Varieties of Mathematical Understanding

Explore the diverse facets of mathematical understanding in this 55-minute lecture by Jeremy Avigad from Carnegie Mellon University, delivered at the Fields Institute as part of the 2022 Fields Medal Symposium honoring Akshay Venkatesh. Delve into conceptual understanding and examine challenging mathematical problems, including Hales' theorem, packing tetrahedra, higher-dimensional sphere packing, and the Keller conjecture. Analyze reactions to these challenges and trace the historical development of mathematical concepts, such as the inverse square law. Gain insights into how views of understanding evolve over time, the importance of incorporating mathematical history, and the value of philosophical reflection in mathematics. Leave with a renewed sense of optimism about the field's ongoing progress and the multifaceted nature of mathematical comprehension.

Intro
Conceptual understanding
Challenges: Hales' theorem
Challenges: packing tetrahedra
Challenges: higher-dimensional sphere packing
Challenges: the Keller conjecture
Reactions: packing tetrahedra
Reactions: the Kepler conjecture
Reactions: higher-dimensional sphere packing
History: the inverse square law
Views of understanding change
change happens for good reasons
mathematics incorporates its past
the value of history
the value of philosophical reflection
optimism