Thinking in Numbers: How Maths Illuminates Our Lives

  And therefore, like a cipher [zero]

  (Yet standing in rich place), I multiply

  With one ‘We thank you’ many thousands more

  That go before it

  Recorde’s book abounded with exercises. Shakespeare and his classmates’ sheets of paper would have quickly turned black with sums. They measured cloth and purchased loaves and counted sheep and paid clergy. But William’s mind returns ceaselessly to the zero. He thinks of ten and how it differs from his father’s ten. To his father, ten (X) is twice five (v): he counts, whenever possible, in fives and tens. To his son, ten (10) is a one (1) displaced, accompanied by a nought. To his father, ten (X) and one (i) have hardly anything in common: they are two values on opposite ends of a scale. But for the boy, ten and one are intimately linked: there is nothing between them.

  Ten and one, one and ten.

  With a retinue of zeroes, the boy sees, even the humble one becomes enormously valuable. Imagination can reconcile even one and one million, as Shakespeare affirmed in his prologue to Henry V, when the chorus stake their claim to represent the multitudes on the field of Agincourt.

  O, pardon! Since a crooked figure [digit] may

  Attest in little place a million,

  And let us, ciphers to this great accompt,

  On your imaginary forces work

  But it is perhaps in his poetry that the boy – now a young man – would most clearly express the impact of Recorde’s teaching on his mind. In Sonnet 38, Shakespeare wrote about the relationship between himself and his beloved Muse, comparing their couple to a ten: the poet, the zero, and his beloved, the one.

  O, give thyself the thanks, if aught in me

  Worthy perusal stand against thy sight . . .

  Be thou the tenth Muse, ten times more in worth . . .

  This relationship, as everyone knows, would prove remarkably fruitful: his poems and plays multiplied. In the Globe Theatre, round as an O, an empty cipher filled with meaning, Shakespeare’s loquacious quill drew crowds with his dreams.

  ‘Shakespeare was the least of an egotist that it was possible to be,’ wrote the nineteenth-century critic William Hazlitt. ‘He was nothing in himself; but he was all that others were, or that they could become.’ That nothing, who had once been a dazzled schoolboy, labouring for the breakthrough moment when he comprehended the paradoxical fullness of the empty figure of a zero, would surely have been delighted with this description.

  Shapes of Speech

  We know next to nothing with any certainty about Pythagoras, except that he was not really called Pythagoras. The name by which he is known to us was probably a nickname bestowed by his followers. According to one source, it meant ‘He who spoke truth like an oracle’. Rather than entrust his mathematical and philosophical ideas to paper, Pythagoras is said to have expounded them before large crowds. The world’s most famous mathematician was also its first rhetorician.

  It is easy to imagine the atmosphere of intrigue and anticipation that would have attended such a spectacle. To believe later accounts, his lectures were always full. People would come from far and wide to hear the legendary figure speak. Citizen after citizen: male and female, young and old, rich and poor, politicians and lawyers and doctors and housewives and poets and farmers and children. Latecomers, red-faced from running, jostled for a place at the back. Waiting for the event to start, they would have shared gossip. Pythagoras, someone whispered, has a thigh made of gold. His words can soothe even wild bears, said another. He communes with nature, affirmed a third. Even the rivers know his name!

  Pythagoras was a handsome, forty-something-year-old man when, in around 530 BCE, he founded his school of disciples on the Greek colony of Croton in southern Italy. On this distant outpost, many hundreds of miles from Athens, its residents treated the newcomer’s teachings with the highest respect. Their appetite for novelty, excitement, ‘the next big idea’, would have been considerable. Prestige, and perhaps some educational and economic advantage over the neighbouring colonies, was also on many minds.

  By all accounts, Pythagoras’s ideas were more than up to his students’ expectations. Mathematics, for him, was nothing less than a way of life. He ‘transformed the study of geometry into a liberal education,’ wrote Proclus, the last major Greek philosopher, ‘examining the principles of the science from the beginning and probing the theorems in an immaterial and intellectual manner’. All existing objects, Pythagoras is reputed to have taught, depended for their identity on their form rather than on their substance, and could consequently be described using numbers and ratios. The entire cosmos constituted some vast and glorious musical scale. The Pythagoreans thus became the first to understand the world not via tradition (religion), or observation (empirical data), but through imagination – the prizing of pattern over matter.

  Pythagoras had star quality, that much is clear. Appearing neither too soon nor too late at his lecture, his timing was impeccable as ever. The crowd was all ears yet the speaker took his time before addressing them. He was in no hurry. Everyone had the feeling that Pythagoras was speaking to him alone. Not a single phrase went above the listener’s head. He understood completely. ‘Yes,’ he would think to himself, ‘yes, it is just as he says. It cannot be any other way.’ But of course this flash of certainty, this grasping of an absolute truth, was an illusion. The listener’s mind had duly accompanied the speaker down one carefully laid path of reasoning, though many alternate paths existed. The listener simply forgot about them, those other ways of thinking and seeing the world. The audience was swept, logical step by logical step, from their old certainties to new and unexpected ones. This was the power of Pythagoras.

