The Grapes of Math
By Alex Bellos
Doubleday, 337 pages, $32.95
How Not to be Wrong
By Jordan Ellenberg
Penguin Press, 468 pages, $32.95
At this year's World Cup, Brazil has so far crushed Cameroon 4-1, tied Mexico 0-0, and dispensed with Croatia 3-1. What are the chances that the hosts will win in 2014? One odds maker puts it at 15-4, another 3-1, and worldsoccer.com says 55 per cent.
Grapes of Math by Alex Bellos and How Not to be Wrong by Jordan Ellenberg deploy a mathematical microscope in examining how numbers prop up these sorts of mathy phenomena, the multifarious calculations of life. Each is an enlightening compendium zooming in and out on lottery plays, the theist v. atheist wager, cheating on tax returns, skateboarding, rollercoaster design, cutting cake, dinner party seating, and the likelihood of imminent apocalyptic global overpopulation. And although there is some overlap in content (Pythagoras, prime numbers), the two authors offer differing and complimentary takes.
Ellenberg says math is a mental prosthesis, "the extension of common sense by other means." A mathematics professor at the University of Wisconsin-Madison, he holds an MFA in creative writing and his first book was a novel, The Grasshopper King.
"Mathematics," says Ellenberg, "is not just a sequence of computations to be carried out by rote until your patience or stamina run out – although it might seem that way from what you've been taught in courses called mathematics. Those integrals are to mathematics as weight training and calisthenics are to soccer. If you want to play soccer… you've got to do a lot of boring, repetitive, apparently pointless drills. Do professional soccer players ever use those drills? Well, you won't see anybody on the field curling a weight or zigzagging between traffic cones. But you do see players using the strength, speed, insight and flexibility they built up by doing those drills, week after tedious week…If you want to play soccer for a living, or even make the varsity team, you're going to be spending lots of boring weekends on the practice field… But now here's the good news. If the drills are too much for you to take, you can still play for fun, with friends. You can enjoy the thrill of making a slick pass between defenders or scoring from a distance just as much as a pro athlete does" – just as much as, say, Brazil's beloved Neymar and Chile's Alexis Sánchez when they face off this weekend.
There's a mathematical theory of communication, of probability, of infinity – could there be a mathematical theory of consciousness and society and aesthetics? "People are trying, that's for sure, with only limited success so far," says Ellenberg. "You should distrust all such claims on instinct. But you should also keep in mind that they might also end up getting some important things right."
He recounts the parable of the fraudulent Baltimore stockbroker, who sends you a weekly newsletter with 10 correct predictions, but he's also distributed a newsletter containing different predictions to 10,240 others, and it was only with your tip sheet that the broker got all 10 tips right, hence you should not jump to invest. "It really is improbable that 10 stock picks in a row would come out the right way, or that a magician who bet on six horse races would get the winner right every time…The mistake is in being surprised by this encounter with the improbable…Impossible things never happen. But improbable things happen a lot."
As for God versus No God, Ellenberg says go with your qualitative gut. And, with a salty sense of humor sprinkled throughout the book, he also suggests another alternative to the either-or scenario. "What about GODS, where the world was put together in a hurry by squabbling committee?...You can't deny that there are aspects of the natural world – I'm thinking pandas here – that seem more likely to have resulted from grudging bureaucratic compromise than from the mind of an all-knowing deity with total creative control."
While for Ellenberg math is a tool, for Bellos it's also a joke that tickles the brain. "Jokes are stories with a setup and a punch line. You follow them carefully until the payoff, which makes you smile…We usually call a mathematical story a 'proof,' and the punch line a 'theorem.' You follow the proof until you reach the payoff. Whoosh! You get it! Neurons go wild! A rush of intellectual satisfaction justifies the initial confusion, and you smile… Not only does math help you understand the world better, it helps you enjoy it more, too."
Based in London, Bellos is also the author of Here's Looking at Euclid, as well as Futebol, The Brazilian Way of Life (just revised and reissued with a chapter on the 2014 World Cup). And he's a maths blogger for the Guardian. It was there he first reported the results of his ingeniously poignant international poll of people's favourite numbers – the number seven triumphed, beloved for being "magical, unalterable, intelligent, awkward, overconfident, masculine."
The full reveal and analysis of the data is detailed in the book. "Seven is special not because of planets, orbits, or [head] orifices, but because of arithmetic. Seven is unique among the first ten numbers because it is the only number that cannot be multiplied or divided within the group," he explains. "Psychologists have studied the uniqueness of seven for decades. When people are asked to think of a digit off the top of their heads, they are most likely to think of a 7. When asked to think of a number between 1 and 20, the majority will think of 17. Such is the subconscious drive toward numbers ending in 7 that it is the basis for a classic trick, in which the magician asks a volunteer to think of a two-digit odd number between 1 and 50 whose digits are different…and correctly predicts that he or she is thinking of… the number in the footnote…"* Bellos also provides a numerical leg up with the Law of Anomalous Numbers: Count the leading digits in a collection of numbers – numbers in the newspaper on any given day, for instance, or numbers in any honestly prepared tax return. "They will always follow a sliding scale," he says, "with 1 far and away the most abundant, 2 the next in line, 3 after that, and so on all the way down to 9, the rarest of the pack." This is also known as Benford's law, and because it's a law it is iron-clad. Hence, all the Canadian Revenue Agency has to do is run a tally of the leading digits in your tax return, and if 1 isn't the most abundant, they know you've fudged your numbers.
Both writers are passionate about their subject, so their exposition can get a little heady at times, but they either wave by less mathematically minded readers and designate a meet-up point, or direct more mathematically minded readers to appendices.
And as for playing the lottery, after exhaustively documenting how doing so is a lousy decision, Ellenberg acknowledges, in terms of behavioral economics, that picking your numbers is fun, so okay, go ahead: "Math gives you permission!"
Permission granted, you can, somewhat guilt free, place your World Cup bets online. Winning is improbable, not impossible. A Swedish fan took the 175-1 odds and bet 80 kroner – CAD $15.70 – that Uruguay's notorious biter Luis Suarez would strike again and now after that shocker of a game with Italy the Swedish fan is 14,000 kroner richer – Do the math!
*"Most people think of 37."
Siobhan Roberts, currently writer-in-residence at Humboldt University's Institut für Mathematik in Berlin, is the author ofGenius at Play: The Curious Mathematical Mind of John Horton Conway, forthcoming next year.