[3] In proving his theorem, Penrose pioneered applications of the branch of mathematics known as topology (from the Greek words topos and logos, meaning “place

[3] In proving his theorem, Penrose pioneered applications of the branch of mathematics known as topology (from the Greek words topos and logos, meaning “place” and “reason”), a discipline often regarded as part of geometry. But whereas ordinary geometry is about distances, shapes and curvatures, topology is rather different – it is about the ways in which points, curves and surfaces can be connected to each other. A few mathematicians sowed the seeds of the subject in the seventeenth century, but they bore fruit in mainstream mathematics only in the early twentieth century. Theoretical physicists cottoned on to its utility in the real world a few decades later, and began to use it to understand magnetism, the behaviour of particles deep inside atoms and a host of other topics. Penrose was the first to use topology to recast our understanding of gravity. As the American theorist Kip Thorne has pointed out, modern topology explained why there must be an end of space-time somewhere inside a black hole. Penrose demonstrated that singularities are unavoidable features of Einstein’s theory, indicating that the theory contained the seeds of its demise. Ultimately, physicists needed a better, singularity-free theory, though that was much easier said than done. Many of Penrose’s mathematical peers were sceptical of his methods, and many physicists were slow to appreciate his innovations. Predictably, Dennis Sciama immediately grasped their importance and encouraged his group to study them closely. In early 1965, when Penrose gave a talk in Cambridge on his theorem, Sciama made a point of introducing him to his student Stephen Hawking, who was likely to die within months if the diagnosis given to him two years before was correct. Defying the medical odds, Hawking was thriving, exploring Einstein’s theory and learning advanced mathematics, while taking a keen interest in the observations made by astronomers. The Americans Arno Penzias and Robert Wilson made the most impact by detecting radiation that appeared to be the afterglow of a “big bang” billions of years before. This discovery established cosmology as a scientific discipline. Soon, many astronomers – including Sciama and his group – had abandoned the “steady state” picture of the universe and began to espouse the big bang picture. Fred Hoyle was one of the most notable holdouts and usefully put pressure on the new orthodoxy, until his opposition to it became not so much reasoned as cranky. In the final chapter of Hawking’s PhD thesis, he drew on this discovery and on Penrose’s techniques. By extrapolating Penrose’s methods backwards in time, Hawking demonstrated that according to Einstein’s theory of gravity the universe began in an extremely hot and dense state, tying in perfectly with the concept that space and time originated in a big bang. Before long, theoreticians were probing the universe billions of years ago, in the first three minutes of its existence. This was the beginning of one of the epic scientific adventures of the past fifty years. Penrose and Hawking worked separately to study singularities in Einstein’s theory, but joined forces two years later to write their classic account of what became known as the Penrose-Hawking singularity theorems. Having stood on the shoulders of Penrose, Hawking had matured into a world-class theoretical physicist. His greatest discovery, made a few years later, was that black holes are not actually black but continuously emit radiation, a prediction he made by ingeniously jamming together Einstein’s theory of gravity and quantum mechanics. The two theories are quite different, couched in different mathematics, and are notoriously difficult to bring together, let alone to unite. Penrose remained friends with Hawking, but took a different path, working alone on a wide variety of challenges, usually at the interface of mathematics and physics. The relationship between the two disciplines has long fascinated Penrose. In essence, they are both about patterns: mathematics is about patterns of ideas, whereas physics about the underlying order of the universe, the patterns connecting quantities that observers can measure quantitatively. For centuries, mathematics has – in the apt words of the American theorist Eugene Wigner – been “unreasonably effective” in studies of physics. Over the past fifty years, it has become no less clear that research at the frontiers of fundamental physics is unreasonably effective in mathematics. No one fully understands this but, as Penrose has said, it seems to be “no accident”. Among Penrose’s dozens of other contributions to mathematical physics, the most radical is twistor theory, a new approach to fundamental physics. Twistors are mathematical objects – Penrose still calls them his “babies” – from which ordinary space and time emerge as secondary concepts. His approach did not catch on in the physics community, partly because the theory is hard to grasp and is yet to enable physicists to solve a problem that they could not address by more conventional means. Twistors have nonetheless proved useful in several parts of physics, most recently in facilitating a fresh understanding of how fundamental particles scatter off each other. Unconcerned by the unpopularity of his babies, Penrose remains optimistic that they will eventually form the basis of a better understanding of nature via a new perspective on quantum mechanics and Einstein’s theory of gravity. He told me in 2014: “From the word go, I was confident that the idea was too beautiful not to have a place in the great scheme of the natural world”. As do all mathematically-minded researchers in physics, Penrose values above all beauty in a new theory, whether or not it appears at first to account for observations and measurements. In the long run, however, a theory stands or falls by its ability to explain observations and give quantitative agreement with the results of experiments. Penrose has been unimpressed with most of the developments in theoretical physics and cosmology that have caused the most excitement among his peers over the past forty years. One example is string theory, which began to attract a huge following in the mid-1980s. Several of the world’s leading physicists have invested the best years of their working lives in this approach to unifying gravity and the other forces of nature. String theorists achieved impressive results. They shed new light on the structures of mathematical physics and helped to make key discoveries in pure mathematics. Largely as a result of this success, the frontiers of physics and pure mathematics have rarely been more intertwined. Although string theory has not received the support of experiment, it is still the most popular unified framework for understanding the fundamental entities of nature. Penrose is unmoved by this optimism. He believes that because the string framework makes sense only when it is set out in a high number of dimensions, the approach is doomed. Another of his bêtes noires is the idea that the universe expanded briefly at a prodigious rate soon after it began. This concept of cosmic inflation, first proposed around 1980, gives plausible explanations of several observations that were otherwise beyond the ken of astronomers, including the large-scale structure of the universe. Most cosmologists enthusiastically embraced this radical concept, though Penrose thought it was so horrible, and so ugly, that it made him feel ill. In 2016, Penrose set out his views on contemporary theoretical physics in “Fashion, Faith and Fantasy in the New Physics of the Universe”, the most recent of the books he has written for a general audience. None of them is easy reading, but they are all bracingly ambitious. They address big questions – including the nature of consciousness and the nature of reality – and are replete with attractive illustrations that often make even the most obscure arguments palatable.