Chapter 3

When it comes to the rules of chance, Jerome Cardano hasn’t received the credit he deserves. The best-known version of the theory of probability’s genesis begins almost a century after Jerome’s death, with the confusion of a seventeenth-century gambler, the Chevalier de Méré.

Known to his friends as Antoine Gombaud, the Chevalier was a successful accumulator of other people’s money. Back then, he made a good living by betting with less insightful men that he could throw a six in just four separate die rolls. He gave them even odds and pocketed their cash night after night.

Then he became ambitious. Gombaud’s first problem arose when he extended his hustle to throwing a double six in twenty-four dice throws. He reasoned that it should work and tried it out. To his surprise, it was an utter failure. Gombaud took his disappointment, with furrowed brow, to an amateur mathematician called Pierre de Carvaci. He couldn’t understand why it failed, either. De Carvaci then called on Blaise Pascal, the physicist and mathematician, for help. Also confounded, Pascal in turn decided to pass the problem to a Toulouse-based lawyer, Pierre de Fermat, of Last Theorem fame. Pascal and Fermat had a few back and forths on the subject in 1654, and eventually teased out the fact that it would take twenty-five throws to achieve a double six.

That this set of exchanges is widely considered the first effort into understanding probabilities would have horrified Jerome, with his ever-present desire for lasting fame. After all, he had first addressed the problem in a set of notes written more than a hundred years earlier, while still a twenty-year-old medical student in Pavia.

By 1520, according to Giovanni Targio, a tutor at the University of Pavia, Jerome is spending his evenings ‘drinking and gambling in the taverns’. Targio writes of his student that he ‘gave offence too often for his life to be free of enemies’. And the truth is that the young Jerome would often roam the streets of Pavia at night in a crude disguise, sword drawn — in flagrant contravention of the laws concerning naked weapons on public streets — ‘determined to live some fantasy life’, as his long-suffering tutor put it. That said, Jerome was no fool, and adroitly applied his mathematical mind to working out winning strategies for the gambling table.

Gambling is, at heart, about the random appearance of a string of numbers. In Jerome’s time, however, no one believed in randomness. If numbers appeared to be random, that was only due to a lack of information. The prevailing view was that God controlled the die and the result of a roll (or the appearance of a card — cards being merely another way of creating random numbers) was the province of the Almighty. If you were out of luck, you must somehow have offended the Divine.

‘So God knows everything. But what determines lotteries? — is it from Him?’ Jerome asks in On Subtlety. His answer is somewhat confused. ‘Not at all, but from some inspiration. However, our law states that lotteries are introduced and controlled by God.’ Hard to see it now, perhaps, but Jerome’s Book on Games of Chance, published posthumously in 1663, is an audacious first attempt to — as the physicist Stephen Hawking put it four and a half centuries later — know the mind of God.

There are thirty-two chapters. In Chapter 14, the world receives the first attempt at a law of probability to be applied when the die is cast:

So there is one general rule, namely, that we should consider the whole circuit, and the number of those casts which represents in how many ways the favourable result can occur, and compare that number to the rest of the circuit, and according to that proportion should the mutual wagers be laid so that one may contend on equal terms.

In modern terms, Jerome means consider all the possibilities (the ‘whole circuit’), then think about the number of ways in which you might get the result you want. Then find the ratio of those two numbers (the ‘proportion’). That tells you how the bets should be placed.

More than a hundred years were to pass before Gottfried Leibniz came up with the same formula in 1676:

If a situation can lead to different advantageous results ruling out each other, the estimation of the expectation will be the sum of the possible advantages for the set of all these results, divided into the total number of results.

And it would be nearly another century before, in 1774, Pierre-Simon Laplace delivers what is today generally assumed to be the start of probability theory:

The probability of an event is the ratio of the number of cases favorable to it, to the number of possible cases, when there is nothing to make us believe that one case should occur rather than any other, so that these cases are, for us, equally possible.

Bravo Jerome! Yet back in 1520 he did not stop at a simple definition of the probability of a single event. He also worked out the probabilities associated with repeated throws of the die. That isn’t easy. You might struggle, for instance to work out the probabilities — and thus the right way to bet (and accept bets) — with two throws of a die.

If you can only win with a one or a two, what are the odds of your two throws both giving you a win? Here is Jerome’s solution, where a one is an ‘ace’ and a two is a ‘deuce’:

Thus, in the case of one die, let the ace and the deuce be favourable to us; we shall multiply 6, the number of faces, into itself: the result is 36; and two multiplied into itself will be 4; therefore the odds are 4 to 32, or, when inverted, 8 to 1.

