Define the universe, and give two examples.

If that kind of challenge makes your head hurt, then science explainer and Cambridge mathematician Barrow’s latest excursus into the unusual (*The Constants of Nature*, 2003, etc.) will send pain receptors into overdrive. Infinity is a hard enough notion to grasp, the kind of thing that set many a smoky dorm room into far-out reveries back in the day. But what if there are multiple infinities? What if there are many different series of things that have no end, mathematically and logically? What if, as the Indian mathematician once said, below the seemingly endless chain of turtles that held the world up in the sky there were simply more turtles? That’s the kind of talk, Barrow writes, that once “made mathematicians very nervous about infinities. It is easy to see why infinity was regarded as a form of logical plague that destroyed the reliability of everything it touched.” Yes, it is, for infinity is a great underminer of certainty. Barrow has more questions for us to entertain: “Does the Universe have an ‘edge’ or is it simply unbounded in size?” “Is infinity just a shorthand for ‘finite but awfully big’?” Mathematicians, logicians, cosmologists, philosophers and physicists have been preoccupied by such conundrums for a very long time, and some of them, such as the 19th-century German scholar Georg Cantor, became “corrupters of youth” (as one of Cantor’s enemies charged) by showing that infinity was not just a potential but a process—and one, incidentally, that might lead to the doorway of God. (God’s infinity, Cantor said, was different from mathematical infinity and physical infinity. Let the headaches commence.) Mathematicians now take the idea of infinity/infinities for granted. Barrow is a lucid and sometimes even lyrical explainer, and nonspecialist readers with a liking for the history of science and the progress of human thought will find these pages to be most accessible. Prospective time-travelers, too, will want to brush up on the math toward the back of the book.

Another pleasing popularization of science from an old hand.