This work is as much a wide-ranging study in the history of mathematics as it is a study of a central but notoriously obscure aspect of Plato's thought. By clarifying the differences between ancient Greek and modern mathematics, and particularly the different notions of number involved, Pritchard shows that our difficulty derives not so much from Plato's philosophy as from the fact that his is a philosophy of a mathematics which is essentially different from our own, both in its aims and in its objects. The picture of Plato's thinking which emerges turns out to be quite consistent with the mathematics of his time, and shows no sign of any revolutionary mathematical notions. Pritchard argues that Plato has no notion even of natural number, let alone of irrational numbers, as some have claimed on his behalf. Further, his mathematical philosophy in no way anticipates Fregean notions, as has also been claimed. This book should be of interest not only to students of ancient philosophy and historians of mathematics, but also to philosophers of (modern) mathematics.