What We Cannot Know: Explorations at the Edge of Knowledge

by Marcus du Sautoy

Despite so much that is still unknown, scientists are optimistic that these questions won’t remain unanswered forever. The last few decades give us reason to believe that we are in a golden age of science. The rate of discoveries in science appears to grow exponentially. In 2014 the science journal Nature reported that the number of scientific papers published has been doubling every nine years since the end of the Second World War.

Take the statement made by French philosopher Auguste Comte in 1835 about the stars: ‘We shall never be able to study, by any method, their chemical composition or their mineralogical structure.’ An absolutely fair statement given that this knowledge seemed to depend on our visiting the star.

In 1900 Lord Kelvin, regarded by many as one of the greatest scientists of his age, believed that moment had come when he declared to the meeting of the British Association of Science: ‘There is nothing new to be discovered in physics now. All that remains is more and more precise measurement.’ American physicist Albert Abraham Michelson concurred.

There are known knowns; there are things that we know that we know. We also know there are known unknowns; that is to say, we know there are some things we do not know. But there are also unknown unknowns, the ones we don’t know we don’t know.

He perhaps missed one interesting category: The unknown knowns. The things that you know yet dare not admit to knowing. As the philosopher Slavoj Zizek argues, these are possibly the most dangerous, especially when held by those with political power. This is the domain of delusion. Repressed thoughts. The Freudian unconscious.

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.

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.

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.

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.

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.

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.

What Poincaré discovered, thanks to his error, led to one of the most important mathematical concepts of the last century: chaos. It was a discovery that places huge limits on what we humans can know. I may have written down all the equations for my dice, but what if my dice behaves like the planets in the solar system? According to Poincaré’s discovery, if I make just one small error in recording the starting location of the dice, that error could expand into a large difference in the outcome of the dice by the time it comes to rest on the table. So is the future of my Vegas dice shrouded behind the mathematics of chaos?

‘The question of whether where we’re heading is something that happens to all inhabited planets or whether there are other planets where it doesn’t happen is something I think we’ll never know.’

But it isn’t clear how far these outcomes are guaranteed by the model. If there is life on another planet, will it look anything like the life that has evolved here on Earth? This is one of the big open questions in evolutionary biology. As difficult as it may be to answer, I don’t believe it qualifies as something we can never know. It may remain something we will never know, but there is nothing by its nature that makes it unanswerable.

Are there other great unsolved questions of evolutionary biology that might be contenders for things we can never know? For example, why, 542 million years ago, at the beginning of the Cambrian period, was there an explosion of diversity of life on Earth? Before this moment life consisted of single cells that collected into colonies. But over the next 25 million years, a relatively short period on the scale of evolution, there is a rapid diversification of multicellular life that ends up resembling the diversity that we see today. An explanation for this exceptionally fast pace of evolution is still missing. This is in part due to lack of data from that period. Can we ever recover that data, or could this always remain a mystery?

The lottery is a perfect test bed for the occurrence of miracles in a random process. On 6 September 2009 the following six numbers were the winning numbers in the Bulgarian state lottery: 4, 15, 23, 24, 35, 42 Four days later the same six numbers came up again. Incredible, you might think. The government in Bulgaria certainly thought so and ordered an immediate investigation into the possibility of corruption. But what the Bulgarian government failed to take into account is that each week, across the planet, different lotteries are being run.

No one has managed to spontaneously generate anything as extraordinary as DNA in the lab. The chances of that are very small. But that’s the point, because given the billion billion or so possible planets available in the universe on which to try out this experiment, together with the billion or so years to let the experiment run, it would be more striking if that outside chance of creating something like DNA didn’t happen. Keep rolling 36 dice on a billion billion different planets for a billion years and you’d probably get one roll with all 36 dice showing 6. Once you have a self-replicating molecule it has the means to propagate itself, so you only need to get lucky once to kick off evolution.

But it’s not only the mathematics of probability that is at work in evolution. The evolutionary tree itself has an interesting quality that is similar to the shapes that appear in chaos theory, a quality known as fractal.

The evolutionary tree is a picture of the evolution of life on Earth. Making your way through the tree corresponds to a movement through time. Each time the tree branches, this represents the evolution of a new species. If a branch terminates, this means the extinction of that species. The nature of the tree is such that the overall shape seems to be repeated on smaller and smaller scales. This is the characteristic feature of a shape mathematicians call a fractal. If you zoom in on a small part of the tree it looks remarkably like the large-scale structure of the tree. This self-similarity means that it is very difficult to tell at what scale we are looking at the tree. This is the classic characteristic of a fractal.

The evolutionary biologist Stephen Jay Gould has contended that if you were to rerun the tape of life that you would get very different results.

Gould also introduced the idea of punctuated equilibria, which captures the fact that species seem to remain stable for long periods and then undergo what appears to be quite rapid evolutionary change. This has also been shown to be a feature of chaotic systems.

