Introduction
The Ten
Do you want to know the meaning of life?
That’s a little broad. I’ll be more precise. Do you want to take a more balanced approach to life? To feel confident in your actions and the things you say? Do you want to learn from your mistakes? To know when to stick with the path you’re on, and when it is time for a change?
Perhaps you want to influence others around you. Maybe you want to know how popular you really are. Wouldn’t it be nice, at least, to know if others are judging your actions fairly? To know that you’re open-minded and just in turn?
The questions above are about our psychology, our philosophy, and in some cases, our spirituality. They are about our human nature and our place in the world. They are also questions that seem to lack clear-cut answers. If they were straightforward, there might not be so many self-help books on offer, so many daytime TV shows where guests let their feelings out, so many motivational TED Talks. It feels natural, in seeking answers to these questions, to reflect on our own memories and instincts. Our search for better judgment, greater self-confidence, and emotional balance becomes a journey into our own human psyche. A journey of self-development.
I want you to think about another possibility, one that this book explores. I am going to tell you a story about an exclusive society of individuals who have cracked the code. They have discovered a small number of equations—ten of them, in fact—that, when used correctly, offer the seeds of success, popularity, wealth, self-confidence, and sound judgment.
This secret society has been with us for centuries. Its members have passed their knowledge down through the generations. They have taken positions of power in public service, in finance, in academia, and, most recently, inside tech companies. They live among us—studying us, advising us, and sometimes controlling us. They are self-assured, often rich, sometimes even happy. They have discovered secrets the rest of us desire.
In Dan Brown’s book The Da Vinci Code, cryptographer Sophie Neveu discovers a mathematical code while investigating the murder of her grandfather. She is led to Professor Robert Langdon, who reveals that her grandfather was head of a secret society, the Priory of Sion, which understands the world through a single number, the golden ratio, ? ˜ 1.618.
The Da Vinci Code is fiction, but the secret society I have investigated for this book has similarities to the one Brown describes. Its secrets are written in a code that few fully comprehend; its members communicate in arcane scripts. The society has its roots in Christianity, and it has been rent by internal moral conflicts. It differs from the Priory of Sion, however, in that it has no rituals whatsoever—no initiation rites, no talismans of membership. This makes it far more difficult to detect, even as it is much more pervasive in its activities. It is invisible to those on the outside.
It has no name, so I have to give it one. I will call it TEN, after the equations one must learn to become a full member.
How do I know about TEN? The answer to that is simple. I am a member. I have been involved in its workings for twenty years and have gravitated closer and closer to its inner circle. I have studied its products and put its equations into practice. I have experienced firsthand the rewards access to its code can bring. I have worked at the world’s leading universities and was appointed full professor in applied mathematics the day before my thirty-third birthday. I have solved scientific problems in fields ranging from ecology and biology to political science and sociology. I have been a consultant in government, artificial intelligence, sports, gambling, and finance. (In order of respectability, perhaps.)
Membership in this club has brought me into contact with others like me. Like the data scientists who study how sport stars control the field of play; and the technical experts employed by Google, Facebook, Snapchat, and Cambridge Analytica, who control our social media and are building our future artificial intelligence. People like Marius and Jan, young professional gamblers who have found an edge on the Asian betting markets; those like Mark whose microsecond calculations skim profits from small inefficiencies in share prices. I have witnessed firsthand how researchers like Moa Bursell, Nicole Nisbett, and Viktoria Spaiser use equations to detect discrimination, understand our political debates, and make the world a better place. I have learned from the older generation, like ninety-six-year-old Oxford professor Sir David Cox, who discovered the society’s source code, piece by mathematical piece.
We know these equations are important because they’ve already changed the world. Chapter 9 tells the story of three engineers in California who used the learning equation to increase the time viewers spent watching YouTube by 2000 percent. The betting equation, the influencer equation, the market equation, and the correlation equation have reshaped, respectively, betting, technology, finance, and advertising to generate billions of dollars of profits for a small number of the society’s members.
