There is a familiar narrative about how mathematics has provided us with increasing knowledge of the world about us.

In ancient times, Pythagoras supposedly believed that the properties of whole numbers were the key to understanding the universe. When Isaac Newton put forward his laws of motion in the 17th century, we had a mathematical system that could predict the movement of the planets and the behaviour of physical objects on Earth. In the 19th century, James Clerk Maxwell derived the equations now named after him to show how electromagnetic fields develop, and over the 20th century our mathematical understanding of the universe we live in expanded dramatically through Albert Einstein’s general theory of relativity and Erwin Schrödinger’s equation in quantum theory. Since Newton, mathematics and its equations have proved astonishingly powerful in helping us understand the physical world and have led us to develop technology that, to the ancients, would have seemed truly magical.

Yet the famous equations of Newton, Maxwell, Einstein and Schrödinger do not feature in David Sumpter’s new book. Instead, he describes how mathematics is being used to understand the complex human interactions of the modern world. His “equations” relate to topics such as gambling, markets, the activities of online influencers, advertising, learning and artificial intelligence. They use ideas involving probability, statistics and decision-making – areas of mathematics that provide the tools we need in today’s world of social media, big data and machine learning. The book’s subtitle indicates Sumpter’s intention of presenting his material accessibly, and while he doesn’t shy away from mathematical detail, his presentation is always clear, and the mathematically sophisticated will have no trouble finding more technical details from the references provided.

Let’s get my big gripe out of the way. The “equations” that Sumpter presents are not all, in fact, equations. An equation is a specific mathematical artefact that *equates* two things – it has the form “A = B”, where A and B are both mathematical expressions. Sumpter’s Confidence Equation, Learning Equation and Universal Equation are not given in this way. Indeed, the last of these is simply the notion of conditional logic denoted by “If…then”; this is certainly a big idea, but it is hard to see how it could be expressed in any way that resembles an equation. I am, to my shame, less pedantic than many mathematicians – some would call me sloppy in my use of mathematical language – but even I found this profoundly irritating. Sumpter is presenting expressions, ideas, formulas, but not always equations! But perhaps he is right to choose a terminology that grabs the attention and will help attract for this book the wide readership it deserves.

Sumpter presents 10 mathematical ideas, a chapter to each, drawing on his personal experience and providing context for the mathematics with anecdotes about his own career. As a running theme through the book, he introduces a “secret society” – the “Ten” – named after the number of equations its members need to know. (In places these 10 necessary equations appear to be those that give the book its title, but elsewhere membership of the organisation appears to predate some of them, so I am not sure this conceit is presented consistently.) This rather nebulous group (whose own members are unaware of its existence) is credited with profound achievements over the last 250 years. Yet this plot device gives the author scope for useful commentary on the way mathematicians interact with the rest of the world.

For example, he starts with betting, showing how mathematics (together with computing power) can detect when bookmakers’ odds slightly disagree with the data, giving the gambler a tiny edge that, over many bets, can be exploited to yield profits. While this may make some people rich, these profits ultimately, as Sumpter points out, are made at the expense not of the wealthy bookmakers but of the small, ill-informed punters who are gambling money they possibly cannot afford to lose. The tension between interesting mathematics and the ethics of its applications becomes an increasing theme in the later chapters.

A strength of this book is that it is very personal – Sumpter writes fascinatingly about his experiences as a consulting mathematician, and reflects on the morality of his professional work. He says it is the book he wanted to write, and his personal values emerge clearly. The book concludes with a discussion of the role of mathematical modelling in today’s world – how “the Ten” must listen to others, considering moral values in deciding which problems to tackle while being rigorous in building models and analysing data. Sumpter recommends that we should consider the feelings of others, and our own intuition, in guiding our choices of subject matter but use full mathematical rigour in attacking these problems. “We need to learn how to handle the power that has been handed down to us…We need to be soft when we define our problems and brutally hard when we solve them.”

It is sometimes argued that mathematics is ethically neutral, simply a system that produces inarguable truths, independent of considerations of morality (indeed, for some this is one of its attractions). If the new mathematics of finance led to the crash and damaged the lives of millions, some would claim that this is not the fault of the mathematicians.

But others question the neutrality of mathematics. In a recent tweet, @ZariskiBusiness wrote: “Math is apolitical which is why we only teach undergraduates how to solve differential equations that can compute the trajectories of missiles.” Sumpter argues strongly that how mathematicians’ work is applied, for example by investment bankers to expand their riches, should be a concern for those creating the mathematics, and his book supports the argument that ethics should be part of the mathematics curriculum. Although he does not cover the topic in this book, presumably because of timing, huge changes have been made to our everyday lives over the last few months as a result of the mathematical modelling of the Covid-19 pandemic. The very public nature of recent interactions between government and scientists may (and should) lead to greater public questioning of the uses of mathematics, so these considerations are particularly relevant at this time.

This book gives insights into the working life of mathematicians (even if few of us are likely to have the opportunities to be entertained in expensive restaurants by business executives, advise Senate committees and meet the senior football people who have come to Sumpter, the author of a previous book on the mathematics of football). Young aspiring mathematicians will benefit from insights such as the reminder that even top mathematicians do not expect to understand mathematical papers on first reading, and will be inspired by the range of 21st-century applications featured. Its serious treatment of the new mathematics of social-media influencing, markets and sports betting, developments in mathematical biology and even the value of Pokémon Go show that present-day applications of mathematics are very different from those of even the recent past. I will encourage my mathematics undergraduates to read this book, since it will inspire them by showing the relevance of mathematics to today’s world and make them think about the moral issues they will face as mathematicians.

**Tony Mann is director of the Greenwich Maths Centre at the University of Greenwich. **

**The Ten Equations that Rule the World: And How You Can Use Them Too By David SumpterAllen Lane, 272pp, £20.00ISBN 9780241404546Published 1 October 2020**

### The author

David Sumpter, professor of applied mathematics at the University of Uppsala, Sweden, was born in London, moved to Scotland at the age of 10 months and studied computer science at the University of Edinburgh. It was during his third year, he recalls, that he realised “there was a code underlying it all: and that code was maths. It blew my mind…I just couldn’t believe how exciting this all was. How many questions there were that could be answered using computer simulations and mathematical models.”

Going on to a PhD at the University of Manchester, Sumpter had a supervisor, Dave Broomhead, who kept bees and so “thought it would be fun if we studied honeybee societies…I learned from him that research is about trying to learn something no one else knows, however small a something that might be.”

His work on bees led other biologists to seek Sumpter’s help, then sociologists studying segregation and economic development, and that in turn led to the mathematics of football and gambling. Soon he found himself “meeting with pro gamblers and flying down to FC Barcelona to talk to their data scientists at La Masia [the football club’s youth academy]. I have had meetings with most of the big clubs in the Premier League.”

Asked about the misconceptions he hopes to challenge, Sumpter responds that maths is a subject anyone can learn more about. Many people labelled “no good at reading or writing” at school, he points out, “go on to be very good at writing reports at work or texting funny things to friends and family…Our ability at English improves after school. And so does our ability in maths! We use spreadsheets, understand statistics and uncertainty (in the count of the US election, for example), we make charts for reports, talk about exponential growth of Covid and we play games involving probability.”

**Matthew Reisz**

## POSTSCRIPT:

Print headline: *Formulas for understanding*

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