  Rhetoric, the art of speaking, gave shape and solidity to Pythagoras’s words and ideas. It would also mark the beginning of truly mathematical thought. ‘A (mathematical) proof,’ says Steven G. Krantz, a mathematician based at Washington University in St Louis, ‘is a rhetorical device for convincing someone else that a mathematical statement is true or valid.’ Philip J. Davis and Reuben Hersh, two other leading academics in the States, support this view. ‘Mathematics in real life is a form of social interaction where “proof” is a complex of the formal and the informal, of calculations and casual comments, of convincing argument and appeals to the imagination and the intuition.’

  Ancient Greece, with its passionate debates, litigious citizens and rowdy assemblies, proved an ideal environment for just this kind of social interaction, and consequently, for the development of both rhetoric and mathematics. In fact, without the refinement of rhetoric there would have been no logic, and thence none of the mathematics that forms one of the cornerstones of our empirically minded Western civilisation. Before any of these cultural and intellectual endeavours there was the practice of persuasion, through argument and the evaluation of evidence. And it was in the law courts with their public trials, that these building blocks of our system of thought were honed.

  Court cases were a daily occurrence in ancient Athens. Hundreds, sometimes thousands, of free citizens filled theatres to hear the parties speak. These citizens composed a vast and anonymous jury: only their age (above thirty years old) and maleness united them. Since the jury was always made up of an odd number of people, no tied judgements were possible. Every decision was final, and without appeal.

  For the space of several hours, the defendant and his accuser took centre stage. The accuser would speak first, leaving the defendant to repel his arguments. Each wanted to be a Pythagoras. Each tried to dazzle the men of the jury with the order, rhythm and precision of his words. For the less gifted or confident, eloquence could be bought at a price; professional speechwriters were very much in demand. The art of public speaking grew prestigious; everyone wanted to learn. A single tightly argued presentation, every Greek knew, could make the difference between liberty and imprisonment, between life and death.

  Let us picture a trial. The defendant, it is claimed, killed the accuser’s son for his one hundred gold c
oins. What argument does the grieving father bring to the court? Perhaps he points to a similar case, known to everyone present, of a man who killed another for his ten gold coins. If a man would be prepared to risk his skin for ten coins, says the father, then he would surely risk it for one hundred coins. The father’s argument, by which he establishes motive, is an exercise in basic mathematical logic: if x is true, then x² is also true.

  A real example of Ancient Greek court rhetoric has come down to us from the orator Antiphon’s First Tetralogy. It involves the case of a man accused of killing his victim (along with the victim’s slave) in cold blood. Anticipating a likely defence of ‘someone else did it’, the plaintiff methodically goes through the possible scenarios, eliminating each in turn, in a style of argument that resembles what mathematicians call ‘proof by exhaustion’.

  Malefactors (thieves) are not likely to have murdered him, as nobody who was exposing his life to a very grave risk would forgo the prize when it was securely within his grasp; and the victims were found still wearing their cloaks. Nor again did anyone in liquor kill him; the murderer’s identity would be known to his boon companions. Nor again was his death the result of a quarrel; they would not have been quarrelling at the dead of night or in a deserted spot. Nor did the criminal strike the dead man when intending to strike someone else; he would not in that case have killed master and slave together. As all grounds for suspecting that the crime was unpremeditated are removed, it is clear from the circumstances of the death themselves that the victim was deliberately murdered.

  The plaintiff renders the potential defence precisely as: ‘death either by (1) thieves (2) drunks (3) quarrel (4) accident’ in order to refute each part. But he does something more. Each scenario represents a miniature proof. Rejecting, for instance, the possibility that a thief committed the murder, his argument proceeds:

  A thief (claims the defendant) killed the victim.

  But thieves steal the victim’s cloak.

  Thus a thief did not kill the victim.

  Which self same structure we can find throughout the

  Elements of Euclid.

  CA and CB are each equal to AB.

  But things equal to the same thing are also equal to one another.

  Thus CA is also equal to CB.

  Written in the third century before the Christian era, it is hard to overstate the Elements’ importance to the history of intellectual progress, or the extent to which it embodied the flourishing of rhetoric and logic that permitted its creation. Proposition 21, in Book IX of the treatise (among its oldest pages, dating back to the Pythagoreans) illustrates the lawyerly manner that its author would adopt throughout.

  If as many even numbers as we please be added together, the whole is even . . . For, since each of the numbers . . . is even, it has a half part; so that the whole [number] also has a half part. But an even number is that which is divisible into two equal parts; therefore [the whole number] is even.

  We might summarise this argument as:

  Proposition: Even numbers (of any quantity) added together make an even number.

  Clarification: Since even numbers have half parts, their sum will also have a half part.

  Axiom: An even number is that which is divisible into two equal parts.