In more modern terms, we start with the first roll and divide the number of winning outcomes by the total number of possible outcomes, that is, two divided by six, or one third. The same is true for the second roll, and so we multiply the two results together: one third times one third is one ninth. That is, there is one favourable outcome in nine throws. Put another way — Jerome’s way — you lose eight times for every one time that you win. In other words, your opponent’s odds are eight to one.

And then three throws:

If three throws are necessary, we shall multiply 3 times; thus, 6 multiplied into itself and then again into itself gives 216; and 2 multiplied into itself and again into 2, gives 8; take away 8 from 216: the result will be 208; and so the odds are 208 to 8, or 26 to 1. And if four throws are necessary, the numbers will be found by the same reasoning …

From this, Jerome deduced a general rule: Cardano’s Formula, which is a means of working out and understanding probabilities.

Nor did Jerome stop there. Instead, he went on to deduce what we now call the ‘law of large numbers’, which shows that large numbers of repetitions of a probabilistic process produce a predictable overcome. Jerome has, to all intents and purposes, pioneered statistics.

Imagine flipping a coin four times. You know you should expect two heads and two tails, but you wouldn’t be surprised if you got three heads and one tail, or vice versa, or even four heads or four tails. We know that, in processes that depend on chance, flukes happen.

Now imagine flipping that coin a thousand times. If you ended up with one thousand heads, you would assume the coin was weighted. Flukes are reasonable only with small numbers. Even six hundred heads out of a thousand tosses would be suspicious because, with such a large number of coin tosses, we expect something close to a fifty-fifty ratio.

Jerome worked out that one thousand tosses of a fair coin, where the odds of a head is one half, should give five hundred heads. In other words, you take the number of repetitions and multiply it by the probability of a particular outcome. That gives you the rough number of times that that outcome should occur. If it doesn’t, someone is probably cheating.

Sadly for Jerome, this law of large numbers has been attributed to a mathematician, Jacob Bernoulli, who only worked it out — calling it his ‘golden theorem’ — 150 years later. But at least Jerome reaped its benefits, for it is his hard-won understanding of how the odds play out that tells Jerome, during a visit to Venice a few years after his graduation, that something is amiss at the gambling table of Senator Thomas Lezun.

It is 8 September 1526, the day of the Blessed Virgin’s birthday. Those of a more devout disposition have spent the day in Venice’s fine churches, reciting the appropriate prayer:

Let it be thy glory, O Virgin who destroyest all heresies, to restore unity and peace once more to all the Christian people.

Jerome, though, is not at prayer. Virgins are not really his thing — especially virgin mothers. In a quarter of a century, in On Subtlety, he will even dare to write of the ‘notorious faked feat, that children are born to women without sexual intercourse’. It may not have been his wisest move, given his eventual arrest by the Inquisition.

Today he is in the Senator’s house, trying to win back the property he had previously lost to his host. He is also hoping to win a night with a beautiful prostitute, a stake that had been on the table the night before. As a twenty-five-year-old man suffering from debilitating impotence, that had been a particularly alluring bet. It may even have distracted him from watching the Senator’s play more carefully. Yesterday was a bad day. Jerome lost his clothes and his rings, as well as his chance of curing his problem through the attentions of a beautiful woman. But things are different today. On the Virgin’s birthday, he has regained his footing. He knows now why he has done so badly. After so many hands of cards, it has become clear that the law of large numbers is not setting the scores straight. After all, this law says that all the cards should come up equally often and that no outcomes should be particularly guessable. Barring a few minor anomalies, everything should therefore follow the laws of probability that Jerome worked out a few years ago. But the statistics are skewed. The best explanation is always the most obvious: the Senator has been cheating. Jerome’s brain is working well today. He has worked out that the cards are marked, and how, and in a few well-judged hands he wins back his clothes and jewellery. He sends them home with his servant because he knows he is about to make a hurried exit and doesn’t want to be encumbered. He carries on with the game and wins enough money that he needs to bag it up. He puts all but one bag into the pockets of his cloak.

His heart is racing, but he is determined to make his accusation. He takes a moment to weigh his options. The front door is locked and the Senator has two servants in the room. The house is festooned with weapons, not all of them decorative. He glances up. Two lances hang from the beamed ceiling within easy reach of the servants. This is a brave — maybe foolhardy — thing to do, but the adrenaline is pumping and rationality is receding from his brain. So he draws his dagger.

