Thinking Better

DEPARTURE

YOU HAVE A CHOICE. The obvious path is a long slog, with no beautiful vistas on the way. It is going to take you forever and sap all your energy, but it will eventually get you to your destination. There is a second path, however. You’ve got to be sharp to spot it veering off the main path, seemingly taking you away from your destination. But you spot a signpost that says SHORTCUT. It promises a quicker off-road route that will get you to your destination faster and with minimal energy expenditure. There might even be the chance of a stunning view on the way. It’s just that you are going to have to keep your wits about you to navigate this path. It’s your choice.

This book is pointing you toward that second path. It’s your shortcut to the better thinking you’ll need to negotiate this unorthodox route and get you to where you want to go.

It was the lure of the shortcut that made me want to become a mathematician. As a rather lazy teenager, I was always looking for the most efficient path to my destination. It was not that I wanted to cut corners; I just wanted to achieve my goal with as little effort as possible.

So when my mathematics teacher revealed to me at the age of twelve that the subject we were learning in school was really a celebration of the shortcut, my ears pricked up. It started with a simple story featuring a nine-year-old boy named Carl Friedrich Gauss. Our teacher transported us back to 1786 in the town of Brunswick, near Hanover, where the young Gauss grew up. It was a small town, and the local school only had one teacher, Herr Büttner, who had to somehow teach the town’s one hundred children in just one classroom.

My own teacher, Mr. Bailson, was a rather dour Scot who kept strict discipline, but it sounded like he was a softy compared to Herr Büttner. Gauss’s teacher would stride up and down the benches brandishing a cane to maintain discipline among the rowdy class. The classroom itself, which I’ve subsequently visited on a recent mathematical pilgrimage, was a drab room with a low ceiling, little light, and uneven floors. It felt like a medieval prison, and Büttner’s regime sounded as if it matched the setting.

The story goes that during one arithmetic lesson Büttner decided to set the class a rather tedious task that would occupy them long enough so that he could take a nap. “Class, I want you to add up the numbers from 1 to 100 on your slates. As soon as you are done, bring your slates to the front of the class and place them on my desk.”

Before he’d even finished the sentence, Gauss was on his feet and had placed his slate on the desk, declaring in Low German, “Ligget se”—there it is. The teacher looked at the boy, shocked at his impertinence. The hand holding the cane quivered with anticipation, but he decided to wait until all the students had submitted their slates for inspection before upbraiding the young Gauss. Eventually the class had finished and Büttner’s desk was a tower of slates covered in chalk and calculations. Büttner began to work his way through the pile, starting with the last slate placed on the top. Most of the calculations were wrong, the students having invariably made some arithmetic slip on the way.

Eventually he arrived at Gauss’s slate. He began preparing his rant at the young upstart, but when he turned over the slate, there was the correct answer: 5050. And there were no extraneous calculations. Büttner was shocked. How had this nine-year-old found the answer so quickly?

The story goes that the precocious young student had spotted a shortcut that helped him avoid the hard work of actually doing much arithmetic. What he had realized was that if you add up the numbers in pairs:

then the answer for each pair was always 101. But there were 50 pairs. Hence the answer was

I remember being electrified by this story. To see Gauss’s insight into how to shortcut all this horribly tedious and labor-intensive work was a revelation.

Although the story of Gauss’s schoolroom shortcut is probably more legend than fact, it nonetheless captures beautifully an important point: mathematics involves not tedious calculation, as so many think, but rather strategic thinking.

“That, my dear students, is mathematics,” my teacher announced at the end of the Gauss story. “The art of the shortcut.”

Hello, my twelve-year-old self thought. Tell me more!

Getting Further Faster

Humans use shortcuts all the time. We have to. We have a short amount of time to make a decision. We have limited mental capacity to navigate complex problems. One of the first strategies that humans developed as a pathway to solving complicated challenges is the idea of heuristics—the process by which we make problems less complex by ignoring, either consciously or unconsciously, some of the information that’s coming into the brain.

