Philosophy of mathematics in the twentieth century

Philosophy of mathematics in the twentieth century has primarily been shaped by three influences. The first of these is the work of Kant and, especially, the problematic he laid down for the subject in the late eighteenth century. The second is the reaction to Kant’s conception of geometry that arose among nineteenth-century thinkers and which centred first on the discovery of non-Euclidean geometries in the 1820s. The third is the new discoveries in logic that emerged with increasing rapidity and force during the latter half of the nineteenth century. In one way or another, the main currents of twentieth-century philosophy of mathematics-and, in particular, the so-called logicist, intuitionist and formalist movements-are all attempts to reconcile Kant’s revolutionary plan for mathematical epistemology with the equally revolutionary ideas of Gauss, Bolyai and Lobatchevsky in geometry, and the powerful ideas and techniques developed by Boole, Peirce, Peano, Frege and other nineteenth-century figures in logic.