The Maths That Made Us: how numbers created civilisation

By Michael Brooks

Quadratic equations, Pythagoras’ theorem, imaginary numbers, and pi — you may remember studying these at school, but did anyone ever explain why? Never fear — bestselling science writer, and your new favourite maths teacher, Michael Brooks, is here to help.

In The Maths That Made Us, Brooks reminds us of the wonders of numbers: how they enabled explorers to travel far across the seas and astronomers to map the heavens; how they won wars and halted the HIV epidemic; how they are responsible for the design of your home and almost everything in it, down to the smartphone in your pocket. His clear explanations of the maths that built our world, along with stories about where it came from and how it shaped human history, will engage and delight. From ancient Egyptian priests to the Apollo astronauts, and Babylonian tax collectors to juggling robots, join Brooks and his extraordinarily eccentric cast of characters in discovering how maths made us who we are today.

• Language: English
• Publisher: Scribe Publications
• Release date: Sep 8, 2022
• ISBN: 9781922586797

Michael Brooks

Michael Brooks is the author of the bestselling non-fiction title 13 Things That Don't Make Sense [9781861976475]. He holds a PhD in quantum physics, is a consultant at New Scientist and writes a weekly column for the New Statesman.

Book preview

THE MATHS THAT MADE US

Michael Brooks is a science writer with a PhD in quantum physics, and the author of several books, including the bestselling 13 Things That Don’t Make Sense and The Quantum Astrologer’s Handbook, a Daily Telegraph Book of the Year.

Scribe Publications

18–20 Edward St, Brunswick, Victoria 3056, Australia

2 John St, Clerkenwell, London, WC1N 2ES, United Kingdom

Published by Scribe as The Art of More 2021

This edition published 2022

Copyright © Michael Brooks 2021

All rights reserved. Without limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any form or by any means (electronic, mechanical, photocopying, recording or otherwise) without the prior written permission of the publishers of this book.

The moral rights of the author have been asserted.

Scribe acknowledges Australia’s First Nations peoples as the traditional owners and custodians of this country, and we pay our respects to their elders, past and present.

978 1 913348 98 4 (UK edition)

978 1 922586 79 7 (ebook)

Catalogue records for this book are available from the National Library of Australia and the British Library.

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Contents

Author’s Note

Introduction

Why our skill with numbers is the greatest human achievement of all

Chapter 1: Arithmetic

How we founded civilisation

Chapter 2: Geometry

How we conquered and created

Chapter 3: Algebra

How we got organised

Chapter 4: Calculus

How we engineered everything

Chapter 5: Logarithms

How we launched science

Chapter 6: Imaginary Numbers

How we fired up the electric age

Chapter 7: Statistics

How we made everything better

Chapter 8: Information Theory

How we created the modern era

Conclusion

Maths is a many-splendoured thing

Acknowledgements

Endnotes

Author’s note

All are welcome here, whether you love mathematics, have always hated it, or just wish you understood it better. There’s a whole rainbow of experiences with this subject and, right from the start, I wanted this book to be accessible to every part of the spectrum. To this end, I’ve kept things as straightforward as possible, but I’ve occasionally thought it was worth investing a tiny bit of effort to properly understand something. That means there are bits of actual maths in here: some graphs, equations, and calculations that I’ll walk you through gently. But if any of it bothers you, and you don’t feel like being bothered, just skip that bit. Life’s short enough already.

Introduction

Why our skill with numbers is the greatest human achievement of all

In June 1992, the American psychologist Peter Gordon travelled to a village of leaf-thatched houses on the banks of the Maici River in Amazonia, Brazil. ¹ He was there to meet his friend Daniel Everett, who was living as a Christian missionary among the remote and isolated Pirahã people. Everett had told Gordon that the Pirahã have a somewhat relaxed attitude to numbers: essentially, they just don’t bother with them. Intrigued, Gordon had come to find out more.

