# Xah Talk Show 2019-08-07 story of math crisis, math foundation, proof theory, and homotopy type theory

At 1:02:45, my explanation of “law of excluded middle” is incorrect. What i said there is “principle of bivalence”, which mean a proposition is either true or false. Law of Excluded Middle is the principle for believing the validity of Proof By Contradiction, at 1:14:16. Intuitionist do not believe Law of Excluded Middle, thus do not buy any proof done by contradiction.

topics covered:

math crisis in history:

- discovery of irrational numbers.
- discovery of negative number?
- discovery of complex numbers.
- discovery of non-euclidean geometry.
- discovery of countable and non-countable infinity.
- godel's theory

schools of thought:

- formalism
- logicism
- intuitionism (contructivism)

bleeding edge: homotopy type theory