The trouble is that most heuristics that humans use lead to bad judgments and biased decisions, and generally aren’t good enough to do the job. We might know one thing from experience and then try to extrapolate to all other problems by comparing them to this one thing we know. We judge the global by the knowledge of the local. This was fine when our environment didn’t extend too much beyond the small region of savannah we inhabited. But as our neighborhood expanded, these heuristics didn’t give us good ways of understanding how things worked beyond our local knowledge. This is the moment we had to start to develop better shortcuts. Those tools are what we today call mathematics.

To find good shortcuts requires the ability to lift yourself out of the geography you’re trying to traverse. If you are in the landscape, then often you can rely only on what you see immediately around you. Although each step feels like it is taking you in the right direction, the cumulative result of the path might take you the long way around to your destination or lead you astray completely. That’s why humans developed a better way of thinking: the ability to lift yourself out of the minutiae of the task at hand and understand that there might be an unexpected path that could get you to your destination faster and more efficiently.

This is what Gauss did with the challenge his teacher set the class. While the other students starting plodding from one number to the next, adding each new number they encountered to the tally, Gauss surveyed the problem in its totality, understanding how to use the beginning and end of the journey to his advantage.

Mathematics is all about this ability to use higher-level thinking to see structure where before we just saw random meandering pathways, to mentally lift yourself out of the landscape and look down from a great height to see the true lay of the land. Shortcuts emerge when problems are mapped in this way. And once we started to exploit the ability to see structure in our mind’s eye without physically encountering it, this capacity for abstract thinking unleashed human civilization’s extraordinary advances over the centuries.

The journey to better thinking began five thousand years ago around the Nile and the Euphrates. Humans wanted to find cleverer ways of building the city-states that were blossoming alongside these rivers. How many blocks of stone would be needed to build a pyramid? What area of land needs to be cultivated with crops to feed a city? What changes in river height were indicators of a forthcoming flood? Those who had the tools to find shortcuts to solving these challenges were those who rose to prominence in these emerging societies. The success of mathematics as a shortcut to the rapid development of these civilizations launched the subject as a powerful tool for those wishing to get further faster.

Time and again a gear change in civilization was effected by the discovery of new mathematics. The explosion of mathematics during the Renaissance and beyond, which gave us tools such as calculus, offered scientists extraordinary shortcuts to efficient engineering solutions. And today mathematics is behind all the algorithms being implemented on our computers to assist us through the modern digital jungle, offering shortcuts that help us find the best routes to our destinations, the best websites for our internet searches, and even the best partners for a journey through life.

It is interesting to note, however, that humans weren’t the first to exploit the power of mathematics to access the best way to tackle a challenge. Nature has been using mathematical shortcuts to solve problems long before we arrived. Many of the laws of physics are based on Nature always finding a shortcut. Light travels along the path that gets it to its destination fastest, even if that involves bending around a large object like the sun. Soap films create the shapes that cost the lowest amount of energy—the bubble makes a sphere because this symmetrical shape is the one with the smallest surface area and therefore costs the minimum energy. Bees make hexagonal cells in their hives because the hexagon uses the least amount of wax to contain a fixed area. Our bodies have found the most energy-efficient way of walking to transport us from A to B.

Nature is lazy, like humans, and wants to find the lowest-energy solutions. As the eighteenth-century mathematician Pierre-Louis Maupertuis wrote: “Nature is thrifty in all its actions.” It is extremely good at sniffing out shortcuts. Invariably it has a mathematical rationale to it. And often the discoveries of shortcuts by humans materialize out of our observations of how Nature solves a problem.

The Journey Ahead

In this book I want to share with you the arsenal of shortcuts that mathematicians such as Gauss have developed over the centuries. Each chapter will introduce a different sort of shortcut with its own particular flavor. But all of them have the aim of transforming you from someone who has to slog through the hard work of solving a problem to someone who can hand in their slate with the answer before everyone else.

I have chosen to take Gauss as a companion on our journey. His classroom success launched him on a career that marks him out for me as the prince of the shortcut. Indeed, the plethora of breakthroughs he made during his lifetime span many of the different shortcuts that I will introduce throughout the book.

By telling the stories of the shortcuts that mathematicians have amassed over the centuries, I hope this book will act as a tool kit for all those who want to save time doing one thing so that they can spend more time doing something more exciting. Very often these shortcuts are transferable to problems that don’t at first glance seem mathematical in nature. Mathematics is a mindset for navigating a complex world and finding the pathway to the other side.

