Abstract

Mary Leng has published many spirited, insightful defences of mathematical fictionalism, the view that the claims of mathematics are not literally true. I offer as an alternative an anti-realist approach to mathematics that preserves many of Leng’s valuable insights while ridding fictionalism of its most unpalatable feature, the claim that substantive mathematical claims are “in error”. In making my argument, I first present the virtues of Leng’s fictionalism by considering how she defends it against influential objections due to John Burgess. Leng’s view is roughly that indispensability in science is necessary but not sufficient for believing in the reality of something, and that philosophical analysis can make clear why some things, including mathematics, are necessary for science but not real. I suggest we can accept this without adopting error theory. Marrying features of Leng’s view with constructivism, a quite different sort of anti-realism about mathematics, allows us to: maintain that mathematical assertions are literally true, but that it is a mistake to understand them as referring to abstract entities; to be anti-realists about mathematics; and to make use of the fictionalist toolkit Leng supplies for explaining why mathematics is indispensable, even if not real.