Galileo famously wrote that the book of nature is written in the language of mathematics, and physicists since Isaac Newton have taken advantage of what Eugene Wigner called ¡°the unreasonable effectiveness of mathematics in the natural sciences¡±, using mathematics to understand the physical world. Conversely, pure mathematicians seek mathematical beauty, working in an abstract Platonic world of pure thought?that seems to many to be unencumbered by the realities of the universe in which we live. And there is (perhaps surprisingly) general agreement about what is beautiful in mathematics. But is mathematical beauty a guide to the truth of ideas about the workings of the universe?

In this passionate book, Graham Farmelo first shows how the great triumvirate of Isaac Newton, James Clerk Maxwell and Albert Einstein developed elegant mathematical theories, each of which enormously advanced our understanding of the natural world, and then discusses the role of mathematics in contemporary physics, giving an account of the history of quantum theory and string theory.

Farmelo is interested in whether mathematics is simply a tool for interpreting experiments or whether physicists should seek inspiration in pure mathematics rather than in experiment. Does the mathematical elegance of a theory matter? Does the remarkably beautiful mathematics underlying string theory testify to its correctness even in the absence of experimental evidence to support it? Einstein, originally uninterested in higher mathematics, changed his mind when working on his theory of gravity, coming to believe that, in Farmelo¡¯s words, ¡°theoreticians should not focus on the results of new experiments but instead use pure thought, guided by advanced mathematics¡±. Key to Farmelo¡¯s account is a talk on ¡°The relation between mathematics and physics¡± by Paul Dirac (of whom he has written an acclaimed biography), delivered to the Royal Society of Edinburgh in 1939, in which the pioneer of quantum theory advocated that physicists should seek to make use of beautiful mathematics: a principle?that certainly applies to the development of today¡¯s string theory.

However, Dirac¡¯s suggestion was not immediately adopted by most of his colleagues.?After the early successes of quantum theory, there followed what Freeman Dyson calls ¡°the long divorce¡± in which physics and pure mathematics went their separate ways. But from the late 1970s, theoretical physicists have increasingly pursued advanced new ideas in abstract pure mathematics, leading to powerful results in both subjects. Indeed the physicist Edward Witten was awarded the Fields Medal ¨C the top international award for mathematicians ¨C in 1990, reflecting the major contributions he has made to mathematics, even if his methods do not meet mathematicians¡¯ normal expectations of rigorous proof.

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The author is equally at home in mathematics as in physics. For example, he provides an insightful account of the rise and fall of the Bourbaki mathematical collective, whose emphasis on rigour and abstraction powered a mathematical revolution in the middle of the 20th century but whose approach is now less fashionable. His later chapters show the recent interplay between mathematics and physics, and the growth of interest in ¡°physical mathematics¡± ¨C the pure mathematics being used in contemporary physics ¨C as physicists attempt to improve on the Standard Model of quantum theory?that emerged in the 1970s.

We are told how the beauty of the mathematics has convinced many physicists of the importance of string theory, although the theory is not yielding the testable predictions that were traditionally a requirement for any new physical idea. Unexpectedly leading physicists into highly abstract areas of pure mathematics, like group theory and topology, the new ideas initially seemed to promise breakthroughs in the unification of modern physics. But new experimental evidence has not given theorists anything more to work with ¨C even the Large Hadron Collider has not yet produced anything to challenge the Standard Model.

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The failure of string theory to deliver fully, so far, on its original promises has led some to question whether Dirac¡¯s vision is the right way forward. A prominent critic of string theory is Peter Woit, whose work in the scientific literature, popular book Not Even Wrong: The Failure of String Theory and Continuing Challenge to Unify the Laws of Physics (2007) and blog of the same name argue that the focus on a kind of physics that cannot be tested because it makes no predictions is damaging for the future public opinion of science. And recently Sabine Hossenfelder has written Lost in Math: How Beauty Leads Physics Astray (2018), arguing, as the title indicates, that mathematical beauty should not be a measure of the quality of a physical theory: there is no reason to suppose that the laws that determine the workings of matter should conform to expectations of mathematical elegance, so such a focus is preventing physicists from finding ways forward.

