Pythagoras discovered that the most harmonious notes result from the ratios of whole numbers. A vibrating string exactly halved or doubled, for example, produces an octave (ratio 1:2 or 2:1). If precisely one-third of the string is held down, or when the string is tripled in length, a perfect fifth (an octave higher) results. A perfect fourth can be obtained by holding down one quarter of the string, or stretching it out four times longer. The whole harmonic scale was constructed in this way. Pythagoras observed that all of music depended on the first four numbers and their interrelations. Ten he worshipped as the most perfect number, reflecting the unity of all things, it being the sum of one and two and three and four.
According to Hippolytus, one of the great theologians of the Early Church, Pythagoras taught that the cosmos sang and was composed of music and ‘he was the first to put down the movement of the seven stars to rhythm and melody.’ He even attempted to reproduce this soothing universal music for his disciples, rousing them in the morning to the sound of his lyre. In the evening, too, he would play for their benefit, ‘in case too turbulent thoughts might still inhabit them.’
From Pythagoras’s lyre it is an easy hop to Einstein’s violin. ‘If I were not a physicist,’ he once told an interviewer, ‘I would probably be a musician. I live my daydreams in music. I see my life in terms of music. I get most of my joy in life out of music.’ His case accompanied him on many trips, but Einstein played with discretion and few reliable reports of his musical ability survive. It appears that his amateur technique was respectable, if somewhat limited. Though there may be an affinity between the laws of maths and music, they cannot be conflated. Even prodigious mathematical gifts such as Einstein’s did not translate into exceptional musicianship, but they certainly sharpened and reinforced his appreciation.
If maths is the secret underpinning the harmonies of cricket and music, mathematical beauty is also key to feats of magic. Ever since I was a small child, playing cards that jump queue, white handkerchiefs that take wing and top hats that turn to rabbit holes have fascinated me. The visions touch some nerve deep inside.
One evening several years ago I attended the London performance of a young conjuror. The magician was playing to a full house. I was seated in one of the middle aisles, in a sea of heads, next to a rather elderly gentleman with his stomach sat on his lap. From this distance I had a good view of the stage. A blend of high technology spotlights and music hall gloom favoured illusion.
People come to magic shows for all sorts of reasons: some for the theatrics, others for the performer’s comedy, and some (like my neighbour), it would seem, to cough. The main draw for me, though, is an experience of the unexpected. It is this that gives a conjuror’s performance its peculiar beauty, a beauty akin to a well-turned equation.
I am not talking here about the end result, the ‘effect’, of a trick. I am talking about the method. Every mind-read drawing or woman sawn in half looks pretty much alike, whereas the hidden ideas that make each possible can vary as much as their performers. Dozens if not hundreds of ways exist to levitate a spoon or make the Statue of Liberty vanish, in the same manner that numerous hundred people (not all of them professional mathematicians) have shown that the square of a right-angled triangle’s hypotenuse equals the sum of the squares of the other two sides. But few of these theoretical demonstrations, or magical methods, would be considered up to Einstein’s test of beauty. The truly beautiful are those that foster surprise.
A genuine experience of the unexpected, in maths as much as in magic, demands of its performer at once originality of insight and a lightness of touch. Even a single step too many in a method renders ugly and clumsy the theorem or the trick.
It is sometimes said that a magician will go to great pains to conceal his workings from the public. The reality is that only a poor method requires such attention; a fine one, by its beauty, conceals itself. We might call this rule the coquetry of good technique.
One part of the magic performance from that evening in London well illustrates this point. A shy-looking woman from the audience was invited up onto the stage. At its centre, on a pedestal, stood a glass bowl containing large multi-coloured buttons. The buttons, we were told, totalled one hundred. Following instructions, the woman dipped her hands into the bowl, netting as many of the buttons as she wished. Unclenching her fingers, she next poured the buttons onto a tray and covered them with a tea towel. The magician approached the tray, peeked under the towel for all of two seconds, and then turned to the public.
‘Seventy-four,’ he declaimed.