Using the clutch of AA batteries he had brought with him to the jungle, Gordon set up an experiment. He laid some out in a line and asked Pirahã villagers to create another line next to it with the same number of batteries. One, two, or three batteries was not a challenge. But the villagers struggled to correctly match a line of four, five, or six. Ten proved almost impossible. They had the same problem when asked to reproduce marks made on a piece of paper. If there were one or two marks they could do it, but six was the most anyone managed. The Pirahã, it seemed to Gordon, didn’t have any kind of handle on numbers — probably because they didn’t need to. Their way of life meant that their brains had never had any reason to form number concepts.

To most of us, it comes as a surprise to learn that people can happily get by without numbers. That’s because we unconsciously appreciate that numbers are deeply embedded in our daily lives. What we don’t appreciate, until it is brought to our attention, is that our way of life, our institutions and our infrastructures were built on numbers. Whether we’re talking about business, housing, medicine, politics, warfare, farming, art, travel, science, or technology, almost every aspect of our existence is built on mathematical foundations. And that is all the more astonishing when you appreciate that mathematics didn’t have to happen.

When it comes to a natural ability with numbers, we’re no better off than many other species. ² Humans are born only with what is now known as an ‘approximate number sense’. ³ This means that, in its raw state, your brain doesn’t bother specifying when there are more than three of something. So, when a human baby sees four apples, the sight is logged as ‘many’ or ‘more’. Our natural counting system is ‘1, 2, 3, more’. The brains of rats, chimpanzees, birds and monkeys also use an approximate number system. Reward a rat for pushing a lever five times, and it will occasionally return to the apparatus, performing a varying number of pushes close to five in the hopes of a treat. People have managed to teach chimps to do more sophisticated number-related tasks — remembering sequences of numbers, for example — and they can sometimes be better at this than untrained adult humans. But the training requires rewards: chimps don’t start to do maths for the fun of it. And neither did you; you learned to count because of cultural pressures. Those pressures came from an interesting place: a deeply ingrained cultural wisdom that tells us that mathematics matters.

The Tudor mathematician and mystic John Dee called mathematics ‘a strange participation between things supernatural, immortal, intellectual, simple, and indivisible and things natural, mortal, sensible, compounded, and divisible’. ⁴ This might seem like mumbo-jumbo, but mathematics is supernatural, in that we have used it to go beyond the natural. Developing maths allowed us to dissect and dismantle nature’s patterns and symmetries and, like gods, recast them in ways that serve our interest. Through maths, we shape the world around us to give ourselves a better experience of being human. The first leap was to count to four, and eventually we found ourselves establishing civilisations. Once our brains are schooled in the art of ‘more’, they become able to cope with complicated abstractions. They grow comfortable in a world where numbers can be applied not just to things that need counting, but to shapes, points, lines and angles — geometry, in other words. This gives us the ability to reimagine — on paper, on a wooden sphere, or just in our minds — a huge and complicated object like the Earth, say, and to navigate our way around it. We can also reimagine numbers — the ones we know and the ones we don’t — as symbols that we can manipulate to control and re-engineer the world, performing astounding feats of ordering, optimisation, and transportation. That’s algebra, in case you were wondering. We can even do calculations that predict the future that will result from the change that is happening around us. We call these calculus, and they enable us to realise a range of human aspirations, from free market capitalism to moon landings.

We learn this maths — or we are supposed to — early in our lives. In school, we are assured that maths is an essential skill; a passport for success; something that we have to pick up. And so we obediently, though often reluctantly, gather the tools of maths and do our best to learn how to use them. Some enjoy it; most don’t. And then, at some point, almost every one of us gives up.

Few of us will learn any more maths after that moment. Over the ensuing years, our hard-won skills wither away and we are left with only the basics at our fingertips. Without technological assistance — such as our mobile phone’s calculator, an essential tool for dividing up restaurant bills these days — we find ourselves able to reliably add and subtract only relatively small numbers, and maybe to multiply and divide a little. The rest is lost. We might even become ‘maths-phobic’, actively avoiding any encounter with numbers. Or we might just think of mathematics as beyond our grasp; something that’s ‘not for me’.

