Alex Bellos – When Math Takes Us By Surprise

For the uninitiated, math is the boring exercise of manipulating numbers, a practice that most people would consider outdated ever since the invention of the calculator.

To Alex Bellos, author of Here’s Looking at Euclid, math is a fascinating subject that can be used to evaluate almost any problem that can be solved using a series of rigid rules.

It is a system that we can use in order to discover why the real world defies common sense, and even to explore realms outside the universe as we know it.

Here’s what he has to say about it.

The Interview

Good talking to you, Alex.


So, people write non-fiction books for two reasons: to share information that they already know, or to give themselves a reason to learn about something. Would you say that this book was a learning experience or an attempt to share your enthusiam with a wider audience?

It was an attempt to communicate my childhood love of math to a general audience. But in order to communicate that love I needed to rediscover it, since it had been almost 20 years since I had last thought seriously about math. So the book was also an excuse for me to go back to my (square) roots, and it did involve a fair bit of relearning.

Math is definitely not like riding a bike. I’m currently revisiting my multivariable calculus and discovering that some relearning is an absolute necessity in order to move forward.

Of course, when most people think about math they think about numbers. You argue that numbers are good but letters are better. Talk to me about abstraction in math, where it came from, and where it’s taking us.

…now anything is allowed into the mathematical domain so long as it can be coherently defined with rules.

One of the great narratives in math – if not the defining narrative – is the move from it being the science of quantity to being the science of patterns and rules. We started with the counting numbers, and a fixed geometry of straight lines, but gradually this led to abstractions like negative numbers and then abstractions of abstractions like imaginary numbers, and now anything is allowed into the mathematical domain so long as it can be coherently defined with rules.

Mathematicians used to be concerned with solving real world problems – geometry was originally about calculating areas for the tax inspector, for example – but for the last century or so pure maths has been about examining abstract systems with little concern for real-world applications.

I do find it interesting just how much math has evolved since its inception. It’s also interesting how this deviation from real-world applications has played out. Playing with abstract ideas like this has actually led us to unanticipated real-world applications. Nobody foresaw that non-Cartesian coordinates, for example, would eventually be used by Einstein to describe gravity and the real shape of the universe.

Mentioning the universe also invokes the notion of the infinite. Infinity is an elusive concept, to be sure. There are different kinds of infinities, for example. Tell me a little about this, and what infinities tell us about the world we live in.

To be sure! I studied math and philosophy at Oxford and I have always been drawn to the more philosophical side of math. The concept of infinity is a BIG issue in this area, literally, and as a result I dedicate a chapter in my book to it.

…you can have one infinity that is bigger than the other.

I focus on the work by the German mathematician Georg Cantor who redefined our understanding of infinity by considering it not as number in the traditional sense but as the amount of objects in a set. He showed with one of the most elegant proofs in math that there are some infinite sets that you can count – such as the set of “counting numbers” i.e 1,2,3,4,… – and some sets that are so big that you cannot count all the elements in them – like the set of numbers on a continuum. So, you can have one infinity that is bigger than the other. This is fascinating and counterintuitive.

Absolutely. There are all kinds of surprises, like the fact that shapes with infinite length can have a finite area, or that there are an infinite number of points in a finite area. But there is one field of mathematics that might be even more counterintuitive than infinity.

Probability and statistics is arguably the most important field of math in the modern world, just as much because they’re abused as because they can be enlightening. I’d like to hear a few examples of the counterintuitive nature of statistics and odds.

Statistics is based on probability, and probability is arguably the trickiest and most misleading part of maths. This is because probability is the study of randomness, and our brains find randomness hard to process. There are so many counter-intuitions in probability.

…coincidences are a lot more likely than you might think.

One of the most famous is the so-called Birthday Paradox: How many people do you need in a room for it to be more likely than not that two people share the same birthday? Since there are 365 days in a year, it seems reasonable to imagine that the number is quite large, maybe 100 people or more. In fact, you only need 23 people! This is stunningly low. The moral here is: coincidences are a lot more likely than you might think.

Since the human brain evolved to look for patterns, it definitely has trouble coping with the unpredictable and the purely random. One of my favorite examples is the Monty Hall problem, which took me forever to wrap my brain around.

We’ve touched on some of the interesting concepts in your book, but before I let you go I’d like to ask what you think is the most important lesson your book has to offer? Do you think it has the potential to change the way some of your readers look at the world?

My aim was to show how it evolved from the notion of counting to the mind-boggling, abstract discipline it is today…

The most important lesson is the realization that math is a fascinating part of our cultural heritage. My aim was to show how it evolved from the notion of counting to the mind-boggling, abstract discipline it is today, using history but also interviewing people who use maths in their day to day lives. Since the world runs on mathematical principles, by changing your view of math you change your view of the world.

If you have never considered why the digits of pi are so special, or why we use ten numerals and not, say, 12, or why the Greeks never had negative numbers or why slot machines are such brilliant money-making machines, then there is something in the book which will make you go “wow”. That was the idea anyway!

The Author of Mind Hacks on the Latest in Cognitive Science