on his mother¡¯s experiences and how the future might be different for women in mathematics, he says, ¡°Unquestionably things are better now. My mom didn¡¯t even start as a statistician - she was a woman at Harvard in the 1960s who was interested in math, which meant she was advised to major in psychology so she could become a high school math teacher. She got her PhD in statistics later, when I was a little kid.
¡°What the future will look like is harder to say. There is still improvement to come, since the people who still think women don¡¯t have a place in math are rapidly ageing out of the profession. But there is still tons of tacit bias; we may expect a mathematician to look a certain way (dude with glasses and probably a beard, i.e. me) and we may then pay more attention to people who fit the description. We may pay more attention to people who pretend to know exactly what they¡¯re talking about, something men in academia are trained to do.¡±
There are, he says, things individual academics can do to improve the climate for female mathematicians. ¡°In our undergraduate courses (and when we work with still younger kids) we can push back on the idea that math is fundamentally about being the cleverest or the fastest in the room, or about being the best at solving canned contest problems; I kind of love contest problems, but they are almost nothing like the real stuff of mathematics. Math among precocious students involves a lot of shouting over each other; grown-up math does not. So one thing people like me can strive for - and I am not yet perfect at this - is to avoid interrupting people (although we¡¯ve been trained to do it) and to let ourselves be interrupted (although we¡¯ve been trained to fend this off.)
He adds: ¡°Those are indirect, cultural things. More directly, if you are in a position of organising conferences or activities or job shortlists or whatever, you can notice if you find yourself making lists with no women on them, and ask yourself, why were these five men the first people I thought of? Is it because these are clearly the five best people or because these five bearded spectacled men have an easier time swimming to the front of my mind?¡±
Does Ellenberg believe he could persuade anyone, given enough personal attention, to learn to love mathematics? Or are there some people who would be immune even after a dose of his mathematical charm?
¡°No. I feel strongly about this. Loving mathematics is weird. Loving ANY particular thing is weird. People love different things and always will.
¡°But love is a high bar to clear. Not everyone loves to read, but most people we know CAN read. They are comfortable reading, they recognise the value of reading, they know how to recognise situations where it might be useful to read something.?Or: I don¡¯t love physical exercise but I¡¯m certainly capable of enjoying it and I get why it¡¯s a good idea to have it be part of your life. I actually find it a bit of a turnoff when people insist that if only I would exercise THEIR way, I would fall in love with it and become a zealot. No!¡±
He adds: ¡°As a teacher I want people to do mathematics, to enjoy mathematics, to admire mathematics, to master mathematics. But loving mathematics? That¡¯s my job.¡±
And now that he is a parent, has Ellenberg finally discovered what it is to be wrong?
¡°My son¡¯s favourite phrase is a highly sarcastic ¡®Seriously, Daddy?¡¯ This started at about age 6. It¡¯s been a long time since he believed I knew everything, but we haven¡¯t yet gotten to the point where he thinks I don¡¯t know anything.¡±
Karen Shook
How Not to Be Wrong: The Hidden Maths of Everyday Life
By Jordan Ellenberg
Allen Lane, 480pp, ?20.00 and ?11.99
ISBN 9781846146787 and 9780718196059 (e-book)
Published 3 June 2014
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