If that’s you, I hope this book will change your mind. The extraordinary achievement that is mathematics belongs to everyone, no matter how good they are — or aren’t — with numbers. We all benefit from the way the human brain has put maths to work over millennia, and we all have the right to engage with it, whatever our educational achievements. Why shouldn’t you be able to see how Newton’s calculus is as beautiful as the Taj Mahal, and why there is as much beauty in the Babylonians’ algebra as there once was in their Hanging Gardens? And the proper appreciation of mathematics is not just about some equivalence to what we traditionally see as beauty; it’s about seeing how we built the things we value as beautiful. Wherever we look in art or architecture, whether it’s one of Vermeer’s paintings or Istanbul’s majestic Hagia Sophia, we find mathematics facilitating its creation. This influence goes beyond aesthetic issues; the human story is itself inextricably interwoven with mathematics. Columbus’s journey to the Americas relied on understanding the properties of triangles, and the modern corporate world is a consequence of what a grasp of numbers makes possible. Mathematics provides the sculptor’s chisel that shaped the Renaissance and the ammunition that has engendered centuries of military success. It is the interpreter that allowed people with no common language to establish mutually beneficial trade and the fuel that took humans to the moon. It is the spark that electrified the world at the beginning of the 20th century, and the power behind every throne in the ancient world. No wonder King Shulgi of Ur was worshipped for his mathematical skills four millennia ago.

I learned none of this at school. I did learn how to pass all my maths exams, and sometimes how to apply that maths in order to work out the acceleration of a car or the force required to send a rocket into orbit. But I never learned what mathematics had done for us as a species, or how we came to invent it. It’s not too late, though. We can still find joy and meaning in maths, even if decades have passed since we gave up on learning its technicalities.

I still remember where and when I hit my mathematical limit: it was October 1987, and I was in a lecture hall at the University of Sussex in the south of England, having just started my undergraduate physics degree. I don’t remember the exact subject, but this lecture was the first in a course on advanced mathematical techniques. The topic just felt too hard for me, and the course was optional, so I walked out. Your story will be different, but at some point we all left our last maths class. Fortunately, the door never really closed behind us. So let’s go back in.

Chapter 1:

ARITHMETIC

How we founded civilisation

Humans didn’t evolve with a compulsion to count. But after we invented numbers and arithmetic, we eventually became reliant upon them. Numbers enabled people to govern, tax, and trade with each other, opening up the possibility of living in large interdependent communities. Eventually, arithmetic and its creations — fractions, negative numbers and the concept of zero — became the driving force behind economic and political success: those who can crunch the numbers are those that decide the future of workers, of nations, and even of the planet.

And it all started with a mental leap to the number 4.

In the first half of the 15th century, the Medici Bank was the toast of Florence and the envy of Europe. ¹ The secret of its success was simple: its chief accountant, Giovanni Benci, was an enthusiast for bookkeeping and a stickler for protocol. He audited the accounts of all the bank’s branches every year, checking on the status of debtors and the likelihood of payment defaults. If you managed one of the bank’s branches, and your accounts didn’t add up, Benci would call you in and tear you apart. And then, in 1455, Benci died and everything fell apart.

The Medici Bank’s employees were suddenly free of Benci’s prudence, and began promising far too generous a return to depositors, akin to a modern bank guaranteeing a 10 per cent return on any investment. The need to find the money for those guaranteed interest payments led to a toxic lending policy. The bank offered loans at exorbitant interest rates and, desperate to finance their wars, European kings and noblemen took up the Medici’s offers with no intention of paying their debts. The bank had no way of enforcing repayments, and so the money was lost. Meanwhile, the partners in the bank cast their eyes over books that were inflated by the promise of these never-to-be-seen payments, and took the non-existent profits out of the business for their own private spending. Their extravagant lifestyles ran riot, draining the bank of cash. In 1478, the Medici bank began to collapse. Faced with personal ruin, Lorenzo de’ Medici, great-grandson of the bank’s founder, bailed himself out by raiding public funds. The Florentine public was outraged, and stormed the Medici palace in 1494, setting fire to all its banking records. A century-long domination of Europe’s cultural, political, and financial capital went up in smoke.