Farmelo¡¯s book is a response to these contrarians. He is confident that the beauty of the mathematics is significant in indicating that we are on the right track, and that eventually, even if we have to wait for many years, we will be able to test string theory against new experimental evidence. As a spirited defence of the idea that beautiful mathematics should be a guide for physicists, Farmelo¡¯s book is a timely response to critics such as Woit and Hossenfelder, defending what science writer Jim Baggott has called ¡°fairy-tale physics¡±. Ultimately I am not sure, however, that he makes his case anything more than a matter of faith. Beauty inspires pure mathematicians, and striking recent developments in physical mathematics have emerged from the recent coming together of the two subjects, but despite the past successes of Newton, Maxwell and Einstein, whether mathematical beauty will be the best guide to new physical truth in the coming years remains to be seen.

There is a big problem for anyone attempting to write a popular book about string theory. The subject requires too much background knowledge for any short explanation to convey more than a very broad outline of the details. Consequently, the later chapters of Farmelo¡¯s books tell us a lot about the people who have made breakthroughs, but the details of these discoveries are inevitably rather vague. We learn about various dualities that have been discovered and the excitement that these created, but although our author gives us a sense of why these are important clues to something deep that is not yet understood, it¡¯s hard for a non-specialist reader to have any real sense of what is going on.

One of the strengths of this book is the insights Farmelo offers into the thinking of the leading figures of modern physics and pure mathematics ¨C the late Sir Michael Atiyah, Freeman Dyson, Sheldon Lee Glashow, Sir Roger Penrose and Nima Arkadi-Hamed. A great many more were also interviewed, and the result gives a valuable panorama of how today¡¯s top physicists see string theory.

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Our modern understanding of the quantum world is one of the great intellectual triumphs of human history. String theory is giving us wonderful developments in pure mathematics, even if it were to turn out to be a dead end for physics. Despite the difficulties of writing about such specialised technical material, Farmelo has succeeded in writing a book for the general reader?that gives insights into the motivation behind a theory developed by many of today¡¯s leading thinkers. His book provides as clear an account of the subject as I can imagine for a non-specialist reader.

Tony Mann is director of the Greenwich Maths Centre at the University of Greenwich.

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The Universe Speaks in Numbers: How Modern Maths Reveals Nature¡¯s Deepest Secrets
By Graham Farmelo
Faber and Faber, 336pp, ?20.00
ISBN 9780571321803
Published 16 May 2019

Graham Farmelo is a fellow of Churchill College, Cambridge, though he often spends time at the Institute for Advanced Study in Princeton. He was born in Camberwell, London, but spent his early years in Orpington, Kent, and studied at the University of Liverpool.

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It was there, he recalls, that he ¡°fell in love with theoretical physics, learned the ropes and started to write and talk about it publicly¡±. After producing science articles and reviews for the British national press, as well as New Scientist and Nature, a publisher suggested he should try his hand at a popular science book. This initially led to his edited collection, It Must be Beautiful: Great Equations of Modern Science (2002), to which he contributed a chapter on ¡°The Planck-Einstein Equation for the Energy of a Quantum¡±. He then went on to his celebrated biography, The Strangest Man: The Hidden Life of Paul Dirac (2009), winner of both the Costa Biography Award and the Los Angeles Times Science Book Prize, and Churchill¡¯s Bomb: A Hidden History of Science, War and Politics (2013).

In his new book, Farmelo has returned to the beauty of equations and key developments in contemporary science. But why should the general reader be interested in something both as fiendishly complex and as controversial as string theory?

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¡°String theory is the basis of the most promising framework we have for understanding nature at the finest level,¡± replies Farmelo, ¡°and for answering truly fundamental questions such as ¡®Why does gravity exist? What are the true natures of space and time?¡¯ I think many people would appreciate
having some insights into this great intellectual adventure, one of the imaginative enterprises that goes some way to redeeming our species.¡±

Matthew Reisz