What We Cannot Know. Marcus du Sautoy
didn’t have any clue how to tackle such a dynamic problem. Their mathematics was a static, rigid world of geometry, not one that could cope with the dice tumbling across the floor. They could produce formulas to describe the geometric contours of the cube, but once the dice started moving they were lost.
What about doing experiments to get a feel for the outcomes? The anti-empiricist attitude of the ancient Greeks meant they had no motivation to analyse the data and try to make a science of predicting how the dice would land. After all, the way the dice had just landed was going to have no bearing on the next throw. It was random and for the ancient Greeks that meant it was unknowable.
Aristotle believed that events in the world could essentially be classified into three categories: ‘certain events’ that happen by necessity following the laws of nature; ‘probable events’ that happen in most cases but can have a few exceptions; and finally ‘unknowable events’ that happened by pure chance. Aristotle put my dice firmly in the last category.
As Christian theology made its impact on philosophy, matters worsened. Since the throw of the dice was in the hands of God, it was not something that humans could aspire to know. As St Augustine had it: ‘We say that those causes that are said to be by chance are not non-existent but are hidden, and we attribute them to the will of the true God.’
There was no chance. No free will. The unknowable was known by God, who determined the outcome of the dice. Any attempt to try to predict the roll was the work of a heretic, someone who dared to think they could know the mind of God. King Louis XI of France even went as far as prohibiting the manufacture of dice, believing that games of chance were ungodly. But the dice like the one I have on my desk eventually began to yield their secrets. It took till the sixteenth century before dice were wrestled out of the hands of God and their fate put in the hands, and minds, of humans.
FINDING THE NUMBERS IN THE DICE
I’ve put another two dice next to my beautiful Las Vegas dice. So here’s a question: if I throw all three dice, is it better to bet on a score of 9 or a score of 10 coming up? Prior to the sixteenth century there were no tools available to answer such a question. And yet anyone who had played for long enough would know that if I was throwing only two dice then it would be wise to bet on 9 rather than 10. After all, experience would tell you before too long that on average you get 9 a third more often than you get 10. But with three dice it is harder to get a feel for which way to bet, because 9 and 10 seem to occur equally often. But is that really true?
It was in Italy at the beginning of the sixteenth century that an inveterate gambler by the name of Girolamo Cardano first realized that there are patterns that can be exploited in the throw of a dice. They weren’t patterns that could be used on an individual throw. Rather, they emerged over the long run, patterns that a gambler like Cardano, who spent many hours throwing dice, could use to his advantage. So addicted was he to the pursuit of predicting the unknowable that on one occasion he even sold his wife’s possessions to raise the funds for the table stakes.
Cardano had the clever idea of counting how many different futures the dice could have. If I throw two dice, there are 36 different futures. They are depicted in the following diagram.
Only in three of them is the total 10, while four give you a score of 9. So Cardano reasoned that, in the case of two dice being thrown, it makes sense to bet on 9 rather than 10. It did not help in any individual game, but in the long run it meant that Cardano, if he stuck to his maths, would come out on top. Unfortunately, while a disciplined mathematician, he wasn’t very disciplined when it came to his gambling. He managed to lose all the inheritance from his father and would get into knife fights with his opponents when the dice went against him.
He was nevertheless determined to get one prophecy correct. He had apparently predicted the date of his death: 21 September 1576. To make sure he got this bet right he took matters into his own hands. He committed suicide when the date finally struck. As much as I crave knowledge, I think this is going a little far. Indeed, the idea of knowing the date of your death is something that most would prefer to opt out of. But Cardano was determined to win, even when he was dicing with Death.
Before taking his life, he wrote what many regard as the first book that made inroads into predicting the behaviour of the dice as it rolls across the table. Although written around 1564, Liber de Ludo Aleae didn’t see the light of day until it was eventually published in 1663.
It was in fact the great Italian physicist Galileo Galilei who applied the same analysis that Cardano had described to decide whether to bet on a score of 9 or 10 when three dice are thrown. He reasoned that there are 6 × 6 × 6 = 216 different futures the dice could take. Of these, 25 gave you a 9 while 27 gave you a 10. Not a big difference, and one that would be difficult to pick up from empirical data, but large enough that betting on 10 should give you an edge in the long run.
AN INTERRUPTED GAME
The mathematical mastery of the dice shifted from Italy to France in the mid-seventeenth century when two big hitters, Blaise Pascal and Pierre de Fermat, started applying their minds to predicting the future of these tumbling cubes. Pascal had become interested in trying to understand the roll of the dice after meeting one of the great gamblers of the day, Chevalier de Méré. De Méré had challenged Pascal with a number of interesting scenarios. One was the problem Galileo had cracked. But the others included whether it was advisable to bet that at least one 6 will appear if a dice is thrown four times, and also the now famous problem of ‘points’.
Pascal entered into a lively correspondence with the great mathematician and lawyer Pierre de Fermat in which they tried to sort out the problems set by de Méré. With the throw of four dice, one could consider the 6 × 6 × 6 × 6 = 1296 different outcomes and count how many include a 6, but that becomes pretty cumbersome.
Instead, Pascal reasoned that there is a 5⁄6 chance that you won’t see a 6 with one throw. Since each throw is independent, that means there is a 5⁄6 × 5⁄6 × 5⁄6 × 5⁄6 = 625⁄1296 = 48.2% chance that you won’t get a 6 in four throws. Which means there is a 51.8% chance that you will see a 6. Just above an evens chance, so worth betting on.
The problem of ‘points’ was even more challenging. Suppose two players – let’s call them Fermat and Pascal – are rolling a dice. Fermat scores a point if the dice lands on 4 or higher; Pascal scores a point otherwise. Each, therefore, has a 50:50 chance of winning a point on any roll of the dice. They’ve wagered £64, which will go to the first to score 3 points. The game is interrupted, however, and can’t be continued, when Fermat is on 2 points and Pascal is on 1 point. How should they divide the £64?
Traditional attempts to solve the problem focused on what had happened in the past. Maybe, having won twice as many rounds as Pascal, Fermat should get twice the winnings. This makes no sense if, say, Fermat had won only one round before the game was interrupted. Pascal would get nothing but still has a chance of winning. Niccolò Fontana Tartaglia, a contemporary of Cardano, believed after much thought that it had no solution: ‘The resolution of the question is judicial rather than mathematical, so that in whatever way the division is made there will be cause for litigation.’
Others weren’t so defeated. Attention turned not to the past, but to what could happen in the future. In contrast to the other problems, they are not trying to predict the roll of the dice but instead need to imagine all the different future scenarios and to divide the spoils according to which version of the future favours which player.
It is easy to get fooled here. There seem to be three scenarios. Fermat wins the next round and pockets £64. Pascal wins the next round, resulting in a final round which either Pascal wins or Fermat wins. Fermat wins in two out of these three scenarios so perhaps he should get two-thirds of the winnings. It was the trap that de Méré fell into. Pascal