My aim is to understand the practice of mathematics in a way that sheds light on the fact that it is at once a priori and capable of extending our knowledge. The account that is sketched draws first on the idea, derived from Kant, that a calculation or demonstration can yield new knowledge in virtue of the fact that the system of signs it employs involves primitive parts (e.g., the ten digits of arithmetic or the points, lines, angles, and areas of Euclidean geometry) that combine into wholes (numerals or drawn Euclidean figures) that are themselves parts of larger wholes (the array of written numerals in a calculation or the diagram of a Euclidean demonstration). Because wholes such as numerals and Euclidean figures both have parts and are parts of larger wholes, their parts can be recombined into new wholes in ways that enable extensions of our knowledge. I show that sentences of Frege's Begriffsschrift can also be read as involving three such levels of articulation; because they have these three levels, we can understand in essentially the same way how a proof from concepts alone can extend our knowledge.
