The Great Unknown: Seven Journeys to the Frontiers of Science

by Marcus du Sautoy

ZERO THE KNOWN UNKNOWNS

In the last ten years alone we’ve landed a spaceship on a comet,

the world’s population continues to escalate, scientific advances provide the best hope of feeding the 9.6 billion people who are projected to be alive in 2050.

But understanding is different from a list of facts. Is

For example, the stuff that makes up the physical universe we interact with seems to account for only 4.9 percent of the total matter content of our universe. So what is the other 95.1 percent of so-called dark matter and dark energy made up of? If our universe’s expansion is accelerating, where is all the energy coming from that fuels that acceleration?

universe infinite? Are there infinitely many other infinite universes parallel to our own? If there are, do they have different laws of physics? Were there other universes before our universe emerged from the Big Bang? Did time exist before the Big Bang? Does time exist at all, or does it emerge as a consequence of more fundamental concepts?

Mathematical unknowns abound. Are there any patterns in prime numbers, or are they outwardly random? Will we be able to solve the mathematical equations for turbulence? Will we ever understand how to factorize large numbers efficiently?

In 2014 the science journal Nature reported that the number of scientific papers has been doubling every nine years since the end of World War II.

Ray Kurzweil believes that the same applies to technological progress: that the rate of change over the next hundred years will be comparable to what we’ve experienced in the last 20,000 years.

That is the challenge I’ve set myself in this book. I want to know whether there are things that, by their very nature, we will never know.

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.

And yet if one unpacks the statement, Rumsfeld very concisely summed up different types of knowledge. He perhaps missed one interesting category: the unknown knowns, the things that you know yet dare not admit to knowing.

There are seven of them, and each one represents the horizon beyond which we cannot see. My journey to the Seven Edges of knowledge will pass through the known knowns, to demonstrate how we have traveled beyond what we previously thought were the limits of knowledge. This

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.

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

Mathematics is the science of patterns. Being able to spot a pattern is a powerful tool in the evolutionary fight for survival. The pattern of the sun means that I can rely on its rising in the sky tomorrow or on the moon running through twenty-eight sunrises before it becomes full again. The caves in Lascaux show how counting thirteen 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.

If nature were not beautiful it would not be worth knowing, and if nature were not worth knowing, life would not be worth living. —Henri Poincaré

This is the signature of chaos theory: sensitivity to very small changes in the initial conditions.

Massachusetts Institute of Technology in 1963, Lorenz had been running equations for the change of temperature in a dynamic fluid on his computer when he decided he needed to rerun one of his models for longer. So he took some of the data that had been output earlier in the run and re-entered it, expecting to be able to restart the model from that point.

When Lorenz sought to explain his findings to a colleague, he was told, “Edward, if your theory is correct, one flap of a seagull’s wings could alter the course of history forever.” The seagull would eventually be replaced by the now famous butterfly

‘Like causes produce like effects.’ This is only true when small variations in the initial circumstances produce only small variations in the final state of the system.” The discovery of chaos theory in the twentieth century revealed this maxim to be false.

May explored the dynamics of a mathematical equation describing population growth from one season to the next. He revealed how even a quite basic equation can produce extraordinarily complex results.

“We’re better at predicting events at the edge of the galaxy or inside the nucleus of an atom than whether it’ll rain on auntie’s garden party three Sundays from now.”

May’s last point relates to the challenge that chaos theory poses for knowing something about the past as much as the future. At least with the future we can wait and see what the outcome of chaotic equations produces. But trying to work backward and understand what state our planet was in to produce the present is equally if not more challenging. The past, even more than the future, is probably something we can never truly know.

Chaos theory is usually a limiting factor in what we can know about the future. But it can also imply limits on what we can know about the past. We

On September 6, 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 it failed to take into account is that each week, across the planet, different lotteries are being run. They have been running for decades. If you do the math, it would be more surprising not to see such a seemingly anomalous result.

Fractals are the geometric signature of a chaotic system, so it is suggestive of chaotic dynamics at work in evolution: small changes in the genetic code can result in huge changes in outcome. This model isn’t necessarily a challenge to the idea of convergence, as there can still be points in chaotic systems toward which the model tends to evolve. Such points are called attractors.

Gould introduced the idea of punctuated equilibria to capture the fact that species seem to remain stable for long periods and then undergo what appears to be quite rapid evolutionary change. This has been shown to be a feature of chaotic systems. The implications of chaos at work in evolution could well fall under the umbrella of things we cannot know.

hold the key to answering the question Poincaré first tackled when he discovered chaos: Will there even be a stable Earth orbiting the sun for evolution to continue playing its game of dice? How safe is our planet from the vagaries of chaos? Is our solar system stable and periodic, or do I have to worry about a grasshopper disrupting our orbit around the sun?

We can now measure how big an effect a small change will have on the outcome of a closed system using something called the Lyapunov exponent. If the Lyapunov exponent is positive, it means that if I make a small change in the initial conditions then the distance between the paths will diverge exponentially.

Everyone takes the limits of his own vision for the limits of the world. —Arthur Schopenhauer

But Hippasus proved that there was no fraction whose square was exactly 2. This length would have to be expressed by a new sort of number. The proof uses one of the classic tools in the mathematician’s arsenal: proof by contradiction. Hippasus began by assuming that 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.

challenged the idea that matter was made up of the four elements of fire, earth, air, and water. These might be good descriptions of the states of matter, he argued, but not of its constituent