It is not just about the money. The power of TEN applies to questions all of us confront on a daily basis. Should you quit your job (or your relationship) and try something else? Are you really less popular than your friends? How much effort should you put into climbing the social ladder? Is it possible to cope with the vast flood of information from social media? Should you let your kids spend six hours a day staring at their phones? How many episodes of a Netflix series should you watch before trying something else?
These might not be the problems you expect a secret mathematical code to resolve. But here’s the thing. The same small set of formulas can provide the answers to questions ranging from the trivial to the profound, and about you as an individual and society as a whole. The confidence equation, introduced in chapter 3, which helps you decide whether you should quit your job, also reveals subtle racial and gender biases at work. The reward equation, discussed in chapter 8, shows how social media has driven society to the tipping point—and why this isn’t necessarily a bad thing.
I believe in TEN because it has worked for me. These equations have shaped my thinking in ways that have helped me find happiness and fulfillment. They’ve also made me a better person: more balanced in my outlook and better able to understand the actions of others. Learning TEN involves challenging many of our everyday assumptions, and changing the way we make decisions. As you learn the secrets in this book, more and more aspects of the world will start to make sense. Math can be used to solve problems you didn’t know had solutions.
That said, this is not the kind of self-help book that tells you the one and only way to do things, nor the kind that can be distilled into bullet points. It is not the Ten Commandments. It is not a list of dos and don’ts. There are instructions on how to use each equation, but none of these come à la carte. You can’t skip to page 157 and find out the exact number of Netflix episodes you need to watch. The secret code of TEN is more like a language; first you learn the basic vocabulary, then some grammar, and soon the whole library is yours to explore.
Which is to say, the equations in this book build on one another. We start with the three key statistical techniques for understanding data: Bayes’s rule (equation 1), logistic regression (equation 2), and confidence intervals (equation 3). We then look at how to build and understand mathematical models: the Markov assumption (equation 4), stationary distributions (equation 5), and stochastic differential equations (equation 6). After that, we develop the mathematics used in modern data science: equations for correlations (equation 7), rewards (equation 8), and learning (equation 9). Finally, equation 10, the universal equation, tells us what questions we can and can’t answer using mathematics.
This is also a book about people, as we explore how the code of TEN has evolved over and shaped the last 250 years of human history. It is the story of a small group of thinkers who have passed the secrets of rational thinking itself from generation to generation. They have asked big questions. They wanted to know how to think more clearly and more precisely. They wanted to be able to evaluate the truth of what we say. They have asked what it even means for something to be true or false. Theirs is a story about the really big issues: the nature of reality and their place in it.
When handing down his power to a new generation, Spider-Man’s uncle tells him that “with great power comes great responsibility.” With so much at stake, the hidden powers of TEN require even greater responsibility than those bestowed by a Spider-Man suit. You are about to learn secrets that can transform your life. And you will also be forced to think about the effect these secrets have had on the world we live in.
For too long, the code has only been accessible to a chosen few. Now we are going to talk about it, openly and together.
1
The Judgment Equation
Imagine you are an experienced traveler, having flown one hundred times before. But the flight you are now on is different. As you descend, the plane starts to rattle and shake in a way you have never experienced. The woman next to you lets out a gasp; the man sitting across the aisle grips his knees. Everyone is visibly scared. Could this be it? Could the worst possible scenario be about to unfurl?
When safely on the ground, most of us understand that flying isn’t dangerous. When you step onto a commercial plane, the probability you will die in a fatal crash is less than one in ten million; but this can feel very different in the air. I envision a crash every time my plane takes off and lands. If I am with my family, I imagine holding my children’s hands, telling them I love them and not to worry. I imagine holding myself together for their sake as we plunge to our deaths. When I am alone on a plane, surrounded by strangers, I see years stretching ahead, years of my family without me. The funeral passes by quickly, and I see my wife alone with our kids, coping and telling them stories about our time together.
This is when I need the judgment equation.
In situations like the one above, mathematicians will take a deep breath, put their trays in the upright position, and collect up all the information they have. In mathematical notation, we write the baseline probability of plane crashes as P(CRASH), with the P standing for probability and CRASH denoting the worst possible scenario of a fatal (for you) landing. We know from statistical records that P(CRASH) = 1/10,000,000, a 1 in 10 million chance.
