Reconciling the Realist/Anti Realist Dichotomy in the Philosophy of Mathematics

In the philosophy of mathematics, the realist vs. anti-realist debate continues today with differing positions on the status of mathematical objects. For realists, objects sit in “Plato’s heaven”, immovable, objective, eternal, and we contemplate them, whereas anti-realists (or Constructionists) are the opposite, and emphasize epistemology over ontology, saying that we construct mathematical objects. There are numerous results in mathematics which can be arrived at both from a realist and an anti-realist viewpoint. In other words, they can be contemplated (proved) via methods deemed unsuitable by anti-realists- or simply arrived at it through methods (or construction) as the anti-realist would say. In this chapter, we argue that realism and anti-realism can be seen as two sides of the same coin, or different ways of knowing the same thing, and therefore the so called dichotomy between these positions is reconcilable for particular mathematical objects.