This introductory chapter begins with a discussion of the philosophical problem posed by arithmetic: accounting for our knowledge of arithmetic proper in a manner that explains the relationship between it and its applications. It is argued that the challenge of accounting for the applicability of arithmetic to the world participates in the wider puzzle of explaining the link between experience, language, thought, and the world. It is therefore natural to try to resolve the issue concerning the grounding of the truths of arithmetic on one of four elements: that arithmetic is based not on experience itself but on a feature of the way we humans experience the world; that language itself supplies the necessary intuition; that what grounds arithmetic is a feature not of the way we experience the world but of the way we think; and that arithmetic is reducible to logic.
Keywords: mathematics, arithmetic, a priori, human experience, logic, reason