This is why mathematics really deserves to be a core subject in the educational curriculum. Not because it is absolutely essential that we all know how to solve a quadratic equation; frankly, when has anyone ever needed to know that? The essential skill is understanding the power that algebra and algorithms play in solving such a problem.

I begin the journey to better thinking with one of the most powerful shortcuts mathematicians have developed: patterns. A pattern is often the best sort of shortcut. Spot the pattern and you’ve found the shortcut to continuing the data into the future. This ability to spot an underlying rule is the basis of mathematical modeling.

Quite often the role of the shortcut is to understand the foundational principle that unites a whole slew of seemingly unrelated problems. The beauty of Gauss’s shortcut is that even if the teacher tries to make it harder by asking you to add the numbers up to 1000 or 1,000,000, the shortcut still works. While adding numbers up one by one would get increasingly time-consuming, Gauss’s trick is unaffected by an increase in the number of numbers you’re adding up. To add the numbers up to 1,000,000, just pair them up again (1 + 1,000,000, 2 + 999,999…) to get 500,000 pairs that each add up to 1,000,001. Multiply these two together (500,000 × 1,000,001) and bingo: you’ve got your answer. The tunnel that provides a shortcut through the mountain is unaffected by the mountain getting taller.

The power of creating and changing language turns out to be a very effective shortcut. Algebra helps us recognize the underlying principles behind a whole range of different-looking problems. The language of coordinates turns geometry into numbers and often reveals shortcuts that were not visible in the geometric setting. Creating language is an amazing tool for understanding. I remember wrestling with an extraordinarily complex setup that needed many conditions to pin down. My doctoral supervisor’s suggestion that I “give it a name” was a revelation—it truly allowed me to shortcut thought.

Whenever I mention the idea of the shortcut, invariably people think I am trying to cheat somehow. The word “cut” sounds like you could be cutting corners, so it’s important right from the outset to distinguish between shortcuts and cutting corners. I’m interested in the clever path to get to the correct solution. I’m not interested in finding some shoddy approximation to the answer. I want complete understanding, but without unnecessary hard work.

That said, some shortcuts are about approximations that are good enough to solve the problem at hand. In some sense, language itself is a shortcut. The word “chair” is a shortcut to a whole host of different sorts of things we can sit on. But it is not efficient to come up with a different word for every distinct instance of an example of a chair. Language is a very clever low-dimensional representation of the world around us that allows us to efficiently communicate to others and facilitates our path through the multifaceted world we live in. Without the shortcut of single words for multiple instances, we would be overwhelmed by noise.

In mathematics too I will reveal how throwing away information is often essential to finding a shortcut. The idea of topology is geometry without measurement. If you are on the London Underground, a map showing how stations are connected is much more useful for finding your way around London than a geometrically accurate map. Diagrams are also a powerful shortcut. Again, the best diagrams discard anything that is extraneous to navigating the problem at hand. But as I shall illustrate, there’s often a fine line between a good shortcut and the dangers of cutting corners.

Calculus is one of humans’ greatest inventions for finding shortcuts. Many engineers depend on this bit of mathematical magic to find the optimal solution to an engineering challenge. Probability and statistics have been a shortcut to knowing a lot about a huge data set. And mathematics can often help you find the most efficient path through a complex geometry or tangled network. One of the staggering revelations I had as I fell in love with mathematics was its ability to find shortcuts to navigate even the infinite—a shortcut to get from one end of an infinite path to the other.

Each chapter begins not with an epigram but a puzzle. Often these puzzles involve a choice: the long slog or, if you can find it, the shortcut. Each puzzle has a solution that takes advantage of the shortcut that is at the heart of that puzzle’s chapter. They are worth tackling before you read the chapter, as often the more time you spend battling to get to your destination the more you appreciate the shortcut when it is finally revealed.

