A mathematical object is an object studied by mathematics. This notion gets involved as soon as one asks the question what mathematical objects consist of, or—more generally—what it does mean for such an object to “exist”. A further question, depending on the answer to the first, is how we can know about mathematical objects.
From a formalist viewpoint, a mathematical object is a term in whatever system of formal logic we happen to be using at the moment. We can also consider types as mathematical objects by interpreting them as terms in some universe.
Using the Platonic viewpoint, one can identify mathematical objects that do not fit the formalist paradigm.
A good example is the notion of an (∞,1)-category, which exists at the Platonic level, and whose “shadows” (using Plato’s terminology) in the world of rigorous mathematics have been exhibited as various models of (∞,1)-categories: relative categories, quasicategories, Segal categories, complete Segal spaces, simplicial categories, etc.
However, the concept of (∞,1)-categories per se resists formalization in a satisfactory way. Homotopy type theory may succeed one day on this path, but so far no working definition has been exhibited yet.
For an overview of Platonism in the philosophy of mathematics see
Last revised on February 1, 2021 at 02:51:44. See the history of this page for a list of all contributions to it.