Roger Penrose is one of the most creative and original mathematical physicists in Britain. This remarkable book is the result of many decades of reflection on our scientific understanding of the physical universe. We have discovered some major things about the nature of reality over the past three hundred years. First, everything in the world around us, including ourselves, is made up of identical material particles with regular properties which obey physical laws. Second, these physical laws can be described well using mathematical equations, giving us the ability to make accurate predictions of the way things will be in the future and the way they were in the past. How this can be so is the topic of *The Road to Reality*. It also contains a deeply thought-out critique of much that is happening in present-day theoretical physics.

Penrose believes that since mathematics is the true language of physics, one cannot get to grips with the nature of physics without understanding the relevant maths. As a consequence, he has decided to use equations as appropriate throughout the book. This is to take the task of scientific explanation seriously in a way that few other semi-popular science writers do: Penrose tries to show what theoretical physicists do and how they do it, rather than merely describing the results of their work in picturesque language. If the book is more demanding than most, readers can rest assured that the difficulty is irreducible.

As it cannot be taken for granted that a reader will already know the necessary mathematics, the first third of the book contains a systematic presentation of it, from the bottom up. Starting with such foundations as Pythagoras’ theorem and the nature of number systems, Penrose goes on to introduce the elements of undergraduate mathematics, and then still more advanced topics.

You might think that because of this, anyone who is not either a professional scientist or a dedicated masochist should just lay the book aside and tackle something easier. Things aren’t that bad, however, both because the text is highly informative, and because the concepts are illustrated by splendid diagrams. One of the deep features of mathematics is that most ideas can be represented either geometrically, and so comprehended visually, or else analytically, i.e. represented through equations and inequalities. Many branches of mathematics are essentially ways of bridging between these two approaches. Readers can therefore grasp much of what is going on through the drawings, without having to struggle through the equations. But Penrose is also a skilled calculator, and presents the detailed calculations underlying many of the things he discusses: in the end, the broad ideas and pictures mean nothing if they don’t correspond correctly to the mathematical details.

All branches of mathematics involve abstraction, which entails a formalisation of representation and argument enabling the ‘proof’ of mathematical propositions. Abstraction occurs, for example, in the way the study of solutions of equations inexorably leads to the need to introduce negative numbers, irrational numbers such as pi (which can’t be represented as fractions), and complex numbers, involving the square root of –1 (which can be usefully represented in geometrical ways). Penrose marvels at the nature of complex numbers, claiming that ‘there is not only a special magic in the mathematics of these numbers, but . . . nature herself appears to harness this magic in weaving her universe at its deepest levels.’ Abstraction enables mathematics to deal with ever more general concepts: from numbers to variables to functions and on to ‘tensors’ and ‘spinors’, each associated with a suitable symbolic representation and a set of rules for manipulating those symbols.

Often there are quite different-looking ways of representing the same mathematical structure. There is then a hidden symmetry relating these different representations, i.e. they can be transformed into each other by a suitable change of variable. Consequently, there are a surprising number of ways of representing physical theories: classical mechanics can be described in terms of forces or fields, and quantum field interactions can be described according to Dirac equations or Feynman diagrams. Multiple viewpoints abound describing the same physical reality.

Physics can be thought of as comprising four major domains: classical physics (which underpins everyday phenomena and much of astronomy), relativity theory (associated with high energies and velocities, as well as very strong gravitational fields), quantum theory (associated with the very small), and quantum gravity (applicable only in extreme circumstances, such as those obtaining in the early universe or at the end of the life of a collapsing star). Each is associated with particular mathematical concepts, and in many ways the prime problem for theoretical physics is discovering the appropriate concept to use in each domain.

In the last part of the book, Penrose comments on three crucial issues in present-day theoretical physics, providing a good corrective to much recent writing that suggests we already have the answers in these areas. The key cosmological problem Penrose focuses on is the origin of the arrow of time. The fundamental physical laws are time-symmetric: for each solution of the equations, there is an equal and opposite solution in which the direction of time is reversed. For example, Maxwell’s equations for electromagnetism allow a radio signal to be received before it is transmitted, and in principle people can grow younger. The ‘arrow of time problem’ is that these things don’t happen: what is it that disallows the time-reversed solution? Penrose’s elegant discussion of this issue leads to the conclusion that there must have been special conditions in place at the start of the universe.

