Albert Lautman : dialectics in mathematics

Albert Lautman (1908-1944) is a rare example of a twentieth-century philosopher whose engagement with contemporary mathematics goes beyond the ‘foundational’ areas of mathematical logic and set theory. He insists that (what were in his day) the new mathematics of topology, abstract algebra, class field theory and analytic number theory have a philosophical significance that distinguishes them from the mathematics of earlier eras. Specifically, these new areas of mathematics reveal underlying dialectical structures not found in earlier mathematics. In a series of short papers and two longer theses (Essay on the unity of the mathematical sciences in their current development and Essay on the notions of structure and existence in mathematics 1), Lautman argues this claim from a philosophical perspective rooted in certain of the later dialogues of Plato. However, Lautman was not satisfied with Plato’s conception of the relation between dialectical Ideas and the matter in which they are realised. In one of his last papers, ‘New research on the dialectical structure of mathematics’2, Lautman bolsters his Platonism with an appeal to Heidegger’s ‘ontological’ distinction between phenomenology and science.3 We may therefore regard this paper as the most advanced expression available of Lautman’s philosophy of mathematics. In this paper, I shall first explore Lautman’s conception of dialectics by a consideration of his references to Plato and Heidegger. I shall then compare the dialectical structures that he found in contemporary mathematics with the model that emerges from his philosophical sources. I shall argue that the structures that he discovered in mathematics are richer than his Platonist model suggests, and that Heidegger’s ‘ontological’ distinction is less useful than Lautman seemed to believe.