The woman duly proceeded to count the scattered buttons on the tray, one by one. It took quite some time. After a minute or so, she prodded the final button and the expression of her face lengthened. There were precisely seventy-four buttons on the tray. Gasps popped around my ears, followed immediately by a brisk weather front of applause. The ‘counting buttons’ trick was a highlight of the show.
I imagine some people applauded the magician for mental powers that flirted with the supernatural. To recognise seventy-four items (and not mistake them for seventy-three, or seventy-five) in the space of two seconds is, it must be said, quite something. Neurologists tell us that the human brain ‘subitises’ (to count at a glance) no better than in fours or fives. This figure remains constant across all categories of people, irrespective of training or synaptic quirks – neither mathematicians nor autistic savants exceed it. In two seconds, even the most practised eye can enumerate only eight or ten, and no higher.
The explanation of divination did not cross my mind; even admitting that it was possible (which I do not), it would have felt somewhat disappointing. A laborious counting of every button, even if accomplished at the most remarkable clip, would be totally lacking in finesse or beauty. I groped with my imagination for the magician’s insight.
How might a person count some (relatively) large quantity in no time at all? This question stayed with me that night as I climbed into bed and tossed and turned, finally to sleep. In my dreams the gleaming transparent globe and its outsized buttons and the shy woman holding the tray all returned to me. I looked and looked but failed to see.
Early in the morning I woke to a beautiful sensation of absolute clarity. The night had seemingly done its work. Now every moment of the magician’s trick, from beginning to end, made perfect sense. Had I unveiled the trick’s machinery? I could not say. In its simplicity and economy the solution felt inevitable, but I have no idea whether it was in fact the magician’s or merely my own. In any event my mood that morning was immediately elevated. I wanted, like Archimedes in his bath, to leap and cry Eureka. Quite possibly I did. I felt like a mathematician who receives the sudden, shocking ecstasy of a proof.
Of the school of nocturnal thoughts that swam anonymously through my brain, one image had stuck. It was a modest household object that I used every day: my kitchen scales. Now to the question ‘how might a person count some large quantity in no time at all?’ came at once the delicious answer: weigh it! What if the identical buttons weighed exactly one gram each? And what if the glass bowl’s pedestal concealed a weighing scale? When the woman lifted seventy-four buttons from the bowl, the reading (displayed backstage) would fall instantly from ‘100’ to ‘26’. Suffice then to relay this number, via an earpiece or some pre-arranged signal, to the performer. That the maths involved might be the simplest of simple subtractions only accentuates the solution’s charm.
This pure beauty that we call mathematical, and that we find in games, music and magic tricks, is something like a rumour or a longing that lingers in the person, hinting at significance and depth. We go back to it time and time over: it is beautiful, because it stays. It is we who change.
Problems, in magic or mathematics, are wonderful things. Without problems, we would have no proofs, and the shimmering pleasure of elucidation is a thing of beauty. Einstein’s equations possessed this special quality in abundance, of course. E = mc² (energy is equal to mass times the speed of li
ght squared) answered riddles – such as the behaviour of light – that most other scientists had not even seen.
I have spoken about mathematical beauty with hardly any reference to numbers, but, of course, numerical problems also provide many instances of beauty. A personal example, from arithmetic, is the multiplication 473 × 911. The solution – 430,903 – may at first sight appear merely banal. Its repeating threes and zeroes, however, reversed as in a mirror, hint at some attractive pattern lurking beneath. And so we begin to loiter around the answer. From a closer look, we might observe the relationship: 903 – 430 = 473. Viewed this way, the solution makes the problem all the more, not less, interesting to us. If now we modify the original question slightly, 473 × 910, simplifying the sum, we reach the answer: 430,430. And we ask ourselves: how is this possible? Again we return to the original problem and dissect its numbers. 473 equals 43 × 11. The number 910 comprises 7 × 13 × 10. We toy with these smaller constituents, until at last we discover that our original sum equates to (43 × 1001) + (43 × 11).
