The infinite is a concept which corrupts and upsets all others, said Borges. For the mathematician David Hilbert, the infinite is a topic that, more than any other, needs to be further clarified. Or, more prosaically, it is worth fifty ducats, the prize offered by the Berlin Academy of Sciences for a theory of infinity, one hundred years after Leibniz’s pioneering work on calculus. In mathematics one can do nothing without the infinite, which involves the natural numbers 1, 2, 3... or the relationship between the diagonal and a side of a square or regular pentagon, as discovered by the bewildered Pythagoreans. This book invites the reader on a journey to the infinite hidden in the folds of Zeno's paradoxes, Galileo’s reckonings, Archimedes’ method, 17th-century mathematicians’ indivisibles, the philosophies of Aristotle, Democritus, Bruno and Spinoza, Cantor’s paradise of transfinite numbers, and the paradoxes of set theory.

Umberto Bottazzini teaches History of Mathematics at the University of Milan.