IF WE ever establish contact with intelligent aliens living on a planet around a distant star, we would expect some problems communicating with them. As we are many light years away, our signals would take many years to reach them, so there would be no scope for snappy repartee. There could be an IQ gap and the aliens might be built from quite different chemistry. Yet there would be much common ground too. They would be made of similar atoms to us. They could trace their origins back to the big bang 13.7 billion years ago, and they would share with us the universe’s future. However, the surest common culture would be mathematics. Mathematics has been the language of science for thousands of years, and it is remarkably successful. In a famous essay, the great physicist Eugene Wigner wrote about the “unreasonable effectiveness of mathematics”. Most of us resonate with the perplexity expressed by Wigner, and also with Einstein’s dictum that “the most incomprehensible thing about the universe is that it is comprehensible”. We marvel at the fact that the universe is not anarchic – that atoms obey the same laws in distant galaxies as in the lab. The aliens would, like us, be astonished by the patterns in our shared cosmos and by the effectiveness of mathematics in describing those patterns. Mathematics can point the way towards new discoveries in physics too. Most famously, British theorist Paul Dirac used pure mathematics to formulate an equation that led to the idea of antimatter several years before the first antiparticle was found in 1932. So will physicists’ luck hold as they aim to probe still deeper levels of structure in the cosmos? Are limits set by the intrinsic capacity of our brains? Can computers offer insights, rather than just crunch numbers? These are some of the questions that exercise me. The precedents are encouraging. The two big breakthroughs in physics in the 20th century owed much to mathematics. The first was the formulation of quantum theory in the 1920s, of which Dirac was one of the great pioneers. The theory tells us that, on the atomic scale, nature is intrinsically fuzzy. Nonetheless, atoms behave in precise mathematical ways when they emit and absorb light, or link together to make molecules.