Foolish and extreme as it may seem, Jerome has done his calculations and is confident enough to strike. His blade slashes across the Senator’s face. He throws a bag of money across the table to make amends for his violence. ‘Your master’s cards are marked,’ he announces to the servants. ‘Unlock the door and let me go, or I will kill you.’ They look to their master, who is weighing the bag of money in his hand. Jerome waits for the Senator’s response, his heart still beating wildly. He has gambled well. The Senator puts down the money bag, considers his own reputation, and, with a hand wiping the blood from his cheek, orders the door to be unlocked. Jerome is free to go.

Outside, Venice is in darkness. Jerome begins to doubt. What sentence, he wonders, might he receive for an attack upon a respected public figure? It would be wise, he decides, to lay low until he can be sure the Senator doesn’t have revenge in mind.

He dons a cloak, and — apparently — some leather armour. He keeps his weapon to hand. He wanders through Venice, keeping to the shadows. If the Senator has reported him to the magistrate, the peace keepers will be out looking for him.

And then, after a few hours of careful shadow hopping, it happens. With his nervous glances to left and right, his peering through the dark, Jerome is not paying enough attention to the ground beneath his feet. Slipping on a wet board, he fails to regain his balance and falls into the freezing, stinking waters of a Venetian canal. His clothing makes it hard — too hard — to swim. He begins to think he might drown.

Then, above his splashing, he hears the sound of an oar slapping at water. Out of the darkness looms a boat. Jerome makes a grab for one of the oars. He catches it, and the attention of the man wielding it. Shouts go up, and the crew begin to pull him, slippery, dripping, and heavy as he is, out of the water. Eventually, after much effort, he is lying facedown on the deck. He turns himself over, looks up, and finds the owner of the boat staring down at him. He recognises the man immediately, despite the bandages that cover half his face. Of all possible rescuers, it is Senator Thomas Lezun who looks down upon him. Jerome’s eyes widen and his mind begins a new calculation of probabilities. Will Lezun, afforded this opportunity by fickle fate, take his revenge?

To Jerome, everything happens for a reason. If the dice land a certain way, it is because the Prince of Fortune decrees it. If he is rescued from a canal by a man whom he earlier insulted, it is God’s way of making a point.

To me, it is just coincidence. Coincidence makes fools of us all — we cannot help but read it as significant, but we are easily deluded. How many times have you heard someone declare that ‘everything happens for a reason’? It doesn’t. Not if quantum theory is to be believed.

One of the most famous quotes in all of modern science is Einstein’s comment that ‘I cannot believe that God plays dice with the universe.’ This was his response to the contention that some of the material world’s occurrences are not preceded by a cause. A single atom of a radioactive material, for example, will emit a particle at random. There is no way to predict when this will happen and there is no known trigger. You can observe lots of atoms emitting particles and use the law of large numbers to give you an average time for emission to happen. But that tells you nothing about what makes it happen in any one atom.

Here it is worth stressing that this is only true, as far as we know, on very small scales. If I use a billiard cue to hit a ball into a pocket and know all the angles and forces involved, I can use the laws of physics developed by Newton to predict the paths of both cue and ball. Nonetheless, if I take one atom from one of those balls and fire it at two suitably sized and suitably separated openings — rather like two billiard-table pockets placed next to one another — there is no way to tell into which pocket it will drop. This is another manifestation of the double slit experiment that haunts our imagination. After a few dozen perfect repetitions I will get the sense of the most probable outcome, but each individual atom seems to make up its own mind. In each event, the effect has no cause.

The same is true if I fire a particle of light — a photon — at a mirror. There is a small probability it will pass straight through, and a larger probability that it will be reflected. If I fire a million photons at the mirror, perhaps only three will go through unreflected. But there is nothing special about those three. It is simply that random chance has dictated that they are not reflected. It is another cause-free event, nothing more significant than an outcome of the laws of probability. This is just like you winning the lottery when you have bought no more tickets than anyone else. God, it turns out, does play dice.

There is an obvious question to confront here. Why are the smaller particles — atoms and photons, for example — subject to purely random outcomes and events, whereas billiard balls are not? Nobody knows. Something happens to make the events of our world, the ‘macro’ world, deterministic, predictable, not random. All we can say is that we do not follow the same rules as the ‘micro’ world of atomic and subatomic particles, where probability theory, the theory birthed by Jerome, is the only way to predict the future. There is an unsatisfactory dark space in our knowledge.

