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

of an electron on the other side of the universe could influence the whole system, we need to have complete, holistic knowledge of the whole system – the whole universe – to be able to steer things. We cannot successfully isolate a part of the universe and hope to make predictions based on that part. So it would require knowledge of the whole to act via this chink in that which is unknown to us.

      Chaos theory is deterministic, so this isn’t an attempt to use the randomness of something like quantum physics as a way to have influence. Polkinghorne’s take on how to square the circle of determinism and influence the system is to use the gap between epistemology and ontology, between what we know and what is true. Since we cannot know a complete description of the state of the universe at this moment in time, this implies that from our perspective there is no determinacy. There are many different scenarios that coincide with our impartial description of what we currently know about how the universe is set up. Polkinghorne’s contention is that at any point in time this gives a God the chance to intervene and shift the system between any of these scenarios without us being aware of the shift. But, as we have seen, chaos theory means that these small shifts can still have hugely different outcomes. Polkinghorne is careful to assert that you allow shifts between systems where there is change only in information, not energy. The rule here is not to violate any rules of physics. As Polkinghorne says: ‘The succession of the seasons and the alternations of day and night will not be set aside.’

      Even if you think this is rather fanciful (which I certainly do), a similar principle is probably key to our own feeling of agency in the world. The question of free will is related ultimately to questions of a reductionist philosophy. Free will describes the inability to make any meaningful reduction in most cases to an atomistic view of the world. So it makes sense to create a narrative in which we have free will because that is what it looks like on the level of human involvement in the universe. If things were so obviously deterministic, with little variation to small undetectable changes, we wouldn’t think that we had free will.

      It is striking that Newton, the person who led us to believe in a clockwork deterministic universe, also felt that there was room in the equations for God’s intervention. He wrote of his belief that God would sometimes have to reset the universe when things looked like they were going off course. He got into a big fight with his German mathematical rival Gottfried Leibniz, who couldn’t see why God wouldn’t have set it up perfectly from the outset:

      Sir Isaac Newton and his followers have also a very odd opinion concerning the work of God. According to their doctrine, God Almighty wants to wind up his watch from time to time: otherwise it would cease to move. He had not, it seems, sufficient foresight to make it a perpetual motion.

       ON THE EDGE OF CHAOS

      Newton and his mathematics gave me a feeling that I could know the future, that I could shortcut the wait for it to become the present. The number of times I have heard Laplace’s quote about ultimately being able to know everything thanks to the equations of motion is testament to a general feeling among scientists that the universe is theoretically knowable.

      The mathematics of the twentieth century revealed that theory doesn’t necessarily translate into practice. Even if Laplace is correct in his statement that complete knowledge of the current state of the universe together with the equations of mathematics should lead to complete knowledge of the future, I will never have access to that complete knowledge. The shocking revelation of twentieth-century chaos theory is that even an approximation to that knowledge won’t help. The divergent paths of the chaotic billiard table mean that since we can never know which path we are on, our future is not predictable.

      Chaos theory implies that there are things we can never know. The mathematics in which I had placed so much faith to give me complete knowledge has revealed the opposite. But it is not entirely hopeless. Many times the equations are not sensitive to small changes and hence give me access to predictions about the future. After all, this is how we landed a spaceship on a passing comet. Not only that: as Bob May’s work illustrates, the mathematics can even help me to know when I can’t know.

      But a discovery at the end of the twentieth century even questions Laplace’s basic tenet of the theoretical predictability of the future. In the early 1990s a PhD student by the name of Zhihong Xia proved that there is a way to configure five planets such that when you let them go, the combined gravitational pull causes one of the planets to fly off and reach an infinite speed in a finite amount of time. No planets collide, but still the equations have built into them this catastrophic outcome for the residents of the unlucky planet. The equations are unable to make any prediction of what happens beyond this point in time.

      Xia’s discovery is a fundamental challenge to Laplace’s view that Newton’s equations imply that we can know the future if we have complete knowledge of the present, because there is no prediction even within Newton’s equations for what happens next for that unlucky planet once it hits infinite speed. The theory hits a singularity at this point, beyond which prediction makes no sense. As we shall see in later Edges, considerations of relativity will limit the physical realization of this singularity since the unlucky planet will eventually hit the cosmic speed limit of the speed of light, at which point Newton’s theory is revealed to be an approximation of reality. But it nevertheless reveals that equations aren’t enough to know the future.

      It is striking to listen to Laplace on his deathbed. As he sees his own singularity heading towards him with only a finite amount of time to go, he too admits: ‘What we know is little, and what we are ignorant of is immense.’ The twentieth century revealed that, even if we know a lot, our ignorance will remain immense.

      But it turns out that it is isn’t just the outward behaviour of planets and dice that is unknowable. Probing deep inside my casino dice reveals another challenge to Laplace’s belief in a clockwork deterministic universe. When scientists started to look inside the dice to understand what it is made of, they discovered that knowing the position and the movement of the particles that make up the dice may not even be theoretically possible. As I shall discover in the next two Edges, there might indeed be a game of dice at work that controls the behaviour of the very particles that make up my red Las Vegas cube.

SECOND EDGE: THE CELLO

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      Everyone takes the limits of his own vision for the limits of the world.

      Arthur Schopenhauer

      When I started at my comprehensive school, I remember my music teacher asking the class if there was anyone who wanted to learn a musical instrument. Three of us put up our hands. The teacher led us into the storeroom cupboard to see what instruments were available. The cupboard was bare except for three trumpets stacked up on top of each other.

      ‘It looks like you’re learning the trumpet.’

      I don’t regret the choice (even if there wasn’t one). I had a great time playing in the local town band and larking around in the brass section of the county orchestra as we counted bars rest. But I used to look over with a little envy at the strings who seemed to be playing all the time, getting all the good tunes. A few years ago, during a radio interview, I was asked what new musical instrument I would choose to learn, given the opportunity, and which piece of music I would aspire to play.

      ‘The cello. Bach’s suites.’

      The question has been nagging at the back of my mind since that interview: could I learn to play those beautiful cello suites? Perhaps it was too late to pick up such new skills, but I needed to know. So I bought a cello.

      It sits behind me as I write about trying to predict the outcome of the dice. When I need a break from analysing the equations that control the fall of the red cube on my desk, I massacre one of the gigues from the first suite for cello. I can feel Bach turning in his grave but I am enjoying myself.

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