He is refering to the late 19th and early 20th century work on the foundations of mathematics amoung mathematicans and philosophers. They attempted to fully formalise mathematics. Serveral approaches and schools in mathematics all hit various paradoxes, often of the same kind, and this ultimately culminated in Godel's incompleteness theorems:

https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_...

https://plato.stanford.edu/entries/goedel-incompleteness