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“What kind of physical world do we live in?” asks John Barrow. “Will we always be able to find ever smaller, more elementary particles inside any that we have, like a never-ending sequence of Russian dolls? Or is there a limit, a smallest thing, a smallest size, or a shortest time, where division comes to a full stop?” And does any of it matter?

Barrow, who definitely believes it does, is a mathematical and literary phenomenon. Not only is he the author of 15 highly readable books on cosmology and mathematical physics but he has also been feted in Italy for his weird and hugely popular play Infinities. The production ran in Milan in 2002 to packed audiences, then transferred to Valencia before returning to Milan last year.

Among other things, the production (a coat-trailing exercise for this book) brought dramatically to life, with the aid of ingenious sets on five stages, a theory known as “the infinite replication paradox”. This argues that in a universe of infinite size (a proposition some academics find perfectly feasible) anything that has a probability of occurring must occur infinitely often. “Thus,” as Barrow explains, “at any instant of time — for example the present moment — there must be an infinite number of identical copies of each of us doing precisely what each of us is now doing.” He adds, as if to demonstrate that nothing is ever as simple as all that: “There are also infinite numbers of identical copies of each one of us doing something other than what we are doing at this moment.” Those who saw Martin Rees expounding on television before Christmas the theory of the multiverse (the idea of an infinite series of universes) may well feel that the concept of infinity is having its moment and that Barrow’s book is bang on time.

Infinity, Barrow is at pains to tell us, is by no means a new topic: it has been debated by philosophers for millennia, and the citations come tumbling out, from Aristotle to Aquinas, from Friedrich Nietzsche to Bernard Williams. Mathematicians have also wrestled with the notion of infinity down the centuries, turning their normally serene discipline into a kind of gangland warfare over the issue. In the first half of the 20th century, mathematicians were routinely dismissed from their posts and ostracised for their attempts to ban infinities from their calculations.

Infinity is very much a live issue today among physicists, although the appearance of infinities in their mathematics, Barrow tells us, “can be a warning that you have entered a blind alley on the road to the truth”. Within the realms of cosmology, scientists are contentiously absorbed by the question: is the universe finite or infinite — in size and in time? Moreover, researchers on the borderlands of philosophy, mathematics and artificial intelligence agonise over whether it is possible to create a computer that can perform an infinite number of tasks in a finite period of time — an apparently impossible paradox. Nor does Barrow neglect questions that intrigue non-specialists who like to ponder the far-fetched implications of future science.

For example, if it were possible to overcome human mortality, would we really welcome the idea of living forever? In his Milan play, Barrow presented dramatically a proposition he argues more closely in his book. A black stage set depicted immortal actors lazily passing their time reading or sitting perpetually under hairdriers, some of them engaged in stunningly boring monologues. Since infinity involves infinite replication and the collapse of diversity, to live forever would mean a hell of monotony. Then there are the implications of time travel, through worm holes, for example, and other such extravagances dreamt up by the cosmologists.

Again there are intransigent philosophical dilemmas, principally the one known as the grandma paradox that points out the impossibility of going back in time if it means you can shoot your grandmother.

No book on infinity would be complete without a discussion of the infinite as God, and Barrow rehearses, accessibly and critically, the so-called “ontological” proofs for God’s existence — from Anselm’s famous conundrum in the 11th century (God being above that of which no greater can be conceived) to Kurt Gödel’s proof in the 20th (published for the first time, it appears, in Barrow’s book). The “proofs” don’t work for us today, Barrow insists, as they depend on the notion of existence being a property, rather than “just a precondition for something”. He has good reasons for saying this, but there are some smart theologians who would argue that he has failed to make a distinction between contingent existence (the created) and necessary existence (God). But that debate is for another occasion and another venue.

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