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Chapter 1
ARITHMETIC
How we founded civilization
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.
