Abstract: This book is a critical examination of the astonishing progress made in the philosophical study of the properties of the natural numbers from the 1880s to the 1930s. It reassesses the brilliant innovations of Frege, Russell, Wittgenstein, and others, which transformed philosophy as well as the understanding of mathematics. The book argues that through the problem of arithmetic participates in the larger puzzle of the relationship between thought, language, experience, and the world, we can distinguish accounts that look to each of these to supply the content we require: those that involve the structure of our experience of the world; those that explicitly involve our grasp of a ‘third realm’ of abstract objects distinct from the concrete objects of the empirical world and the ideas of the author's private Gedankenwelt; those that appeal to something non-physical that is nevertheless an aspect of reality in harmony with which the physical aspect of the world is configured; and finally those that involve only our grasp of language.

Keywords: mathematics, arithmetic, Kant, Frege, Russell, Wittgenstein, Deekind, Carnap, Ramsey, Gödel