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

even when I am creating new mathematics, I will often quote results by fellow mathematicians whose proofs I won’t have checked myself. To do so would mean running in order to keep still.

      And for any scientist the real challenge is not to stay within the secure garden of the known but to venture out into the wilds of the unknown. That is the challenge at the heart of this book.

       WHAT WE DON’T KNOW

      Despite all the breakthroughs made in science over the last centuries, there are still lots of deep mysteries waiting out there for us to solve. Things we don’t know. The knowledge of what we are ignorant of seems to expand faster than our catalogue of breakthroughs. The known unknowns outstrip the known knowns. And it is those unknowns that drive science. A scientist is more interested in the things he or she can’t understand than in telling all the stories we already know how to narrate. Science is a living, breathing subject because of all those questions we can’t answer.

      For example, the stuff that makes up the physical universe we interact with seems to account for only 4.9% of the total matter content of our universe. So what is the other 95.1% 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 is fuelling that acceleration?

      Is our 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 own 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?

      Why is there a layer of fundamental particles with another two almost identical copies of this layer but with increasing mass, the so-called three generations of fundamental particles? Are there yet more particles out there for us to discover? Are fundamental particles actually tiny strings vibrating in 11-dimensional space?

      How can we unify Einstein’s theory of general relativity, the physics of the very large, with quantum physics, the physics of the very small? This is the search for something called quantum gravity, an absolute necessity if we are ever going to understand the Big Bang, when the universe was compressed into the realm of the quantum.

      And what of the understanding of our human body, something so complex that it makes quantum physics look like a high-school exercise. We are still trying to get to grips with the complex interaction between gene expression and our environment. Can we find a cure for cancer? Is it possible to beat ageing? Could there be someone alive today who will live to be a 1000 years old?

      And what about where humans came from? Evolution is a process of random mutations, so would a different roll of the evolutionary dice still produce organisms with eyes? If we rewound evolution and pressed ‘play’, would we get intelligent life, or are we the result of a lucky roll of the dice? Is there intelligent life elsewhere in our universe? And what of the technology we are creating? Can a computer ever attain consciousness? Will I eventually be able to download my consciousness so that I can survive the death of my body?

      Mathematics too is far from finished. Despite popular belief, Fermat’s Last Theorem was not the last theorem. 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?

      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. Computers too are developing at an exponential rate. Moore’s law is the observation that computer processing power seems to double every two years. Engineer Ray Kurzweil believes that the same applies to technological progress: that the rate of change of technology over the next 100 years will be comparable to what we’ve experienced in the last 20,000 years.

      And yet can scientific discoveries maintain this exponential growth? Kurzweil talks about the Singularity, a moment when the intelligence of our technology will exceed our human intelligence. Is scientific progress destined for its own singularity? A moment when we know it all. Surely at some point we might actually discover the underlying equations that explain how the universe works. We will discover the final list of particles that make up the building blocks of the physical universe and how they interact with each other. Some scientists believe that the current rate of scientific progress will lead to a moment when we might discover a theory of everything. They even give it a name: ToE.

      As Hawking declared in A Brief History of Time: ‘I believe there are grounds for cautious optimism that we may be near the end of the search for the ultimate laws of nature’, concluding dramatically with the provocative statement that then ‘we would know the mind of God’.

      Is such a thing possible? To know everything? Would we want to know everything? Science would ossify. Scientists have a strangely schizophrenic relationship with the unknown. On the one hand, it is what we don’t know that intrigues and fascinates us, and yet the mark of success as a scientist is resolution and knowledge, to make the unknown known.

      Could there be some quests that will never be resolved? Are there limits to what we can discover about our physical universe? Are some regions of the future beyond the predictive powers of science and mathematics? Is time before the Big Bang a no-go area? Are there ideas so complex that they are beyond the conception of our finite human brains? Can brains even investigate themselves, or does the analysis enter an infinite loop from which it is impossible to rescue itself? Are there mathematical conjectures that can never be proved true?

       WHAT WE’LL NEVER KNOW

      What if there are questions of science that can never be resolved? It seems defeatist, even dangerous, to admit there may be any such questions. While the unknown is the driving force for doing science, the unknowable would be science’s nemesis. As a fully signed-up member of the scientific community, I hope that we can ultimately answer the big open questions. So it seems important to know if the expedition I’ve joined will hit boundaries beyond which we cannot proceed. Questions that won’t ever get closure.

      That is the challenge I’ve set myself in this book. I want to know if there are things that by their very nature we will never know. Are there things that will always be beyond the limits of knowledge? Despite the marauding pace of scientific advances, are there things that will remain beyond the reach of even the greatest scientists? Will there remain mysteries that will resist our attempts to lift the veils that currently mask our view of the universe?

      It is, of course, very risky at any point in history to try to articulate Things We Cannot Know. How can you know what new insights are suddenly going to pull the unknown into the knowable? This is partly why it is useful to look at the history of how we know the things we do, because it reveals how often we’ve been at points where we think we have hit the frontier, only to find some way across.

      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. What Comte hadn’t factored in was the possibility that the star could visit us, or at least that photons of light emitted by the star could reveal its chemical make-up.

      A few decades after Comte’s prophecy, scientists had determined the chemical composition of our own star, the Sun, by analysing the spectrum of light emitted. As the nineteenth-century British astronomer Warren de la Rue declared: ‘If we were to go to the Sun, and to bring some portions of it and analyse them in our laboratories, we could not examine them more accurately than we can by this new mode of spectrum analysis.’

      Scientists went on to determine the chemical composition