History’s next demonstration of the world-changing power of accounting came with the French Revolution. We can trace its eruption to the sacking of accountant Jacques Necker, who had been trying to fix France’s broken financial system and reduce its crippling national debt. In the process, he had exposed the profligate indulgence of the French royal court. Eventually Necker’s interference was too much for the ruling classes, who were losing money hand over fist in his reforms. Necker lost his job as finance minister — but gained a loyal and dangerous band of admirers.

The historian François Mignet describes the revolution’s inciting moment: the hotheaded Camille Desmoulins stands on a table, pistol in hand. ² ‘Citizens! There is no time to lose!’ the young rebel cries. Necker’s dismissal, Desmoulins says, is an insult and a threat to every patriotic citizen of France. ‘One resource is left; to take arms!’ At this rallying cry, crowds rush into the streets. On their shoulders they carry busts of the sacked accountant. Mignon tells us: ‘Every crisis requires a leader, whose name becomes the standard of his party; while the assembly contended with the court, that leader was Necker.’

Necker’s crusade was focused on something we rarely conceive of as revolutionary: he wanted to balance the books. Necker had pointed out that the English parliament published all its accounts and England’s finances were in a healthy state, despite heavy borrowing to finance wars abroad. He was determined that France should achieve the same transparency. Balanced books, Necker said, were the basis of moral, prosperous, happy, and powerful government. And so he attempted to streamline the French government’s sprawling array of ledgers into a single account based on books that he would audit himself. The idea was not popular among those in power, but extremely popular among those who were not. And so, as historian Jacob Soll has put it, ‘The French Revolution would begin, in part, as a fight about accountability and accurate numbers in government.’ ³

It’s not only France that envied foreign financial systems; the pillars of the United States economy — tax revenues, the dollar and the central bank — were copied principally from Dutch and English banking practices. At the time, America had no banks, and was drowning in debt. Banks, said Alexander Hamilton in 1781, were ‘the happiest engines that ever were invented for advancing trade’. ⁴ Hamilton argued that freedom from British rule would come from understanding and controlling the accounts. ‘Tis by introducing order into our finances — by restoring public credit — not by gaining battles, that we are finally to gain our object,’ he said. ‘Great Britain is indebted for the immense efforts she has been able to make in so many illustrious and successful wars essentially to that vast fabric of credit raised on this foundation. Tis by this alone she now menaces our independence.’

In his role as first secretary of the treasury, Hamilton put all necessary measures in place and lifted the nascent United States out of the mire of bankruptcy. By 1803, Hamilton’s financial nous had enabled the US to raise enough Treasury bonds to purchase the Louisiana Territory from France, doubling the size of America. You might enjoy the musical Hamilton as a celebration of one of America’s founding fathers, but economic historians enjoy it as a celebration of fiscal prudence. And mathematicians see it as testimony to the power that comes from mastering numbers.

Learning to Count

We shouldn’t take mathematics for granted. The modern human — Homo sapiens, the ‘wise man’ — has been around for 300,000 years, and we have found human-created artefacts that are at least 100,000 years old. But our oldest reliable record of human counting is somewhere around 20,000 years old. The markings laid out on the surface of the Ishango Bone, discovered in the Ishango region of what is now known as the Democratic Republic of Congo, are a series of long notches that are grouped into three columns, each of which is subdivided into sets. Though we can’t know anything for sure, it doesn’t seem too much of a stretch to suppose that a single stroke designates an occurrence of ‘one’. Two strokes is ‘two’ and, well, you get the idea. Taken as a whole, the notches look like a tally system for counting lunar cycles. ⁵

The relatively recent creation of this bone suggests that counting is a late-blooming skill, not an inevitable result of intelligence. The brain inside your head is largely the same as the one inside the skull of the first Homo sapiens, and it seems that for most of our species’ history, this wise man did not bother with numbers at all.

