Philosophy of mathematics in the twentieth century

Philosophy of mathematics in the twentieth century has primarily been shaped by three influences. To understand twentieth-century philosophy of mathematics, it is therefore necessary first to have some knowledge of Kant's ideas and of the ideas that were at the heart of the nineteenth-century reactions to his views. It also became a prime force shaping the major movements of twentieth-century philosophy of mathematics. Like Frege's logicism, the intuitionism of the early part of this century was also dominated by the idea that what the mind brings forth purely of itself cannot be hidden from it; and the belief that the existence of non-Euclidean geometries reveals important epistemological differences between geometry and arithmetic. Hilbert consciously adopted a conception of mathematical knowledge that was more in keeping with what he thought of as the ideal of objectivity. Logicism re-emerged in the 1930s and 1940s as the favoured philosophy of mathematics of the logical empiricists.