Kant and formalism: Hilbert, Russell and whitehead

A Kantian account of geometry would be couched in terms of spatial intuition-that ‘lazy limbo of mystery’, as Bertrand Russell called it. David Hilbert’s remarks in Foundations of Geometry go beyond a simple neutrality about Immanuel Kant’s philosophy: they suggest that he saw the Foundations as the carrying out of a Kantian project. In 1898, therefore, Russell reacted strongly against Kant about whom he never said another good word. This was an important part of the revolution in Russell’s philosophy that took place at the end of the 19th century, but it was only part. It is sometimes thought that Russell’s rejection of a Kantian philosophy of mathematics was immediately followed by his embrace of logicism. With the advantages of hindsight, it is natural to see Russell’s surprisingly formalist account of projective geometry in 1899 as a step on the way to logicism.