A man working at a blackboard with various formulas

Jordan Ellenberg: 'How Not to Be Wrong'

For lifelong mathematician Jordan Ellenberg, the weird world of numbers is stranger than you could imagine. And yet it is through maths we can make sense of the apparent paradoxes of daily life.

Ask any group of people if they are in favour of increased governmental spending on public services in general and you can bet your pay-cheque you'll get a majority response in the negative. Now, ask the same people the same question, only break it down from the general into more localised services such as refuse collection, highway maintenance and street lighting. What you will find is that the same people will now be in favour of increased public expenditure. It seems like a paradox based on people's apparent desire to see improvements in their own back yards, while not caring about the world outside their own bubble. But that's just insulting decent citizens and, besides, it's not a paradox.

So says Jordan Ellenberg, author of a terrific new book on popular mathematics called 'How Not to Be Wrong'. "It may seem like a paradox, but actually, when you work out the maths, there's nothing paradoxical at all here. There is consensus that cuts are needed and lack of consensus about where those cuts should come, and so there is no individual programme which a majority supports cutting." The result is that if you were to put a poll out into the field, you'd get irrational polling results, even though each individual is likely to be perfectly rational. For politicians, to take the position supported by the majority – what we think of as being a democratic duty – could well be to adopt a scenario bound up in contradiction.

This is just one example of how common sense can mislead us, and where, as Ellenberg repeatedly says in his book, we need to allow maths (which for him is a supercharged extension of common sense) to take over for a while. This is never more apparent than in the counter-intuitive proposition that it is good for business travellers to occasionally miss their flight at the airport. Of course, we are all thinking that the statement is bordering on madness, because that is what our common sense is screaming at us. But, as with our assumptions about polling voters on public service expenditure cuts, we're wrong again.

It was the economist George Stigler who famously, though perhaps apocryphally, said: "If you never miss a plane, you're spending too much time in the airport." What starts off as a clich'd throwaway comment (akin to "if you have to ask the cost of something, you can't afford it"), quickly emerges in Ellenberg's world as the tradeoff between arriving too early (and so wasting valuable time) and arriving too late (increasing the likelihood of missing your plane.) The crux is that whatever intermediate point you decide is best, there's a small – but not zero – chance you'll miss your plane. "I myself haven't ever missed a plane that I recall. But I've come close. But, I don't think I've reduced my chance of missing a plane to being zero. I think I've just been lucky." Maths can only guide us so far: so when most people are flying for an overseas meeting that simply cannot be missed, the implications of which are too disastrous to think about, most of us will be only too happy to pitch up at Heathrow ludicrously early and while away the time plugged into our laptops.

The world of maths Ellenberg is one of those interesting people that you meet from time to time. Quite apart from being a popular mathematics professor at the University of Wisconsin where he specialises in arithmetic algebraic geometry, he has the unusual distinction of being a prize-winning novelist and a former child prodigy (in maths, of course). When asked about how he got to where he is today, he says: "I was an early starter, very interested in maths from a young age. That's not every mathematician's story. But it's mine. A life in my world? I talk to people, we write on the board, I write email. But of course there's a lot going on, lots of ideas zooming back and forth in all directions."

One of the ideas was to write a book about maths for people who aren't mathematicians. "The book is written for people like you," says Ellenberg, meaning the readers of E&T magazine. "But it's also for people not like you.'I hope it works for a wide range of audiences, from 13-year-olds super-energised about learning maths, to 73-year-olds who always wanted to be. I wrote the book as an extension of my teaching. I get excited about maths in the classroom and I wanted to see if I could convey that to more than a hundred people at a time."

For most of us, the only chance we will ever get to communicate interactively with more than a hundred people simultaneously is via a social networking platform, all of which are, of course, based on mathematics. Ellenberg describes how Facebook is "made of maths. I write in the book about the way many people worry about Facebook knowing everything about you. I argue you might want to worry more about Facebook having only very vague, incomplete knowledge about you, and using it to make decisions that affect you". To Ellenberg this isn't simply a case of the latter being the lesser of two evils: it's a statement of reality based on the maths involved.

Now give us the big answer 

What everyone dreams of is winning the National Lottery. However, we instinctively know that the maths is stacked against us and the chances of winning are... well, let's not even go there because it's too frightening. But perhaps even more frightening is that according to Ellenberg "starting a successful small business is about as likely as winning the lottery". How can that be? It starts with "balancing out a tiny probability of making a fortune, against a modest probability of eking out a living, against a substantially larger probability of losing your pile". When you crunch the numbers, says Ellenberg, the expected financial value, like that of a lottery ticket, is less than zero. Typical entrepreneurs – as with typical lottery punters – overrate their chance of success.

Not wishing to finish on a down note, I ask Ellenberg if any IET member has a chance of winning the lottery. He replies: "Who says you won't? That one of your 153,000 readers will win doesn't sound that unlikely. Unfortunately, you, the one reading this right now, are seriously unlikely to be the one who wins. Sorry!" 

'How Not to Be Wrong' by Jordan Ellenberg is published by Allen Lane, £20

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