  Conclusion: Therefore the sum of any quantity of even numbers is even.

  Which echoes any number of the possible arguments

  that would have passed through the Ancient Greek

  courts.

  Proposition: The accused stole my ox.

  Clarification: Since he said nothing to me about the ox before taking it, the ox was taken without my permission.

  Axiom: An item of property removed without the owner’s consent is stealing.

  Conclusion: Therefore my ox was stolen.

  With their axioms – statements that we accept as being self-evidently true – the Greek plaintiff was able to methodically construct his case, the Greek mathematician, his theorem. Nowhere else on Earth had men thought to agree on what constituted the essence of such and such a thing. Uniquely, the Greeks tore themselves away from the word of rulers, gods or tradition, in favour of logical reasoning. What is wrongdoing? What is murder? What is theft? The Ancient Greeks were the first to ask themselves these kinds of questions. They were the first to distinguish a ‘criminal act’ from ‘misfortune’ or ‘error of judgment’. Definitions – concise, basic and unambiguous – entered the Athenian imagination, as Aristotle tells us.

  It often happens that a man will admit an act, but will not admit the accuser’s label for the act . . . He will admit that he took a thing but not that he ‘stole’ it; that he struck someone first, but not that he committed ‘outrage’; that he had intercourse with a woman, but not that he committed ‘adultery’; that he is guilty of ‘theft’ but not of ‘sacrilege’, the object stolen not being consecrated; that he has encroached, but not that he has ‘encroached on State lands’; that he has been in communication with the enemy, but not that he has been guilty of ‘treason’ . . . Therefore we must be able to distinguish what is theft, outrage, or adultery, from what is not, if we are to be able to make the justice of our case clear ...

  Similarly, Euclid defines a ‘point’, a ‘line’, a ‘square’, a ‘unit,’ and a ‘number’ (among other things), as if answering questions that no one had thought to ask before. What is a point? What is a line? To a scribe in Alexandria, or a logician in Xianyang, these sentences were only puzzles. They had no meaning. Or else for response they would simply make a point or draw a line with a drop of ink.

  Euclid’s books not only posed these questions, they laid down the law (so far as the answers were concerned) for future generations of mathematicians. What is a point? That which has no part. A line? A length without breadth. A square? A quadrilateral, which is both equilateral and right-angled. A unit? That of which each of the things that exist is called one. A number? A multitude composed of units. Et cetera. These foundations allowed the mathematicians to ‘be able to make the justice of [their] case clear.’

  Not to mention the case of others. In the middle of the nineteenth century, more than two millennia after Euclid, a copy of his Elements travelled in the carpetbag of a circuit lawyer from Illinois. His name was Abraham Lincoln.

  The pages and their propositions made a deep impression on Lincoln’s mind. They followed him into his subsequent career in politics. In a speech given to an Ohio crowd in 1859 and in opposition to a pro-slavery rival, one Judge Douglas, Lincoln declared, ‘There are two ways of establishing a proposition. One is by trying to demonstrate it upon reason, and the other is, to show that great men in former times have thought so and so, and thus to pass it by the weight of pure authority. Now, if Judge Douglas will demonstrate somehow that this is popular sovereignty, —the right of one man to make a slave of another, without any right in that other, or anyone else to object; demonstrate it as Euclid demonstrated propositions, —there is no objection. But when he comes forward, seeking to carry a principle by bringing it to the authority of men who themselves utterly repudiate that principle, I ask that he shall not be permitted to do it.’

  Definitions and axioms would shape President Lincoln’s most famous addresses. His powers of rhetoric, persuasion, deduction and logic were all subjected to the severest test. The nation was in crisis. Civil war would shortly come. The President spoke to the entire country in defence of the Union.

  I hold, that in contemplation of universal law, and of the Constitution, the Union of these States is perpetual. Perpetuity is implied, if not expressed, in the fundamental law of all national governments. It is safe to assert that no government proper ever had a provision in its organic law for its own termination. Continue to execute all the express provisions of our national Constitution and the Union will endure forever, it being impossible to destroy it except by some action not provided for in the instrument itself.

  Proposition: The Union of these States is perpetual.

  Cl
arification: Perpetuity is implied in the fundamental law of all national governments.

  Axiom: No government ever had a legal provision for its own termination.

  Conclusion: Therefore continue to execute the Constitution and the Union will endure forever.

  Throughout Lincoln’s four years in office, intense fighting saw approximately 750,000 men killed, and the nation all but tear itself apart, but the president’s proof would ultimately be vindicated.

  ‘We are not enemies,’ the President had said in the same national address, ‘but friends.’ Perhaps he was thinking of a proverb attributed to Pythagoras, one that he took as an axiom: ‘Friendship is equality.’

  On Big Numbers

  In the second of his Olympian Odes, the ancient lyric poet Pindar wrote, ‘the sand escapes numbering’. He was expressing the same idea that would lead his fellow Greeks to coin the term ‘sand hundred’ for an inconceivably great quantity.