However, according to the popular inflationary universe theory, there was no such special state. It proposes that there was an extremely brief period of rapidly accelerating expansion, during which the universe quickly became hugely bigger and cooler. This appears to solve various problems, such as why the universe looks so similar whichever direction we turn our gaze. But Penrose asks whether the inflationary theory can achieve what it is supposed to achieve: could the process it describes have resulted in smoothness, starting from generic initial conditions, when the physical processes underlying this scenario depend on standard thermodynamics, which themselves would have applied only if the universe had started with very special initial conditions? Inflation does not remove the need for special initial conditions at the start of the universe, as is often claimed; rather, it presupposes them.

Penrose’s next target is the measurement problem in quantum theory. There is at present no consistent picture of the process of measurement that takes quantum physics into account. Usually, it is assumed that the measurement apparatus does not obey the rules of quantum theory, but this contradicts the presupposition that all matter is at its foundation quantum mechanical in nature. Penrose gives a clear description of the various ways this paradox can be handled, and how its resolution depends on what one believes about the nature of reality. Many physicists do not believe this is a question one should ask (‘I don’t demand that a theory correspond to reality because I don’t know what reality is,’ Stephen Hawking says), but Penrose believes that such matters are crucial to quantum mechanics and are far from being settled. He explains why standard approaches fail to solve the problem, and proposes instead that quantum measurement processes are associated with quantum gravity. This idea has not gained wide acceptance, but is undeniably deeply thought through.

Finally, Penrose discusses three approaches to quantum gravity, the as yet unestablished theory which, it is hoped, will one day unite Einstein’s theory of gravity (general relativity theory) with quantum theory. The first approach, string theory, is the most popular at present. It represents particles as two-dimensional sheets in versions of space-time with more than four dimensions, and supposes an as yet unproven physical symmetry (‘supersymmetry’) which implies the existence of many more kinds of particle than we have so far detected. Penrose gives a somewhat bemused description of its nature and achievements, together with a substantial critique of its methods and claimed results. The second approach is loop quantum gravity, which, unlike string theory, takes seriously Einstein’s notion that matter curves space-time. The third approach is Penrose’s own twistor theory, based on a complex geometrical description of particle motion and spin.

Is what is discussed in this book indeed ‘The Road to Reality’? Probably so. Penrose considers carefully the relation between evidence and the acceptance of physical theories, making a few wry comments on the role of fashion in theoretical physics. He remarks along the way that either ‘beauty’ or the discovery of ‘miracles’ (surprising hidden mathematical relationships) is often taken as a basis for selecting a physical theory when experimental data is unavailable or testing is impossible, and warns that neither amounts to conclusive proof of the validity of a physical theory. Present-day theoretical physics, he says, has a long way to go before we have a viable ‘theory of everything’, particularly because the foundations of quantum theory are unsatisfactory until the measurement problem is resolved. He proposes that ‘a “fundamental” physical theory that lays claim to any kind of completeness at the deepest levels of physical phenomena must also have the potential to accommodate conscious mentality.’ We have a long way to go in this regard, too.

As for the nature of what is real, Penrose argues that many mathematical phenomena (the numerical value of pi, for example, or the irrationality of the square root of two, or the existence of Mandelbrot figures) are discovered rather than invented. Consequently, he proposes that there is a Platonic reality to mathematics: that it exists in an abstract sense, independent of the human mind, and that we must recognise three different kinds of ‘world’ representing three different forms of existence – the physical, the mental and the mathematical. His discussion of the nature of these different kinds of realities and the relations between them is one of the highlights of this book.

Does he provide ‘A Complete Guide to the Laws of the Universe’? In discussing the fundamental physical laws and their underlying mathematics, he comes close. With regard to how complexity and life arise out of this physics, however, the answer is no; in that sense, a theory of everything remains elusive.

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