Discussions about nominalism in the philosophy of mathematics in the 20th century started roughly with the work that W. V. Quine and Nelson Goodman developed toward constructive nominalism (Goodman and Quine 1947). But, as Quine later pointed out, in the end it was indispensable to quantify over classes (Quine 1960). As will become clear below, responses to this indispensability argument have generated a significant amount of work for nominalists. And it is the focus on the indispensability argument that largely distinguishes more recent nominalist views in the philosophy of mathematics, which I will focus on, from the nominalism developed in the early part of the 20th century by the Polish school of logic (Simons 2010).