Once we did get to grips with numbers, however, the advantage was clear. This is why you probably don’t even remember learning to count. Counting is such a valued skill in most human cultures that you would have started before you began to lay down permanent memories. And I’m willing to bet that you learned to count using your fingers. ⁶

The first time I ever really thought about finger-counting — apart from in embarrassment when I realised I was doing it in public, in a supermarket, as I counted off that night’s dinner party guests — was when I saw Quentin Tarantino’s riotous war movie Inglourious Basterds. During a scene in a basement bar, a British character is pretending to be German. He indicates to the barman that he wants three glasses by holding up his index, middle, and ring fingers. The German officer with whom he is sharing a table knows immediately that his drinking partner is a fraud. ‘You’ve just given yourself away, Captain,’ he says.

Germans use the thumb for ‘one’, so a German would have ordered three glasses using the thumb and the first two fingers. ⁷ In Asia, people finger-count differently. My friend Sonali, who grew up in India, learned to count using the individual segments of her fingers. Merchants in the Indian state of Maharashtra do it differently again. ⁸ They start with the thumb, like the Germans, but when they get to five, they raise the thumb of the other hand — usually the right — to indicate one ‘five’ has been reached. The left fist closes again, and the thumb comes out to indicate ‘six’.

Imagine doing business with a Maharashtran merchant. At first you would probably be confused, but it wouldn’t take you long to figure out, with no language at all, how much you were being asked to pay. Thanks to finger-counting, you can carry out commercial trade with no common written or spoken language. All you need is for both sides to know what currency you’re talking about, and to appreciate the meaning of numbers as they rise from 1 into the hundreds and thousands.

This is why learning finger signs was an essential part of education for almost all members of ancient societies. Even the most isolated communities would barter with passing traders with whom they might have no common language. In his 4th-century BC writings, Aristophanes mentions finger-counting as being a common practice of ancient Greece and Persia. The Roman writer Quintillian talked about the shame that would be heaped upon a lawyer who hesitated over his finger signs for numbers. Aztec paintings depict men using finger signs, and in medieval Europe, finger-counting was so ubiquitous that Luca Pacioli’s 1494 acclaimed mathematics textbook Summa de Arithmetica, Geometrica, Proportiono e Proportionalita contained a complete illustrated guide to the art. Even as late as the 18th century, the German adventurer Carsten Niebuhr describes Asian market traders conducting covert negotiations by grasping each other’s fingers and thumbs in various configurations. To keep their business to themselves, they would do this with their hands hidden inside voluminous sleeves or under a large piece of cloth draped over their wrists.

Because the means of signifying numbers has always varied from culture to culture, students of business had to learn their hand gestures carefully. Poets and teachers created rhymes and aphorisms to help with this, such as this effort from the ancient Arab world. ‘Khalid left with a fortune of 90 dirhams, and when he came back he had only a third of it left.’ Though it doesn’t sound helpful to us, the Arabian finger sign for 90 was an index finger curled tightly against the base of the thumb. One-third of 90 is 30, the sign for which was a much broader circle, with the tip of the index finger held against the tip of the thumb. The implication is that Khalid has been sodomised as well as robbed. I suspect you will now remember these ancient signs for 90 and 30 for the rest of your life.

The reason finger signs are so ubiquitous has a lot to do with the reason that humans became good with numbers, once we realised their value. It’s this: over the first five years of your life, through play, experimentation and stimulation, your brain develops something called finger sense, or ‘gnosis’. This is the ability to treat and sense each digit separately. After a while, your brain begins to hold an internal representation of your fingers, and this representation is used to help when you start to deal with numbers. ⁹ The beauty of fingers is that they can be seen, felt and moved. They come in two collections of five units, each of which can be put into different configurations of flexion. If you were to put together a tool for assigning a concept of ‘how many?’ to a group of objects in front of you, you would struggle to beat your own fingers.

Brain scans show that when most of us are presented with mathematical tasks such as subtracting one number from another, the area of the brain that deals with inputs from the fingers steps up to the plate. If the numbers involved are big, the activation of those brain circuits is even clearer. Interestingly, if you’re particularly good at subtraction, your brain’s finger circuits don’t get quite so active: they’re barely breaking sweat, in other words. But it’s also worth noting that if you weren’t encouraged to use your fingers in play as a child — especially when singing counting songs such as ‘One, two, buckle my shoe’, you may never have really ‘got’ numbers. ¹⁰ Numbers just won’t be represented in your brain in the same way that they are for other people. That’s one reason some people struggle with maths.

