Jakob and Todd talk about set theory, its historical origins, Georg Cantor, trigonometric series, cardinalities of number systems, the continuum hypothesis, cardinalities of infinite sets, set theory as a foundation for mathematics, Cantor’s paradox, Russell’s paradox, axiomatization, the Zermelo–Fraenkel axiomatic system, the axiom of choice, and the understanding of mathematical objects as “sets with structure”.
Jakob and Todd discuss category theory, an important field in modern mathematics that focuses on the relations (morphisms) between mathematical objects. We discuss the importance of abstraction and the development in the history of mathematics beyond solving particular problems to studying the general nature of mathematical structures as such, the kinds of problems that can and can’t be solved, their properties, etc. We also consider the significance of a relation-centered approach to other fields, how things like languages, theories, and beliefs can be analyzed by the relations between their constituent elements.
For the visual aids referred to in the discussion see the video version on YouTube.