Phil Beriault

This volume contains thirteen papers that were presented at the 2017 Annual Meeting of the Canadian Society for History and Philosophy of Mathematics/Société canadienne d’histoire et de philosophie des mathématiques, which was held at Ryerson University in Toronto. It showcases rigorously reviewed modern scholarship on an interesting variety of topics in the history and philosophy of mathematics from Ancient Greece to the twentieth century. A series of chapters all set in the eighteenth century consider topics such as John Marsh’s techniques for the computation of decimal fractions, Euler’s efforts to compute the surface area of scalene cones, a little-known work by John Playfair on the practical aspects of mathematics, and Monge’s use of descriptive geometry. After a brief stop in the nineteenth century to consider the culture of research mathematics in 1860s Prussia, the book moves into the twentieth century with an examination of the historical context within which the Axiom of Choice was developed and a paper discussing Anatoly Vlasov’s adaptation of the Boltzmann equation to ionized gases. The remaining chapters deal with the philosophy of twentieth-century mathematics through topics such as an historically informed discussion of finitism and its limits; a reexamination of Mary Leng’s defenses of mathematical fictionalism through an alternative, anti-realist approach to mathematics; and a look at the reasons that mathematicians select specific problems to pursue. Written by leading scholars in the field, these papers are accessible to not only mathematicians and students of the history and philosophy of mathematics, but also anyone with a general interest in mathematics.

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.

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.