A further illustration of this numerical beauty can be found within the primes. The number 75,007 (as it happens, a rather chic Parisian postcode) poses the problem of whether any smaller number evenly divides it. In other words, is the number 75,007 prime? Deceptively simple to formulate, this question proves to be treacherously difficult to answer. As with the sum above, we must grapple with the number until its secret finally yields.
We begin by assuming (the odds are on our side) that the number 75,007 is composite – that some smaller numbers exist that will cleanly divide it. Not an even number, it cannot be divisible by 2 (the smallest prime). The number 75, we note, divides into 3 and 5 (the next primes), but not when followed by two zeroes and a seven. We might think of 75,007 as a house on a very lengthy street, and observe that, sixty-eight doors down, number 75,075 patently divides by 1,001 (and therefore, by 7 and 11 and 13). But 68 can be split only into two and then a further two to leave seventeen.
Imagine the mathematician, pencil in hand, attempting to winkle out the number’s factors. The insight resists him; he wonders whether it will ever come. He abandons his desk, pacing the floor, and he marches along the mental street and its numbered houses. And then, all of a sudden, the dizzying thought strikes him: 75,007 can be expressed as 74,900 + 107, or (10,700 x 7) + 107, or more precisely still (107 × 100 x 7) + 107, and his blood leaps with joy to recognise the repeated factor: 107. On a crisp sheet of paper, he writes: 75,007 = 107 × 701.
Human beings’ quest for meaning is perpetual; lack of meaning is offensive to the mind, and whatever the scale of the problem, a solution is a thing of beauty. Einstein’s equations solved problems such as ‘What do we mean by the words “time” and “mass’’?’ A mathematician could tell us that the number 75,007 means to travel from 0 to 107, and then repeat the same distance successively 701 times. Other meanings, like those found in music or cricket, while more intimate and inexpressible, can prove just as powerful. Where chaos is subdued and the arbitrary averted, there lies beauty, and it is all around us.
I recall an afternoon I once spent at a summerhouse belonging to friends. We had just returned, tired and hungry, from a long hike among the encircling hills. One of my friends switched on a little radio. We sat here and there around the living room, with views out to the sea, half-listening and half-talking. The broadcaster was reading out the week’s letters sent in by listeners to the programme. Among their litany of compliments and complaints, he recited a short puzzle sent in by a long-time listener in the North.
‘A remarkable species of water lily doubles in size every day. If the lily covers an entire lake in a period of 30 days, after how many days will it cover half of the lake?’
Our various discussions slowed, then continued as before. Someone switched off the radio. The face of the friend opposite me gradually darkened, and her replies became wayward and staccato. Other voices took up the slack. It seemed no one else had taken the problem of the lily to heart.
Minutes passed. My friend cast sidelong glances at the walls and the windows and the flower-lit hills they framed. Her blue eyes narrowed to a squint. Kitchen noises filled the house, followed by the chattering distribution of hot teacups, exhaling steam. The untouched drink trembled when she struck the table absent-mindedly with her shin.
And then I saw it. Suddenly, her features shone. ‘Twenty-nine,’ she said, with a broad smile. If the lily doubles every day, it will also halve every day when we track its history. With a size of 1 (for ‘one lake’) after 30 days, it had a size of 0.5 after 29 days, 0.25 after 28 days, 0.125 after 27 days, and so on.
The moment had surprised her, she said. It had come right out of the blue. I saw the moment my friend beheld the astonishing beauty of mathematics.
A Novelist’s Calculus
The history of the world, Tolstoy said, is the history of little people. Leo Nikolayevich himself, though, was a giant of a man. Just shy of six feet, he was taller than most of his contemporaries. Stronger, too. He could lift 180 pounds with a single hand. He dressed his muscles plainly, in a peasant’s smock, with a belt that cinched the small of his back. His ego-enhancing arguments were just as sturdy. Always resistant to the thinking of his day, he denounced historians as hero worshippers. In War and Peace, over one thousand pages long, he launched his most sustained attack. His primary weapon was drawn from mathematics.
