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

cannot know. The gods in many ancient cultures were always a placeholder for the things we couldn’t explain or couldn’t understand. Our ancestors found volcanic eruptions or eclipses so mysterious that they became acts of gods. As science has explained such phenomena, these gods have retreated.

      This definition has some things in common with a God commonly called the ‘God of the gaps’. This phrase was generally used as a derogatory term by religious thinkers who could see that this God was shrinking in the face of the onslaught of scientific knowledge, and a call went out to reject this kind of God. The phrase ‘God of the gaps’ was coined by the Oxford mathematician and Methodist church leader Charles Coulson, when he declared: ‘There is no “God of the gaps” to take over at those strategic places where science fails.’

      But the phrase is also associated with a fallacious argument for the existence of God, one that Richard Dawkins spends some time shooting down in The God Delusion: if there are things that we can’t explain or know, there must be a God at work filling the gap. But I am more interested not in the existence of a God to fill the gap, but in equating God with the abstract idea of the things we cannot know. Not in the things we currently don’t know, but the things that by their nature we can never know. The things that will always remain transcendent.

      Religion is more complex than the simple stereotype often offered up by modern society. For many ancient cultures in India, China and the Middle East, religion was not about worshipping a Supernatural Intelligence but precisely the attempt to appreciate the limits of our understanding and language. As the theologian Herbert McCabe declared: ‘To assert the existence of God is to claim that there is an unanswered question about the universe.’ Science has pushed hard at those limits. So is there anything left? Will there be anything that will always be beyond the limit. Does McCabe’s God exist?

      This is the quest at the heart of this book. Can we identify questions or physical phenomena that will always remain beyond knowledge? If we can identify things that will remain in the gaps of knowledge, then what sort of God is this? What potency would such a concept have? Could the things we cannot know act in the world and affect our futures? Are they worthy of worship?

      But first we need to know if in fact there is anything that will remain unanswered about the universe. Is there really anything we cannot know?

FIRST EDGE: THE CASINO DICE

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      The unpredictable and the predetermined unfold together to make everything the way it is. It’s how nature creates itself, on every scale, the snowflake and the snowstorm. It makes me so happy. To be at the beginning again, knowing almost nothing.

      Tom Stoppard, Arcadia

      There is a single red dice sitting on my desk next to me. I got the dice on a trip to Las Vegas. I fell in love with it when I saw it on the craps table. It was so perfectly engineered. Such precise edges coming to a point at the corners of the cube. The faces so smooth you couldn’t feel what number the face was representing. The pips are carved out of the dice and then filled with paint that has the same density as the plastic used to make the dice. This ensures that the face representing the 6 isn’t a touch lighter than the face on the opposite side with a single pip. The feeling of the dice in the hand is incredibly satisfying. It is a thing of beauty.

      And yet I hate it.

      It’s got three pips pointing up at me at the moment. But if I pick it up and let it fall from my hand I have no way of knowing how it is going to land. It is the ultimate symbol of the unknowable. The future of the dice seems knowable only when it becomes the past.

      I have always been extremely unsettled by things that I cannot know. Things that I cannot work out. I don’t mind not knowing something provided there is some way ultimately to calculate what’s going on. With enough time. Is this dice truly so unknowable? Or with enough information can I actually deduce its next move? Surely it’s just a matter of applying the right laws of physics and solving the appropriate mathematical equations. Surely this is something I can know.

      My subject, mathematics, was invented to give people a glimpse of what’s out there coming towards us. To look into the future. To become masters of fate, not its servants. I believe that the universe runs according to laws. Understand those laws and I can know the universe. Spotting patterns has given the human species a very powerful way to take control. If there’s a pattern then I have some chance to predict the future and know the unknowable. The pattern of the Sun means I can rely on it rising in the sky tomorrow or the Moon taking 28 sunrises before it becomes full again. It is how mathematics developed. Mathematics is the science of patterns. Being able to spot patterns is a powerful tool in the evolutionary fight for survival. The caves in Lascaux show how counting 13 quarters of the Moon from the first winter rising of the Pleiades will bring you to a time in the year when the horses are pregnant and easy to hunt. Being able to predict the future is the key to survival.

      But there are some things which appear to have no pattern or that have patterns that are so complex or hidden that they are beyond human knowledge. The individual roll of the dice is not like the rising of the Sun. There seems to be no way to know which of the six faces will be pointing upwards once the cube finally comes to rest. It is why the dice has been used since antiquity as a way to decide disputes, to play games, to wager money.

      Is that beautiful red cube with its white dots truly unknowable? I’m certainly not the first to have a complex relationship with the dynamics of this cube.

       KNOWING THE WILL OF THE GODS

      On a recent trip to Israel I took my children to an archaeological dig at Beit Guvrin. It was such a popular settlement in ancient times that the site consists of layer upon layer of cities built on top of each other. There is so much stuff in the ground that the archaeologists are happy to enlist amateurs like me and my kids to help excavate the site even if a few pots get broken along the way. Sure enough, we pulled out lots of bits of pottery. But we also kept unearthing a large number of animal bones. We thought they were the remains of dinner, but our guide explained that in fact they were the earliest form of dice.

      Archaeological digs of settlements dating back to Neolithic times have revealed a disproportionately high density of heel bones of sheep or other animals among the shattered pottery and flints that are usually found in sites that humans once inhabited. These bones are in fact ancestors of my casino dice. When thrown, the bones naturally land on one of four sides. Often there are letters or numbers carved into the bones. Rather than gambling, these early dice are thought to have been used for divination. And this connection between the outcome of the roll of a dice and the will of the gods is one that has persisted for centuries. Knowledge of how the dice would land was believed to be something that transcended human understanding. Its outcome was in the lap of the gods.

      Increasingly these dice assumed a more prosaic place as part of our world of leisure. The first cube-shaped dice like the one on my desk were found around Harappa in what is now northeast Pakistan, where one of the first urban civilizations evolved, dating back to the third millennium BC. At the same time, you find four-faced pyramid dice appearing in a game that was discovered in the city of Ur, in ancient Mesopotamia.

      The Romans and Greeks were addicts of games of dice, as were the soldiers of the medieval era who returned from the Crusades with a new game called hazard, deriving from the Arabic word for a dice: al-zahr. It was an early version of craps, the game that was being played in the casino in Vegas where I picked up my dice.

      If I could predict the fall of the dice, all the games that depend on them would never have caught on. The excitement of backgammon or hazard or craps comes from not knowing how the dice are going to land. So perhaps gamers won’t thank me as I try to predict the roll of my dice.

      For centuries no one even thought that such a feat was possible. The ancient Greeks,