Global crash statistics are clearly not enough for the current situation; it is your plane that is shaking! To combine events, we write P(SHAKE | CRASH) to denote the probability that the plane shakes this much given that it is just about to crash (SHAKE denotes “the plane shaking,” the vertical line, | , denotes “given”). It is reasonable to assume that P(SHAKE | CRASH) = 1—that is, crashes are always preceded by a lot of shaking.
We also need to know P(SHAKE | not CRASH), the probability of this much shaking in an otherwise safe landing. Here you have to rely on your senses. This is the scariest flight you have been on, out of 100 similar trips, so P(SHAKE | not CRASH) = 1/100 is your best estimate.
These probabilities are useful, but they aren’t what you are desperate to know. What you want to know is P(CRASH | SHAKE), the probability that you are about to crash given that the plane is shaking so much. This can be found using Bayes’s theorem:
The · symbol in the equation represents multiplication. Soon I will explain where this equation comes from, but for now let’s just accept it. It was proved true by the Reverend Thomas Bayes in the mid-eighteenth century and has been used by mathematicians ever since. Putting all the numbers we have into our equation, we get:
Even though this is the worst turbulence you have ever experienced, the chance you are going to die is 0.00001. There is a 99.99999 percent chance that you are about to land safely.
The same reasoning can be applied in a whole range of seemingly dangerous situations. If you think you see something scary in the water while swimming on a beach in Australia, the probability that it is a shark remains minuscule. You may worry when a loved one is late coming home and you can’t reach them, but the most likely explanation is that they have simply forgotten to charge their phone. A lot of what we perceive as new information—shaking planes, murky shapes in the water, or failed phone calls—is less scary than we think when we approach the problem correctly.
The power of Bayes’s rule, as we will see, lies in how it clarifies our thinking. When we find ourselves in stressful situations, Bayes allows us to sort out information from thoughts and fears, telling us what’s actually important to know and why. It lets us keep calm when everyone else is panicking. It makes us more forgiving of ourselves and others. It can help reveal the origins of the universe—and it might even let us believe in miracles.
* * *
I see the world in a way I call “cinema.” I spend a lot of my time, when I’m on my own or even when I’m in the company of others, playing films of my future in my head. These aren’t just one film or one future but rather movies with lots of different plot twists and endings.
Most of these are much less dramatic than an airplane crash. They play, continually and in parallel, in an area of my brain just above my left eye. Or at least that is how it feels. I have a meeting with a book editor and play through our discussion in my head, thinking about what I will tell her. I am delivering a seminar and I see how I will present the material, imagining the difficult questions that might come up. Many of the films are abstract—I navigate my way through a scientific article I am writing; I see the structure of the thesis of one of my PhD students; I work through a math problem.
The scientific term for the movies I play in my head is model. A shark attack is a model just as the plan for my scientific research is a model. Models can be anything from vaguely defined thoughts to more formally defined equations, and they are the foundation of a mathematical approach to the world.
It’s not as though the films I create are actually the future. Nor are scientific models always correct. Details are missing, yet to be filled in by reality. Crucially, they are almost always wrong. The book editor takes the discussion in another direction and I forget my questions. A hole appears in the reasoning in a scientific article and I can’t fix it. I make a massive calculation error in the first line of my math and my results come out all wrong.
When I play the airplane crash movie, I estimate the probability of it really happening and find it reassuringly low. This doesn’t stop the film from playing—I still get scared when flying or when swimming in the ocean—but it does help me focus my thoughts. Instead of just being afraid, I think about how much my family means to me, and why I should travel less and swim in the ocean more.
You may not imagine a video projector playing above your eyes, but we all have ways of modeling the future. Do you look ahead in words, films, or computer games? In photographs, sounds, or smells? Is it an abstract feeling, or do you visualize real events? All of these are mathematical engines just waiting to rev up. To use the power of Bayes, and the other nine equations to come, the first step is to become aware of how we use models.
Copyright © 2020, 2021 by David Sumpter
