In 1945, the turn to logic and language that initiated the analytic tradition in philosophy was sixty-six years old. The tradition was founded in 1879 when Gottlob Frege invented the predicate calculus as a necessary prerequisite to his goal of deriving all mathematics (except geometry) from logical axioms and definitions of mathematical concepts. His aim – to identify what numbers are and explain our knowledge of them – fit what he, Bertrand Russell, and G. E. Moore then took to be the main tasks of philosophy – to give a general description of reality, to explain what, and how, we know about it, and to discern moral facts capable of guiding action.1 One part of reality, numbers, were, for Frege, whatever they had to be to explain our arithmetical knowledge. His explanation was based on taking natural numbers to be sets of concepts the extensions of which can exhaustively be paired off, without remainder. Related definitions of arithmetical notions allowed him to derive the axioms of Peano Arithmetic from what he took to be self-evident logical axioms, without which thought might prove impossible. In this way, he thought, he could reduce arithmetical knowledge to logical knowledge.