What I have discovered on my own journey is that there are different sorts of shortcut. Because of this, I spend time highlighting the multiple approaches you might take to the journey you are about to embark on, and show that you will get to your destination faster by using the most effective shortcut. There are shortcuts that are already waiting there in the terrain for us to take advantage of them; it’s just that you might need a signpost to point you in the right direction or a map to show you the way. There are shortcuts that won’t exist if you don’t do a lot of hard work to carve them out—like the tunnel that takes years to dig but once there allows everyone else to follow you through to the other side. There are shortcuts that require totally escaping the space you are in—the wormhole from one side of the universe to the other, or the extra dimension that shows how two things are much closer than you imagine provided you can step out of the confines of the current world. There are shortcuts that speed things up, shortcuts that cut down the distance you need to travel, and shortcuts that reduce the amount of energy you need to expend. Somewhere there is a saving that is worth the time to find the shortcut.

But I’ve also recognized that there are times when the shortcut misses the point. Maybe you want to take your time. Maybe the journey is the thing. Maybe you want to expend energy in an attempt to lose weight. Why go on a walk in Nature for the day if you curtail the pleasure of the walk by taking a shortcut home? Why read a novel rather than a synopsis on Wikipedia? But it’s still good to know you’ve got the option of a shortcut even if you decide to ignore it.

The shortcut is to some extent about our relationship to time. What do you want to spend your time doing? Sometimes it is important to experience something in time and there is little value to finding a shortcut that cheats you of the feeling. Listening to a piece of music can’t be shortcutted. It takes time. But on other occasions life is too brief to spend time getting to where you want to be. A film can condense a life into ninety minutes; you don’t want to witness every action of the character you are following. Taking a flight to the other side of the world is a shortcut to walking there and means you can begin your vacation sooner; if you could shorten the flight even further, you probably would. But there are times when people want to experience the slow version of getting to their destination. Pilgrimage abhors the shortcut, for instance. And I never watch film trailers, because they shortcut the film too much. But it is still worth having the choice.

Shortcuts in literature are invariably paths that lead to disaster. Little Red Riding Hood never would have met the wolf if she hadn’t strayed from the path in search of a shortcut through the wood. In Bunyan’s Pilgrim’s Progress, those who take a shortcut around Difficulty Hill get lost and perish. In The Lord of the Rings Pippin warns that “shortcuts make long delays” (although Frodo counters that inns make even longer ones). Homer Simpson swears after his disastrous detour on the way to Itchy and Scratchy Land, “Let us never speak of the shortcut again.” The dangers inherent in taking shortcuts are well summed up in the film Road Trip: “Of course it’s difficult—it’s a shortcut. If it was easy, it would just be ‘the way.’” This book looks to rescue the idea of a shortcut from these literary tropes. Rather than the road to disaster, the shortcut is the road to freedom.

Human Versus Machine

One thing that sparked my desire to write this book celebrating the art of the shortcut is the increasingly common perception that the human race is about to be superseded by a new species that doesn’t need to bother with shortcuts. We are now living in a world where computers are able to do more computations in an afternoon than I can do in a lifetime. Computers can analyze all of world literature in the time that it takes me to read one novel. Computers are able to analyze many more variations of a game of chess than the few moves I can hold in my head. Computers can explore the contours and pathways covering planet Earth faster than it takes me to walk to the corner store.

Would a computer today come up with Gauss’s shortcut? Why would it bother, when it can add up the numbers from 1 to 100 in the nth of an nth of a blink of an eye?

What hope is there for humankind to keep up with the extraordinary speed and nearly infinite memory of our silicon neighbors? The computer in the film Her declares to its human owner that the pace of human interaction is so slow, it prefers to spend time with other operating systems that can match its speed of thought. The computer looks at the speed at which humans operate the way we look at the slow pace at which a mountain emerges and erodes.

But maybe there is something that gives humans the edge. The limitations on the human brain’s ability to perform computations simultaneously compared to a computer’s and the physical shortcomings of the human body compared to the strength of a mechanical robot will force humans to stop and think whether there is a way to avoid all the steps that a computer or robot finds trivial.

Faced with a seemingly unassailable mountain, humans will instead seek out the shortcut. Rather than trying to go over the top of the mountain, is there perhaps a sneaky way around? And often it is the shortcut that leads to a truly innovative way to solve a problem. While the computer plows on, flexing its digital muscles, the human sneaks to the finish line, thanks to cunning shortcuts that help us avoid all the hard work.

