“To Create More Worlds”: Mathematical Practice as Philosophy

The primary aim of this chapter is to consider, by building on the preceding argument of this study, mathematicians’ working philosophy of mathematics as emerging in and defining actual mathematical thinking and practice, as exemplified in three cases: 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 the second figure—Galois, Riemann, and Grothendieck—in each case, my aim is not to juxtapose these figures. 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 also gives rise to these differences. As throughout this study, the chapter’s approach to their mathematical practice and to creative mathematical practice in general is defined by the invention of new concepts, a practice given a new dimension, that of “mathematical practice as philosophy,” in this chapter.