The Sussex astronomer (Theories of Everything, 1991, etc.) has done it again--i.e., wrought a brilliant summation of ideas about mathematics that shows a depth of scholarship and an analysis that will leave the reader more than a little shaken. For example, Barrow traces the origins of counting and number systems in the Old and New Worlds (very much in parallel with John McLeish, reviewed below). His somewhat startling conclusion is that number systems do not arise like language, common to all human societies, but probably spread from one place to another. And thank God for the Indus culture, for without it--and without the Arabs who later spread the ""word"" about the decimal-place system and the zero--we might be stuck with Roman numerals. But Barrow's real point here is philosophical: Is mathematics a discovery or an invention; the ultimate description of reality or a form of abstract beauty in the eyes of the mathematician-logician beholder? Here, he points to the developments in the late-19th and early-20th centuries of the formalists like David Hilbert and the latter-day French descendants who called themselves the Bourbaki, eschewing all representations or models. Along came GÃ–del to pull the rug out from under, declaring the incompleteness of math and the undecidability of statements in axiomatic arithmetical systems. Barrow contrasts the formalists with constructivist-empiricists and today's ultimate hackers to conclude that there remains a residue of Platonic religious mysticism in our feelings about mathematics. ""All our surest statements about the nature of the world are mathematical statements, yet we do not know what mathematics 'is'...why it works nor where it works; if it fails or how it fails."" Heady stuff this, caviar for the connoisseur--but not for the innumerate.