Slackers, take note. I think laziness is our saving grace, what will protect us against the onslaught of the machine. Human laziness is a really important part of finding good new ways to do things. When a computer is faced with a problem, we know what it will say: Well, I’ve got these computational tools, so I can just bash my way through the problem. But I often look at a problem and think, This is just getting too complicated—let me try to step back and figure out a shortcut. Because a computer doesn’t get tired and it’s not going to be lazy, maybe it will miss things that our laziness takes us to. Because we don’t have the ability to plow right through problems the way a computer would, we’re forced to find clever ways to handle them.

There are many stories of how innovation and progress grew out of laziness and a desire to avoid hard work. Scientific discovery often emerges from a mind left idling. The chemist August Kekulé is said to have come up with the ring structure of the benzene molecule after falling asleep and dreaming of a snake swallowing its own tail. The great Indian mathematician Srinavasa Ramanujan often spoke of his family goddess, Namagiri, writing equations in his dreams. As he recalled: “I became all attention. That hand wrote a number of elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writing.” Jack Welch, as chairman and CEO of General Electric, spent an hour each day in what he called “looking out the window time.” A new invention is often born of someone who can’t be bothered to do things the hard way.

Laziness doesn’t mean that you do nothing. And this is a really important point. Finding shortcuts often requires hard work. It’s something of a paradox. Although the motivation to find a shortcut might come from the desire to avoid boring work, or perhaps to cope with the boredom that idleness brings, the result is often intense periods of deep thought. There is a fine line between idleness and boredom, and it is this that often is the catalyst for the hunt for a shortcut that can then involve a great amount of labor. As Oscar Wilde wrote, “To do nothing at all is the most difficult thing in the world, the most difficult and the most intellectual.”

Doing nothing is often a precursor to great mental progress. A paper published in 2012 in Perspectives on Psychological Science entitled “Rest Is Not Idleness” reveals how important our default mode of neural processing is to cognitive ability. This mode is often suppressed when our attention is too focused on the outside world. The recent surge of interest in mindfulness suggests the value of stilling the mind as a pathway to enlightenment. Often it means you prefer to play rather than work. But play is often the place to foster creativity and new ideas. It is one of the reasons that the offices of start-ups and math departments often contain pool tables and board games as well as desks and computers.

Perhaps society’s disapproval of laziness is a way of controlling and curtailing those who prefer not to conform. The real reason the lazy person is regarded with suspicion is that laziness is the mark of someone not prepared to play by the rules of the game. Gauss’s teacher saw his pupil’s shortcut to doing hard work as a threat to his authority.

Idleness has not always been shunned. Samuel Johnson very eloquently argued in favor of laziness: “The Idler… not only escapes labours which are often fruitless, but sometimes succeeds better than those who despise all that is within their reach.” As Agatha Christie admitted in her autobiography, “Invention, in my opinion, arises directly from idleness, possibly also from laziness. To save oneself trouble.” Babe Ruth, one of the best home-run hitters baseball has ever seen, apparently was motivated to hit the ball out of the stadium because he hated having to run between bases; when he hit a homer, he could take his time rounding the bases.

Choosing to Work

I do not wish to imply that all work is bad. Indeed many people get great value out of the work they do. It defines their identity. It gives them purpose. But the quality of the work is important. Generally, the work we find valuable is not a series of tedious, mindless tasks. Aristotle distinguished between two different sorts of work: praxis, which is action done for its own sake, and poiesis, or activity aimed at the production of something useful. We are happy to look for shortcuts in the second sort of work, but there seems little point in chasing the shortcut if the pleasure is in doing the work for its own sake. Most work seems to fall into the second category. But surely the ideal is to aspire to work of the first kind. That is where the shortcut aims to take you. The shortcut is not about eliminating work; it wants to lead you on a path to meaningful work.

The principle behind the new political movement Fully Automated Luxury Communism is that with advances in AI and robotics, machines can take over our menial work, leaving time for us to indulge in work we find meaningful. Work becomes a luxury. The cultivation of good shortcuts should be added to the list of technologies steering us toward a future of work that is undertaken for the joy of it rather than as a means to an end. This was Marx’s aim with communism: to remove the difference between leisure and work. “In a higher phase of communist society… labour has become not only a means of life but life’s prime want.” The shortcuts we have created promise to take us away from what Marx called the “realm of necessity” and lead us instead into the “realm of freedom.”