‘So what did Lezun do?’ I ask.

‘He handed me a set of his own clothes,’ Jerome says. ‘I can still see his smirking face now.’

‘And were you wearing armour?’

‘Where did you read that I was?’

‘In your autobiography, De Vita Propria Liber: The Book of My Life.’

He smirks, calculating quickly that he has the bet sewn up. ‘Interesting,’ he says. ‘I haven’t written it yet.’

Thanks to my sudden confusion, I forget to ask about the prostitute.

Scientifically problematic, isn’t it, that I am conveying information from Jerome’s future? This is one of the strongest arguments against time travel; if you can influence the past, chronology, cause and effect, and common sense can be broken down far too easily. But, as we have already discussed, quantum theory — and its experiments — have shown time and again that cause-and-effect are not fundamental attributes of the universe. In order to understand why that is, we have to go back to basics, to the origins of quantum theory. And that story begins before history.

Ever since our species’ birth, humans have looked for correlations between phenomena in the heavens and on Earth as a means to progress and success. Correlations between planting seeds at a new moon and reaping a bumper harvest, for example. Or perhaps we linked the appearance of a comet and an unexpected defeat in battle, or a particular arrangement of the planets with the birth of a king.

To make sense of these links, we began to note them down on stone and clay tablets. Later, we used paper. Once we had permanent records, we began systematically analysing that data, mining them for patterns that would help us to predict future events. A small subset of people even began to try to make sense of how the universe itself works. That subset of people — we now call them scientists — eventually showed that the hypothesis that the motions of the stars and planets cause effects on Earth could not be supported by the data. So while the predictors of the future carried on with their practice — unsupported by evidence, but eagerly watched by the general public — the scientists intent on understanding how the universe works abandoned the skies and began to pull the world apart. This is what led us, eventually, to the quantum.

Imagine taking a clock apart piece by piece. You will see its constituent parts — the cogs and wheels — and gain a sense of what they do. But now imagine wanting to know how the cogs and wheels obtain their properties. It won’t be long before you are investigating the properties of iron. Grind away at some iron and you’ll eventually get down to the very fundamental building block of that element: an atom of iron.

The atom has always been controversial. A Greek scientist, Democritus, contended a few hundred years before the birth of Christ that the atom was as far as matter could be split while retaining its essential character. Atoms of salt, he said, are sharp; atoms of water are smooth; atoms of iron are solid and strong. For millennia, though, the existence of the atom was just speculation. Only in the twentieth century did scientists — led, as it happens, by Einstein — reach consensus that they do indeed exist.

Atoms, we now know, are the bricks of our physical environment. They vary in size and have various different properties. There are atoms in the air you breathe. They flow in rivers and make up the oceans. Your body is built from them. However, they are not indivisible. Atoms, too, can be broken down, into the particles that we call electrons, protons, and neutrons. And all of these particles — isolated atoms included — can behave strangely, in ways that you and I cannot.

Somewhat surprisingly, we have the alchemists to thank for the eventual emergence of the quantum rules that govern atoms and subatomic particles. It was they who became obsessed by light being the ‘great primary cause’, as the nineteenth-century scientist Robert Hunt puts it in his 1854 book, Researches On Light.

Hunt traces the science back to Benvenuto Cellini, one of Jerome’s contemporaries. Cellini was a celebrated goldsmith and jeweller — much of his work commissioned by popes and nobles — an ‘eccentric and extraordinary genius’, Hunt claims. His contemporaries might have been less kind. Cellini was undoubtedly a gifted craftsman, but ‘eccentric’ would have to cover various charges of sodomy (brought against him by men and women whom he had used in what was known as ‘the Italian fashion’), and murder, and several hefty fines and prison sentences for his misdemeanours.

Cellini’s Treatise on Jewellery, published two years before Jerome’s arrest, contains a passage discussing observations of a ‘carbuncle glowing like a coal with its own light’. When held up to the lamp, and then put in a darkened room, this stone would light the room.

The publication caused quite a stir among the alchemists and they soon got to work seeking out, and creating for themselves, more and more materials that would ‘phosphoresce’, emitting their own light. A Bolognese shoemaker called Vincenzo Cascariolo made the biggest breakthrough: his alchemical experiments culminated in the synthesis of the first artificial phosphorescent material. Cascariolo managed to roast sulphur-rich barium sulphate for long enough to create a golden glowing mineral.

