The unknown thrills and terrifies me in equal measure. On the one hand it is the things that I don’t know that drive me every morning to get up and head to my desk to battle away with the mathematical conundrums that have eluded me, in some cases for decades. I think many people are surprised to discover that there are still so many things in mathematics that are unknown. Most people get the impression from school that mathematics was handed down to us in some great text book from the sky. That all the answers are there at the back of the book ready to be sneaked at if you get stuck. Two decades ago the newspapers heralded the proof of Fermat’s Last Theorem, discovered by my Oxford colleague Andrew Wiles. Many people I talked to assumed that this was in fact that last theorem. That we had finished mathematics. All was now known.

I think that was partly my motivation for writing my first popular book The Music of the Primes. I was keen to put back into the public imagination a great unsolved problem of mathematics. To show those outside of mathematics that there was still so much we didn’t know. That we didn’t even understand our most basic ingredients: the prime numbers. These indivisible numbers are the building blocks of mathematics yet we don’t understand how they are distributed through the universe of numbers. Their behaviour is still unknown. But we have a hunch about how Nature laid out these numbers and it is encapsulated in what many mathematicians regard as our greatest unsolved problem: The Riemann Hypothesis. Crack this enigma and we will know the primes.

Yet despite over 150 years of endeavor by the world’s greatest minds, the hunch that Riemann had in 1859, the same year that Darwin published On the Origin of Species, has not revealed the origins of what makes the primes tick. They are still part of the unknown. But I think all of the mathematicians that strive to prove this great theorem believe that it won’t remain unknown forever. But how can we be sure? How can we know that we will ultimately discover the answer. The unknown is the life blood of mathematics. The things we don’t know are what make it a living breathing subject not just a subject confined to the static journals on the library shelves of our great institutions. It is the unknown that thrills us as practicing mathematicians.  But what if there is no proof. What if there are things that there is no way we can know. It is the unknowable that fills us with terror.

To know the unknowable has been the journey that I have been on in my latest book The Great Unknown. To understand whether in mathematics and more generally in science there are questions that by their very nature are unanswerable. Not just that we don’t know how to answer them now. There are lots of those. What is dark matter? Is their life on other planets? Is there a cure for cancer? But whether there are known unknowns (to quote Rumsfeld) that will always remain unknowable. Perhaps there are none. Perhaps everything is intrinsically knowable. I think that is the mindset of the scientist. That almost arrogant belief that we can know everything is what keeps us heading into the lab each morning in the pursuit of knowledge.

But maybe it is important to stop and question whether the scientific battles we are engaged in are winnable. Or whether we need to recognize that perhaps we can’t know it all. After all the universe surely has not been set up as an exercise in the philosophy of science. It would be very anti-Copernican if the universe were arranged so that we can know it all.  So what might be the unknowns that will always be beyond science?

The mathematics of chaos theory articulates limits to how much we can know about the future given our current description of the present. Certain dynamical systems reveal how even a small error in the data describing the present can explode into hugely divergent predications for the future. The limits inherent in quantum physics suggest that even a complete description of the present does not determine a unique future. The revelations of quantum physics have put unknowability at the heart of the way we must do science.

Our current description of matter posits that it is made out of electrons and quarks. But are these truly the indivisible particles that make up the universe? Previous generations all thought they’d hit the bottom layer only to find atoms come apart into electrons, protons and neutrons which in turn divided up into quarks.

The question of whether out universe is infinite or not is one that generations of scientists have wrestled with. But given that information travels at the speed of light and the universe has only been going for 13.8 billion years, this implies a cosmic horizon, a bubble which surrounds us beyond which we can receive no information. So how can we ever know if it goes on for ever. And although we think it all began 13.8 billion years ago, can we be sure there was nothing before the Big Bang. That singularity currently masks our ability to look any further back in time.

Even the brains we’ve been using to navigate the universe contain an enigma of their own. How does sticking together a bunch of atoms suddenly give rise to consciousness? Will we ever be able to make artificial consciousness? Will my smart phone suddenly declare “IPhone think therefore iPhone am?” And if it does how will we ever know if it is really conscious or just faking it.

The amazing thing though for me is that my very own subject of mathematics has been able to articulate it’s own limits. Gödel’s Incompleteness Theorem declares that any system of mathematics will have true statement about numbers that cannot be proved true within that system. The conjectures I am working on had promised the thrill of discovery but what if there is no proof. It is the thought of this unknown that terrifies me as I head to my desk each morning.