Calculus was by no means a novel idea in Tolstoy’s time. Its ‘inventors’, Isaac Newton and Gottfried Leibniz, in the late-seventeenth century, were refining theories that had been in development since the time of the ancient Greeks. As geometers study shape, the student of calculus examines change: the mathematics of how an object transforms from one state into another, as when describing the motion of a ball or bullet through space, is rendered pictorial in its graphs’ curves. In these curves, smooth and subtle, girding the infinitesimal movements behind every human life, Tolstoy thought he saw the blindness of contemporary historians.
Alongside formidable intellectual powers, he certainly had a head for exotic ideas. I think of the wackier pronouncements: the dismissal of Shakespeare as a lousy poet, of Darwinism as a transient fad, of marriage as legalised fornication. Like Thomas Jefferson, he took a pair of scissors to the New Testament, trimming its pages of every one of its miracles. His cult of simplicity (as G.K. Chesterton later called it) welcomed flocks of disciples to his estate: men and women, young and old, dressed in bed sheets and bark sandals, who followed his every step and hung on his every word. But more ambitious, more inventive, and more subversive than any of these was the novelist’s view of history itself as a kind of calculus.
We find this view studded throughout the pages of War and Peace, in those portions that resemble the tight and intense arguments of a pamphleteer. It so happens that they are the same parts that most modern readers tend, perhaps understandably, to skip. But this less diligent reader misses a crucial bedrock of Tolstoy’s work.
The movement of humanity, arising as it does from innumerable arbitrary human wills, is continuous. To understand the laws of this continuous movement is the aim of history . . . only by taking infinitesimally small units for observation . . . and attaining to the art of integrating them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history.
Calculus, which Tolstoy defined as ‘a modern branch of mathematics having achieved the art of dealing with the infinitely small,’ offered him a vocabulary in which to voice his disagreement with many historians. He denounced their lamentable tendency to simplify. The experts stumble onto a battlefield, into a parliament or a public square and demand, ‘Where is he? Where is he?’ ‘Where is who?’ ‘The hero of course! The leader, the creator, the great man!’ And having found him, they promptly ignore all his peers and troops and advisors. They close their eyes and abstract their Napoleon from the mud and the smoke and the masses on either side, and marvel at how such a figure could possibly have prevailed in
so many battles and commanded the destiny of an entire continent. ‘There was an eye to see in this man,’ wrote Thomas Carlyle about Napoleon in 1840, ‘a soul to dare and do. He rose naturally to be the King. All men saw that he was such.’
But Tolstoy saw differently. ‘Kings are the slaves of history,’ he declared, ‘the unconscious swarmlike life of mankind uses every moment of a king’s life as an instrument for its purposes.’ Kings and commanders and presidents did not interest Tolstoy. History, his history, looks elsewhere: it is the study of infinitely incremental, imperceptible change from one state of being (peace) to another (war).
The decisions of exceptional men could explain all of history’s great events said the experts. For the novelist, this belief was evidence of their failure to grasp the reality of an incremental change brought about by the multitude’s infinitely small actions. Out of a need to theorise, to locate ‘causes’, the historian privileges one series of events and examines it apart from all the others. Why, all of a sudden, had Napoleonic France and Tsarist Russia rushed to war? What drove millions of men, men who licked their plates and read stories to their sons and worried about their looks, to suddenly thieve and crush and slaughter one another? Napoleon overreached, a victim of his own pride and mania, says one expert. He let himself go, growing fat and moody. With successive battle victories under his belt, he fell inevitably to thinking himself invincible. No, no, says another historian, you forget how weak and high-strung was the Tsar Alexander. Such weakness certainly invited a military strike. The longstanding economic embargoes in Europe, suggests a third, led to strained relations between the different peoples. A fourth points out that hundreds of thousands of soldiers obtained gainful employment. Napoleon himself, near the end of his life, is said to have put the war down to the intrigues of the British.
Thinking in Numbers: How Maths Illuminates Our Lives Page 13