The Foundations of Arithmetic

A short book of 1884 in which Frege asks what numbers are and how we know the truths of arithmetic, written in plain prose without the logical notation of his earlier Begriffsschrift. The Introduction sets out three principles: keep the psychological apart from the logical, ask for the meaning of a word only in the context of a sentence, and never lose sight of the difference between concept and object. Frege then criticises earlier views: that arithmetic rests on experience, as Mill held; on pure intuition, as Kant held; or on mental images and abstraction. He argues that a statement of number is an assertion about a concept: to say how many moons Jupiter has is to say something about the concept 'moon of Jupiter'. Numbers are objects, and Frege explains them through one-to-one correlation, the idea now called Hume's principle, and then defines them as extensions of concepts. He concludes that arithmetic is analytic, derivable from logic alone, the thesis known as logicism.

It is a model of clear philosophical argument and a founding text of analytic philosophy, and its context principle and account of number still frame the philosophy of mathematics.