Once you have numbers at your fingertips, it might seem obvious that the next step is to start writing them down. But if we didn’t have to start using numbers, we certainly didn’t have to start writing them down. After all, when trade was in the moment, involving face-to-face bargaining and immediate transfer of goods or services, there was no need to keep tabs on the transactions. So what made us develop written numerics? By writing numbers down, we could formulate predictions about celestial events that might have religious relevance — new moons or solar eclipses, say. Or we could create inventories of stock and prices paid, and document promises to buy and sell at some point in the future. Writing numbers probably started as a religious practice but it also allowed us to take trade to the next level. Whatever its origins, it led directly to the prosperity we enjoy today.

The Accounting Revolution

We can’t really know who the first people to keep records of numbers were; it may be that the Ishango Bone was notched a long time after humankind’s mathematical journey began. We do know two things for sure, however. The first is that there have been myriad forms of numerical notation, starting with notched bones and moving into Incan knots, Babylonian marks on clay, Egyptian ink on papyrus, and eventually the 20th century’s electrical voltages inside a microchip. The second is that this new ability to keep financial accounts was revolutionary. You might not think of accountancy as anything other than a chore you’re glad someone else can do for you, but its invention shifted human culture on its axis.

Our earliest evidence of commercial accounting comes from around 4,000 years ago, when Mesopotamian traders began making records of agreements to sell sheep. Each agreement was represented by a clay ball. The balls were sealed inside a hollow sphere, which was marked with the number of balls it contained, then baked so that the record could not be altered. It was an insurance against the misremembering — deliberate or otherwise — of what had been agreed.

That system evolved into a simpler record: marks baked onto the surface of a clay tablet. Now it was easy to see what had been agreed, bought, sold, or paid. And by this time, humans were already starting to recognise that manipulating numbers could bring more than trade: it could also bring power.

In 2074 BC, in the region of the world we now call south-west Iran, King Shulgi of Ur introduced what scholars have termed ‘the first mathematical state’. ¹¹ Shulgi began with a military reform, and followed it up with an administrative reform. This required the scribes of Ur to create complex accounts of everything in the kingdom. The overseers of the working population of Ur left us records of hours worked, illnesses, absences, and the output of loaned and borrowed slaves. If they were not able to show that they had pressed 30 days’ worth of work from each of their workers in each month (regardless of how many days were in the month), the deficit would have to be paid to the state. If the overseer scribe died in deficit, the debt passed to his family. King Shulgi’s accounting system was designed around a surprising principle: it should make it as easy as possible to detect attempts to defraud the state. Auditing, it turns out, is the true cradle of civilisation.

If Ur was the first mathematical state, Shulgi was the first mathematical god. He declared himself divine in the twenty-third year of his reign. From this time on, his subjects were instructed to worship him and praise his attributes — and in particular his artistry with numbers. We have records of the hymns that were sung in Shulgi’s praise; one of his divine attributes was, apparently, his extensive mathematical training in the ‘tablet-house’, where he learned addition, subtraction, counting, and accounting.

Such were the advantages of putting mathematics at the centre of Shulgi’s state that, within a generation, mathematics became the highest art in the land, an essential component of a scribe’s training. By the turn of the second millennium BC, a fully qualified scribe would be able to read and write in Sumerian and Babylonian, and know about music and mathematics. The mathematics in question was not the utilitarian number-wrangling of the accountants, but the manipulation of numbers to do extremely difficult — and ostensibly useless — calculations. Essentially, it involved solving riddles such as ‘I have added together the perimeter, the diameter and the area of a circle, and the outcome was 115’ — the scribe’s job was to find the radius. ¹² This was maths for maths’ sake, and it was considered one of the ‘virtues’. Only with mathematical prowess in place could an educated