I can measure how big an effect a small change will have on the outcome using something called the Lyapunov exponent. For example, in the case of billiards played on differently shaped tables, I can give a measure of how catastrophic a small change will be on the evolution of a ball’s trajectory. If the Lyapunov exponent of a system is positive, it means that if I make a small change in the initial conditions then the distance between the paths diverges exponentially. This can be used as a definition of chaos. With this measure several groups of scientists have confirmed that our solar system is indeed chaotic. They have calculated that the distance between two initially close orbital solutions increases by a factor of ten every 10 million years. This is certainly on a different timescale to our inability to predict the weather. Nevertheless, it means that I can have no definite knowledge of what will happen to the solar system over the next 5 billion years.

You might expect that if the solar system was going to be ripped apart it would have to be one of the big planets like Jupiter or Saturn that would be the culprit. But the orbits of the gas giants are extremely stable. It’s the rocky terrestrial planets that are the troublemakers. In 1% of simulations that they ran, it was tiny Mercury that posed the biggest risk. The models show that Mercury’s orbit could start to extend due to a certain resonance with Jupiter, with the possibility that Mercury could collide with its closest neighbour, Venus. In one simulation, a close miss was enough to throw Venus out of kilter, with the result that Venus collides with Earth. Even close encounters with the other planets would be enough to cause such tidal disruption that the effect would be disastrous for life on our planet. This isn’t simply a case of abstract mathematical speculation. Evidence of such collisions has been observed in the planets orbiting the binary star Upsilon Andromedae. Their current strange orbits can be explained only by the ejection of an unlucky planet sometime in the star’s past. But before we head for the hills, the simulations reveal that it will take several billion years before Mercury might start to misbehave.

Chaos theory asserts that I cannot know the future of certain systems of equations because they are too sensitive to small inaccuracies.

Chaos theory implies that our futures are often beyond knowledge because of their dependence on the fine-tuning of how things are set up in the present. Because we can never have complete knowledge of the present, chaos theory denies us access to the future. At least until that future becomes the present.

Polkinghorne has proposed that it is via the indeterminacies implicit in chaos theory that a supernatural intelligence can still act without violating the laws of physics. Chaos theory says that we can never know the set-up precisely enough to be able to run deterministic equations, and hence there is room in Polkinghorne’s view for divine intervention, to tweak things to remain consistent with our partial knowledge but still influence outcomes.

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.

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.

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.

Reading through the history, I am surprised that it is only just over a hundred years ago that convincing evidence was provided for the fact that things like my dice are made of discrete building blocks called atoms and are not just continuous structures. Despite being such a relatively recent discovery, the hunch that this was the case goes back thousands of years.

In India it was believed that matter was made from basic atoms corresponding to taste, smell, colour and touch. They divided atoms into ones that were infinitesimally small and took up no space, and others that were ‘gross’ and took up finite space – an extremely prescient theory, as you will see once I explain our current model of matter.

Aristotle was one of those who did not believe in the idea of fundamental atoms. He thought that the elements were continuous in nature, that you could theoretically keep dividing my dice up into smaller and smaller pieces. He believed that fire, earth, air and water were elemental in the sense that they could not be divided into ‘bodies different in form’. If you kept dividing, you would still get water or air. If you take a glass of water then to the human eye, it appears to be a continuous structure which can theoretically be infinitely divided.

The stage was set for the battle between the continuous and discrete models of matter. The glissando versus the discrete notes of the musical scale. The cello versus the trumpet.

The Babylonians had been fascinated by the challenge of calculating this length. Dating back to the Old Babylonian period (1800–1600 BC), a tablet housed at Yale University has an estimate for the distance. Written using the sexagesimal system, or base 60, they got the length to be: which in decimal notation comes out at 1.41421296296 …, where the 296 repeats itself infinitely often.

Pythagoras’ theorem about right-angled triangles implied that this long side had length the square root of 2 times the length of the short sides. But Hippasus could prove that there was no fraction whose square was exactly 2. The proof uses one of the classic tools in the mathematician’s arsenal: proof by contradiction. Hippasus began by assuming there was a fraction whose square was 2. By some deft manipulation this always led to the contradictory statement that there was a number that was both odd and even. The only way to resolve this contradiction was to realize that the original assumption must have been false: there can be no fraction whose square is 2.

I certainly have the feeling that this length exists. I can see it on a ruler held up against the long side of the triangle. It is the distance between two opposite corners of any side of my dice. Yet try to write down the number as an infinite decimal and I can never capture it. It begins 1.414213562 … and then continues to infinity never repeating itself.

Despite my aversion to things we cannot know, reading about numbers that can’t be captured using simple whole-number ratios or fractions was one of the defining moments that sparked my love affair with mathematics.

The discovery of these expressions pulls these irrational numbers into the known. A fraction is a number whose decimal expansion repeats itself from some point. Couldn’t I regard these expressions as a pattern, not too dissimilar from the repeating pattern of the decimal expansion of the fraction? The repeating pattern of the fraction means that there are two numbers whose ratio captures the number, while in the case of √2 and π I am resigned to needing infinitely many numbers to pin down these lengths. The question of whether something has to be finite to be known will haunt me continually throughout my journey to the frontiers of the unknown.

