Mathematical Practice as Philosophy, with Galois, Riemann, and Grothendieck

Abstract

The primary aim of this chapter is to consider mathematicians’ working philosophy of mathematics as emerging in and defining actual mathematical thinking and practice, as exemplified in three cases stated in my subtitle: Niels Henrik Abel and Évariste Galois, Nikolai Lobachevsky and Bernhard Riemann, and André Weil and Alexander Grothendieck. I speak of “and” and hence “conjunction,” rather than the disjunctive “or,” in all three paired cases considered in this chapter because, while my primary concern is on the (historically) second figure – Galois, Riemann, and Grothendieck – in each case, my aim is not merely to juxtapose these figures, especially given the significance of their thinking for transforming mathematics. The work of Abel, Lobachevsky, and Weil were revolutionary events as well. Instead, while granting the differences between their thinking, my aim is to explore the shared grounding that gives rise to these differences, defining, and defined by, their mathematical practice as philosophy. The chapter’s approach to their mathematical practice and to creative mathematical practice in general is parallel to that of Deleuze and Guattari in creative philosophical practice, which or even true philosophy itself is defined by them as the practice of the invention of new concepts, with philosophical concepts given a particular definition, in part in juxtaposition to mathematical and scientific concepts. The working philosophy of mathematics this chapter considers, under the heading of “mathematical practice as philosophy,” is, analogously, defined as the practice of the invention of new mathematical concepts, defined, against the grain of Deleuze and Guattari’s argument, in affinity with (although not identically to) philosophical concepts in their definition.