Categorical Structuralism and the Foundations of Mathematics
Joost Vecht
Abstract:
Structuralism is the view that mathematics is the science of structure. It has been noted that category theory expresses mathematical objects exactly along their structural properties. This has led to a programme of categorical structuralism, integrating structuralist philosophy with insights from category theory for new views on the foundations of mathematics.
In this thesis, we begin by by investigating structuralism to note important properties of mathematical structures. An overview of categorical structuralism is given, as well as the associated views on the foundations of mathematics. We analyse the different purposes of mathematical foundations, separating different kinds of foundations, be they ontological, epistemological, or pragmatic in nature. This allows us to respond to both the categorical structuralists and their critics from a neutral perspective. We find that common criticisms with regards to categorical foundations are based on an unnecessary interpretation of mathematical statements. With all this in hand, we can describe “schematic mathematics”, or mathematics from a structuralist perspective informed by the categorical structuralists, employing only certain kinds of foundations.