Eventually (and much more slowly than most like to think) the alchemists themselves transmuted into scientists, who began to investigate the properties of materials for their own sake, rather than in the pursuit of riches or the Elixir of Life. As Hunt put it, ‘the hypothesis of the Alchymist has been converted into a probable theory by the discoveries of the modern chemist.’

Now the names become a little more familiar. Robert Boyle and David Brewster, for example, entered the fray, as did Newton, an alchemist through and through who wondered aloud whether solid bodies and light are interconvertible, and whether light is the source of the ‘activity’ of solid bodies. Eventually, it was realised that this light — all light — is a manifestation of energy. We know this energy as radiation and have more modern scientists such as Marie Curie to thank for these discoveries and their applications.

Radiation is not just light, though. There also exists invisible radiation, as Henri Becquerel discovered through his investigations of uranium salts. As the nineteenth century turned into the twentieth, we discovered a range of energetic particles that come from the component parts of atoms. Experiments showed that the atoms within these stones and metals break apart spontaneously, forming into different kinds of atoms and emitting energetic radiation. And as a result of examining this radiation very closely and precisely, we learned something very odd.

Have you ever seen an iron horseshoe in a blacksmith’s fire? If so, you will know that, left in the heat, it glows red, then orange, then eventually white. This radiation comes from the individual atoms, with each one giving out a tiny bit of light that adds to the glow. The white glow is a composition of lots of different colours, just as you can obtain different colours from mixing different coloured dyes. Cellini, as a jeweller, would have seen the rainbow of colours projected onto a nearby wall when white light hits a diamond (a phenomenon, as it happens, that Jerome once discussed with King Edward VI of England). The process in the blacksmith’s forge is, essentially, the reverse: all the colours united together as the heat takes hold.

As the twentieth century dawned, observations through spectrometers showed exactly how much of each colour of light an object like a white-hot horseshoe would give out. Scientists set themselves to explaining this distribution of energy in terms of the atoms’ behaviour. It proved remarkably difficult. In the end, they only managed to solve it using the same technique that brings success in dice games: probability.

Imagine you need to roll a twelve, with two dice, and your opponent needs to roll an eight. You know there is only one roll that allows you to win: two sixes. But your opponent can win with any of five rolls: two fours, or a two and a six, or a three and a five on either of the dice. He is therefore five times as likely to win as you.

Max Planck, a German physicist working at the Friedrich-Wilhelms-Universität in Berlin, did the same calculation with atoms. If you have ten atoms together, as if they were ten dice, certain colours of radiation are more common. That, Planck surmised, is because more combinations of those atoms’ activity lead to those particular colours. Some colours are almost never emitted, much as a sixty is almost never achieved by throwing ten dice.

To match the observations of the colours emitted by hot objects, Planck divided up the energy of atoms into discrete amounts as if each ‘quantum’ of energy was a dot on a die. So there is nothing between 1 and 2 lumps of energy, just as there is nothing between 1 and 2 on a die. He then laid out the probability of the various possible quantum changes in energy. He didn’t accept straight away that that was how atoms really were. He just wanted to see if it worked.

To Planck’s surprise (and distaste, it has to be said) it did, and he discovered a peculiar relationship. According to the wave theory of light, we can quantify its colour in terms of ‘frequency’ (f). Imagine it coming towards you as a sequence of up and down oscillations: the frequency is the number of peaks in the wave that reach your eyes every second. Blue light has more peaks per second than red light; green is somewhere in the middle. Blue is therefore a higher frequency of visible light. Violet — and ultraviolet — is higher still.

Planck divided the amount of energy (E) in a ‘quantum’ packet of light (adopting a term already in use for other small packets of material) by its frequency (f). When he did so, he would always obtain the same number. We now know that number as Planck’s constant. It is usually written as h, and E equals h times f.

Planck’s extraordinary observation — that radiation must exist as indivisible quanta whose energy is related to the properties of its waves — was the start of ‘quantum mechanics’. This theory, which has taken over science as our ultimate explanation of how the universe works, splits every observable quantity — not just energy, but momentum, position, and so on — into indivisible, separable amounts, like the dots on a die.

Why? The answer is both simple and utterly confounding. It does so for no better reason than it fits the observations of what happens when you throw a horseshoe into the fire.