Version: v1, Published online: 2011

Retrieved January 23, 2019, from https://www.rep.routledge.com/articles/thematic/fictionalism-in-the-philosophy-of-mathematics/v-1

## Article Summary

Fictionalism in the philosophy of mathematics is the view that mathematical statements, such as ‘8+5=13’ and ‘*∏* is irrational’, are to be interpreted at face value and, thus interpreted, are false. Fictionalists are typically driven to reject the truth of such mathematical statements because these statements imply the existence of mathematical entities, and according to fictionalists there are no such entities. Fictionalism is a nominalist (or antirealist) account of mathematics in that it denies the existence of a realm of abstract mathematical entities. It should be contrasted with mathematical realism (or Platonism) where mathematical statements are taken to be true, and moreover are taken to be truths about mathematical entities. Fictionalism should also be contrasted with other nominalist philosophical accounts of mathematics that propose a reinterpretation of mathematical statements, according to which the statements in question are true but no longer about mathematical entities. Fictionalism is thus an error theory of mathematical discourse: at face value mathematical discourse commits us to mathematical entities; and although we normally take many of the statements of this discourse to be true, in doing so we are in error (cf. error theories in ethics).

Although fictionalism holds that mathematical statements implying the existence of mathematical entities are strictly speaking false, there is a sense in which these statements are true - they are true in the story of mathematics. The idea here is borrowed from literary fiction, where statements like ‘Bilbo Baggins is a hobbit’ is strictly speaking false (because there are no hobbits), but true in Tolkien’s fiction *The Hobbit*. Fictionalism about mathematics shares the virtue of ontological parsimony with other nominalist accounts of mathematics. It also lends itself to a very straightforward epistemology: there is nothing to know beyond the human-authored story of mathematics. And coming to know the various fictional claims requires nothing more than knowledge of the story in question. The most serious problem fictionalism faces is accounting for the applicability of mathematics. Mathematics, unlike Tolkien’s stories, is apparently indispensable to our best scientific theories and this, according to some, suggests that we ought to be realists about mathematical entities.

It is fair to say that there are serious difficulties facing all extant philosophies of mathematics, and fictionalism is no exception. Despite its problems fictionalism remains a popular option in virtue of a number of attractive features. In particular, it endorses a uniform semantics across mathematical and nonmathematical discourse and it provides a neat answer to questions about attaining mathematical knowledge. The major challenge for fictionalism is to provide an adequate account of mathematics in applications.

Colyvan, Mark. Fictionalism in the philosophy of mathematics, 2011, doi:10.4324/9780415249126-Y093-1. Routledge Encyclopedia of Philosophy, Taylor and Francis, https://www.rep.routledge.com/articles/thematic/fictionalism-in-the-philosophy-of-mathematics/v-1.

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