EDWARD FRENKEL is a Russian mathematician working in representation theory, algebraic geometry and mathematical physics. He is professor of mathematics at the University of California, Berkeley, and the author of “Love and Math”, recently published by Basic Books.You describe math as “beautiful”. What do you mean?Imagine you had an art class in which they taught you how to paint a fence, but never showed you the great masters. Of course, you would say; ‘I hate art.’ You were bad at painting the fence but you wouldn’t know what else there is to art. Unfortunately, that is exactly what happens with mathematics. What we study at school is a tiny little part of mathematics. I want people to discover the magic world of mathematics, almost like a parallel universe, that most of us aren’t aware even exists.How did you discover it?When I was growing up near Moscow I thought mathematics was the most boring and irrelevant subject, but I was fascinated with quantum physics and elementary particles. Luckily for me a professional mathematician was a friend of my family and when I was about 15 years old he said to me; ‘Do you know that this theory of elementary particles is based on mathematics?’ He showed me a book full of formulas and equations I could not understand, but I realised that these were glimpses of this magic world that was hidden from me and this was portal into that world. It was love at first sight. What professional mathematicians do goes to the heart of reality, to the heart of the universe. It’s what enables us to learn how the world works.Can complex mathematical ideas be explained to someone who is not a mathematician?Yes. Take the idea of symmetry. In what sense is a round table more symmetrical than a square table? In a round table any rotation around the centre point preserves the table’s shape and location. But if I turn a square table at a random angle you can see the difference—only four rotations, by multiples of 90 degrees, will preserve it. Transformation of an object which preserves its shape and location is what we call a symmetry. Anyone can understand this. The way physicists learnt about elementary particles and were able to theorise their existence was precisely through a theory of symmetries. The Higgs Boson is this elusive particle recently discovered by the Large Hadron Collider in Geneva, and for this prediction two physicists were awarded the Nobel prize. That prize was really awarded for mathematical prediction because the origins of that theory are in symmetry.Symmetry exists without human beings to observe it. Does maths exist without human beings to observe it, like gravity? Or have we made it up in order to understand the physical world? I argue, as others have done before me, that mathematical concepts and ideas exist objectively, outside of the physical world and outside of the world of consciousness. We mathematicians discover them and are able to connect to this hidden reality through our consciousness. If Leo Tolstoy had not lived we would never have known Anna Karenina. There is no reason to believe that another author would have written that same novel. However, if Pythagoras had not lived, someone else would have discovered exactly the same Pythagoras theorem. Moreover, that theorem means the same to us today as it meant to Pythagoras 2,500 years ago.So it’s not subject to culture? This is the special quality of mathematics. It means the same today as it will a thousand years from now. Our perception of the physical world can be distorted. We can disagree on many different things, but mathematics is something we all agree on.
The only reason the theory means the same is that it describes the reality of the physical world, so mathematics must need the physical world. Not always. Euclidian geometry deals with flat spaces, such as the three-dimensional flat space. For millennia people thought we inhabited a flat, three-dimensional world. It was only after Einstein that we realised we lived in a curved space and that light doesn’t travel in a straight line but bends around a star. Pythagoras theorem is about geometric shapes in an idealised space, a flat Euclidian plane which, in fact, is not found in the real world. The real world is curved. When Pythagoras discovered his theorem there were, of course, inferences from physical reality, and a lot of mathematics is drawn from our experience in the physical world, but our imagination is limited and a lot of mathematics is actually discovered within the narrative of a hidden mathematical world. If you look at recent discoveries, they have no a priori bearing in physical reality at all.The naive interpretation that mathematics comes from physical reality just doesn’t work. The other interpretation that mathematics is a product of the human mind also has serious issues, because it seems clear that some of these concepts transcend any specific individual.Take Evariste Galois, who was killed in a duel at the age of 20. He came up with a beautiful theory on symmetry called Galois theory. His contemporaries didn’t get it but this theory now forms the core of modern mathematics. But what if the work had been burned? Would we never have known Galois theory? No. Someone else would have discovered it because it is inevitable.Because it is simply true? Yes. It’s a difficult philosophical question, to which we still don’t have the answer, but it’s an important question to be aware of. It’s not the same as the mathematics we use to calculate a tip—it goes to the heart of reality and of consciousness. It is all around us, with smart phones and computers and GPS devices and the algorithms that control our lives. The Amazon recommendations we are offered are based on very sophisticated algorithms, which analyse our past purchases, correlating us with other users. Mathematics is invading our world more and more and it communicates timeless, persistent and necessary truths which transcend time and space. The Langlands Programme should be as familiar to us as the theory of relativity.What is the Langlands Programme?Robert Langlands is a mathematician who occupies the office of Albert Einstein at the Institute of Advanced Studies in Princeton. His is one of the biggest ideas in the last 50 years. In the late 1960s he was able to find unexpected links between two fields of mathematics which seemed to be light years apart. One of them is number theory—this is something we can all relate to as it was driven by an impulse to understand the structure of numbers. We start with the simplest possible numbers 1,2,3,4, and so on, but then we realise that there are numbers that cannot be expressed as whole numbers or even as ratios of whole numbers, such as the square root of 2. It exists but doesn’t fit in the narrow framework. That’s one continent on this planet of mathematics. There are other continents—when we hear music we realise that the sound is composed of different instruments, of different notes. Each of those notes can be represented mathematically as a wave. In mathematics we call them harmonics and there is a deep theory that takes this further in which sophisticated signals can be deconstructed; this is called harmonic analysis.These theories are as far apart as Europe and America but what the Langlands Programme does is enable transportation between any point in Europe to any point in America and back. Langlands discovered some secret patterns relating to numbers, which could be expressed using harmonics. That was the initial idea, but since then other mathematicians took this up and found similar and surprising patterns in geometry and quantum physics, in particular the electromagnetic duality. It may all sound very complicated, but in my book I explain [electromagnetic duality] using an analogy based on the recipe of borscht, which is a favourite soup in my home country, Russia.Love and Math: The Heart of Hidden Reality. By Edward Frenkel. Basic Books; 304 pages; $27.99 and £18.99. Buy from Amazon.com. Buy from Amazon.co.uk.