In fact, I need only know 39 digits of π to be able to calculate the circumference of a circle the size of the observable universe to a precision comparable to the size of a hydrogen atom.

Although Newton would concur with Boyle’s proposal for a material world made from indivisible units, the mathematical tools that Newton developed at the same time as Boyle’s work relied heavily on time and space being infinitely divisible.

It was the work of the English chemist John Dalton at the beginning of the nineteenth century that provided the first real experimental justification for thinking of matter as made of indivisible atoms. His discovery that compounds seemed to be made up of substances that were combined in fixed whole-number ratios was the breakthrough, and it led to the scientific consensus that these substances really did come in discrete packages.

The Russian scientist Dmitri Mendeleev is remembered for laying out this growing list of molecular ingredients in such a way that a pattern began to emerge, a pattern based on whole numbers and counting. It seemed that the Pythagorean belief in the power of number was making a comeback. Like several scientists before him, Mendeleev arranged them in increasing relative weight, but he realized that to get the patterns he could see emerging he needed to be flexible. He’d written the known elements down on cards and was continually placing them on his desk in a game of chemical patience, trying to get them to yield their secrets. But nothing worked. It was driving him crazy. Eventually he collapsed in exhaustion and the secret emerged in a dream that, when he woke, gave him the pattern for laying out the cards. One of the important points that led to his successful arrangement was the realization that he needed to leave some gaps – that some of the cards from the pack were missing.

Boltzmann believed that this atomic theory was a powerful way to interpret the concept of heat based on the idea that a gas was made up of tiny molecules bashing around like a huge game of micro-billiards. Heat was just the combined kinetic energy of these tiny moving balls. Using this model, combined with ideas of probability and statistics, he was successfully able to explain the large-scale behaviour of a gas. But most physicists were still committed to a continuous view of matter and were very dismissive of Boltzmann’s ideas.

Boltzmann was plagued by fits of depression, and there is evidence that he was in fact bipolar. The rejection of his ideas by the scientific community is believed to have contributed to the depression that struck in 1906 and that led to him hanging himself during a holiday with his family near Trieste while his daughter and wife were out swimming. It was a tragic end, not least because the most convincing evidence that he was right was just emerging. And it was one of the big names of physics who produced ideas that supported the atomistic view and were very hard to ignore. The work that Einstein and others did on Brownian motion would prove extremely difficult to explain for those who, like Mach, believed in a continuous view of the world.

Carl Anderson, a physicist working at Caltech, had used these cloud chambers in 1933 to confirm the existence of a strange new sort of matter called antimatter that had been predicted some years earlier by British physicist Paul Dirac.

Even stranger particles that had not been predicted at all were soon leaving trails in Anderson’s cloud chamber. Anderson started to analyse these new paths with his PhD student Seth Neddermeyer in 1936. The new particles corresponded to negatively charged particles passing through the cloud chamber. But they weren’t electrons. The paths these new particles were leaving indicated a mass much larger than that of the electron. Just as Thomson had done, mass can be measured by how much the particle is deflected under the influence of a magnetic field. The particle seemed to have the same charge as the electron but was much harder to deflect. Now called the muon, it was one of the first new particles to be discovered in the interactions of cosmic rays with the atmosphere.

The muon appeared to behave remarkably like the electron but had greater mass and was more unstable. When the American physicist Isidor Rabi was told of the discovery, he quipped: ‘Who ordered that?’ It seemed strangely unnecessary for nature to reproduce a heavier, more unstable version of the electron. Little did Rabi realize how much more there was on the menu of particles.

When you are trying to classify things, it helps to recognize the dominant characteristics that can gather a large mess of objects into smaller groups. In the case of animals, the idea of species creates some order in the animal kingdom. In particle physics one important invariant that helped divide the zoo into smaller groups was the idea of charge. How does the particle interact with the electromagnetic force? Electrons would bend one way, protons the other, and the neutron would be unaffected.

There are several mechanisms for this decay, each depending on one of the fundamental forces. Each mechanism has a characteristic signature which helps physicists to understand which fundamental force is causing the decay. Again it’s energy considerations that control which is the most likely force at work in any particle decay. The strong nuclear force is usually the first to have a go at decaying a particle, and this will generally decay the particle within 10–24 of a second. Next in the hierarchy is the electromagnetic force, which might result in the emission of photons. The weak nuclear force is the most costly in energy terms and so takes longer. A particle that decays via the weak nuclear force is likely to take 10–11 seconds before it decays. So by observing the time it takes to decay, scientists can get some indication of which force is at work.

The physicists Abraham Pais, Murray Gell-Mann and Kazuhiko Nishijima came up with a cunning strategy to solve this puzzle. They proposed a new property like charge that mediated the way these particles interacted or didn’t interact with the strong nuclear force. This new property, called strangeness, gave physicists a new way to classify all these new particles. Each new particle was given a measure of strangeness according to whether or not its decay would have to take the long route.