What We Cannot Know: Explorations at the Edge of Knowledge. Marcus Sautoy du

but he is not a mathematician; this is, as you know, a great fault.’ A great fault, indeed!

      Pascal, in contrast, was great on the mathematical front and argued that the spoils should be divided differently. There is a 50:50 chance that Fermat wins in one round, in which case he gets £64. But if Pascal wins the next round then they are equally likely to win the final round, so could divide the spoils £32 each. In either case Fermat is guaranteed £32. So the other £32 should be split equally, giving Fermat £48 in total.

      Fermat, writing from his home near Toulouse, concurred with Pascal’s analysis: ‘You can now see that the truth is the same in Toulouse as in Paris.’

       PASCAL’S WAGER

      Pascal and Fermat’s analysis of the game of points could be applied to much more complex scenarios. Pascal discovered that the secret to deciding the division of the spoils is hidden inside something now known as Pascal’s triangle.

      The triangle is constructed in such a way that each number is the sum of the two numbers immediately above it. The numbers you get are key to dividing the spoils in any interrupted game of points. For example, if Fermat needs 2 points for a win while Pascal needs 4, then you consult the 2 + 4 = 6th row of the triangle and add the first four numbers together and the last two. This is the proportion in which you should divide the spoils. In this case it’s a 1 + 5 + 10 + 10 = 26 to 1 + 5 = 6 division. So Fermat gets ²⁶⁄₃₂ × 64 = £52 and Pascal gets ⁶⁄₃₂ × 64 = £12. In general, a game where Fermat needs n points to Pascal’s m points can be decided by consulting the (n + m)th row of Pascal’s triangle.

      There is evidence that the French were beaten by several millennia to the discovery that this triangle is connected to the outcome of games of chance. The Chinese were inveterate users of dice and other random methods like the I Ching to try to predict the future. The text of the I Ching dates back some 3000 years and contains precisely the same table that Pascal produced to analyse the outcomes of tossing coins to determine the random selection of a hexagram that would then be analysed for its meaning. But today the triangle is attributed to Pascal rather than the Chinese.

      Pascal wasn’t interested only in dice. He famously applied his new mathematics of probability to one of the great unknowns: the existence of God.

      ‘God is, or He is not.’ But to which side shall we incline? Reason can decide nothing here. There is an infinite chaos which separated us. A game is being played at the extremity of this infinite distance where heads or tails will turn up … Which will you choose then? Let us see. Since you must choose, let us see which interests you least. You have two things to lose, the true and the good; and two things to stake, your reason and your will, your knowledge and your happiness; and your nature has two things to shun, error and misery. Your reason is no more shocked in choosing one rather than the other, since you must of necessity choose … But your happiness? Let us weigh the gain and the loss in wagering that God is … If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is.

      Called Pascal’s wager, he argued that the payout would be much greater if one opted for a belief in God. You lose little if you are wrong and win eternal life if correct. On the other hand, wager against the existence of God and losing results in eternal damnation, while winning gains you nothing beyond the knowledge that there is no God. The argument falls to pieces if the probability of God existing is actually 0, and even if it isn’t, perhaps the cost of belief might be too high when set against the probability of God’s existence.

      The probabilistic techniques developed by mathematicians like Fermat and Pascal to deal with uncertainty were incredibly powerful. Phenomena that were regarded as beyond knowledge, the expression of the gods, were beginning to be within reach of the minds of men. Today these probabilistic methods are our best weapon in trying to navigate everything from the behaviour of particles in a gas to the ups and downs of the stock market. Indeed, the very nature of matter itself seems to be at the mercy of the mathematics of probability, as we shall discover in the Third Edge, when we apply quantum physics to predict what fundamental particles are going to do when we observe them. But for someone on the search for certainty, these probabilistic methods represent a frustrating compromise.

      I certainly appreciate the great intellectual breakthrough that Fermat, Pascal and others made, but it doesn’t help me to know how many pips are going to be showing when I throw my dice. As much as I’ve studied the mathematics of probability, it has always left me with a feeling of dissatisfaction. The one thing any course on probability drums into you is that it doesn’t matter how many times in a row you get a 6: this has no influence on what the dice is going to do on the next throw.

      So is there some way of knowing how my dice is going to land? Or is that knowledge always going to be out of reach? Not according to the revelations of a scientist across the waters in England.

       THE MATHEMATICS OF NATURE

      Isaac Newton is my all-time hero in my fight against the unknowable. The idea that I could possibly know everything about the universe has its origins in Newton’s revolutionary work Philosophiae Naturalis Principia Mathematica. First published in 1687, the book is dedicated to developing a new mathematical language that promised the tools to unlock how the universe behaves. It was a dramatically new model of how to do science. The work ‘spread the light of mathematics on a science which up to then had remained in the darkness of conjectures and hypotheses’, declared the French physicist Alexis Clairaut in 1747.

      It is also an attempt to unify, to create a theory that describes the celestial and the earthly, the big and the small. Kepler had come up with laws that described the motions of the planets, laws he’d developed empirically by looking at data and trying to fit equations to create the past. Galileo had described the trajectory of a ball flying through the air. It was Newton’s genius to understand that these were examples of a single phenomenon: gravity.

      Born on Christmas Day in 1643 in the Lincolnshire town of Woolsthorpe, Newton was always trying to tame the physical world. He made clocks and sundials, constructed miniature mills powered by mice, sketched countless plans for buildings and ships, and drew elaborate illustrations of animals. The family cat apparently disappeared one day, carried away by a hot-air balloon that Newton had made. His school reports, however, did not anticipate a great future, describing him as ‘inattentive and idle’.

      Idleness is not necessarily such a bad trait in a mathematician. It can be a powerful incentive to look for some clever shortcut to solve a problem rather than relying on hard graft. But it’s not generally a quality that teachers appreciate.

      Indeed, Newton was doing so badly at school that his mother decided the whole thing was a waste of time and that he’d be better off learning how to manage the family farm in Woolsthorpe. Unfortunately, Newton was equally hopeless at managing the family estate, so he was sent back to school. Although probably apocryphal, it is said that Newton’s sudden academic transformation coincided with a blow to the head that he received from the school bully. Whether true or not, Newton’s academic transformation saw him suddenly excelling at school, culminating in a move to study at the University of Cambridge.

      When bubonic plague swept through England in 1665, Cambridge University was closed as a precaution. Newton retreated to the house in Woolsthorpe. Isolation is often an important ingredient in coming up with new ideas. Newton hid himself away in his room and thought.

      Truth is the offspring of silence and meditation. I keep the subject constantly before me and wait ’til the first dawnings open slowly, by little and little, into a full and clear light.

      In the isolation of Lincolnshire, Newton created a new language that could capture the problem of a world in flux: the calculus. This mathematical tool would be key to our knowing how the universe would behave ahead of time. It is this language that gives me any hope