Popularisers of modern physics face a problem that is possibly insuperable. The mathematics that lies at the heart of the subject is too hard for a non-specialist to grasp, and yet, without the maths, the physics makes no sense. When one tries to express, say, quantum mechanics in words, one ends up with statements which, though written in ordinary language, defy comprehension: an electron is both a wave and a particle, subatomic particles can be in more than one place at the same time, and so on. For generations, we readers of popular physics have tried desperately to convince ourselves that we understand such sentences, but we don't really, and we won't until we really understand the theories of quantum physics. And that requires mastering mathematics that is beyond us.

Like most popularisers of science, Lee Smolin reacts to this challenge by just leaving out the maths. "There are no equations," he says in the preface, "and everything you need to know to follow my arguments is explained." But, as Smolin must surely know, without the equations it is impossible to convey everything one needs to know to follow his arguments. This is especially true in his case, because he is not popularising accepted theories of physics; he is putting forward a speculative new foundation for the whole of theoretical physics that challenges much accepted wisdom.

In order to assess the cogency of his ideas, or even to follow the gist of what he is saying, a lot of difficult stuff needs to have been mastered. One needs, for example, to understand the currently accepted "standard model" of particle physics (bosons, hadrons, fermions and all that). One also needs to have some grasp of quantum mechanics and relativity theory and why it has proved so difficult to bring them together to form a unified theory – the formidable problems, for example, of trying to understand the force of gravity (one of the four fundamental forces of the standard model) in a quantum mechanical way. One needs to understand all this, because Smolin's contributions to physics lie at the cutting edge of the discipline; one cannot understand them without understanding the questions that contemporary physicists are wrestling with. Among physicists, Smolin is best known for his work on "loop quantum gravity", a theory that cannot easily be made accessible to a non-specialist audience.

This has not stopped Smolin from trying. His three previous books – *Life of the Cosmos* (1997), *Three Roads to Quantum Gravity* (2001) and *The Trouble with Physics* (2006) – were, ostensibly at least, aimed at non-specialists. Like them, *Time Reborn* tries not only to convey an understanding of the difficulties faced by contemporary physicists, but also to advance novel solutions to those problems. As it turns out, however, at the heart of Smolin's proposed solutions to what he calls the "crisis of physics" is a philosophical view that should, in principle, be easier to grasp.

The problem here is that the philosophical view for which Smolin is arguing is not one that many non-physicists would find particularly controversial. It is that time is real, a position that Smolin describes as a "revolutionary view", but which, for most people, is just common sense. Of course time is real! For most of us, casting anxious glances at the mirror as the effects of time reveal themselves in the ageing process, it is all too real. To understand why this unexceptional, common sense assertion is regarded as revolutionary, one must, to some extent at least, understand how the world looks to modern physicists.

Fortunately, this can be done without mathematics. Einstein, in a letter to a bereaved friend, wrote (as if it provided some comfort): "People like us, who believe in physics, know that the distinction between past, present and future is only a stubbornly persistent illusion." As Smolin shows, Einstein was not being whimsical here; the view that the passing of time is an illusion is now the orthodoxy among theoretical physicists.

It is Smolin's view that the best hope for a solution to the difficulties that face contemporary physics – for example, the difficulties in bringing gravity into line with the rest of the currently accepted picture of reality – lies in overturning this orthodoxy and reaffirming the view that most of us non-physicists have anyway, namely that "nothing we know or experience gets closer to the heart of nature than the reality of time". In putting his case for it, Smolin says many things that are comprehensible and that, to me at least, seem both true and important.

Among those things is the idea (that Smolin advances brilliantly and persuasively) that the reason physicists have come to reject the reality of time is that they have been bewitched by the beauty and success of the mathematical models they use into mistaking those models for reality. For timelessness, though not really a feature of our world, is a feature of mathematics. Two plus two equals four, but if we ask when or for how long the perplexing (though true) answer seems to be: "Well, always. It is an eternal truth. Time is irrelevant to it." And thus we seem to be driven to accepting the thought that some truths, at least, are eternal. And, if we can have timeless truths in mathematics, why not in physics?

To think like this, Smolin claims, is to forget, or to deny, that the objects of mathematics – numbers, curves etc – do not exist, whereas physics concerns itself with what does exist, and, in reality, in the domain of things that do exist, time is inescapable. So, he insists: "Useful as mathematics has turned out to be, the postulation of timeless mathematical laws is never completely innocent, for it always carries a trace of the metaphysical fantasy of transcendence from our earthly world." He thus presents us with a choice: "Either the world is in essence mathematical or it lives in time."

Some of the most interesting chapters in this book are those in which Smolin traces the history of what the philosopher Edmund Husserl called the "mathematisation of Nature". For Smolin (as for Husserl) the key figures here are Galileo and Newton, the first for discovering that falling bodies are described by a simple mathematical curve, the second for showing that the force that impels those falling bodies along that curve is the same force that impels the earth along its path round the sun and that sends apples crashing from a tree. "By the time Newton had finished," Smolin says, "we lived in a single, unified world," a world "as eternal and divine as a mathematical curve".

Though Newton's theories of gravity were superseded by Einstein's, the world of general relativity, no less than that of Newton's laws of motion, is still, Smolin says, "represented by a mathematical object", and it still invites us to regard the world (mistakenly) as "timeless and pristine". Whether or not Smolin wins his argument with his fellow physicists, the case he makes for saying that when we deny the reality of time, we are confusing a mathematical model with what it is modelling seems to me convincing. Most of us may not need persuading that time is real, but this book goes some way towards explaining why there are those who do.