(1) The examples he gives of "non-mathematical" concepts are pretty mathematizable. Randomized controlled testing -- mathematics can prove why this is a good id

(1) The examples he gives of "non-mathematical" concepts are pretty mathematizable. Randomized controlled testing -- mathematics can prove why this is a good idea. Occam's razor - usefulness can be formalized in the machine learning context. Different conceptions of justice, fairness, etc. can be modeled mathematically (at least certain aspects of them can), which helps one understand and distinguish them. (Examples: utilitarianism, equality of outcome versus equality of opportunity.) (2) The question of "functionally equivalent" is interesting here. If an alien species accurately predicts the trajectory of a complex rocket, many of us would say they must be using math, because we know that our mathematics governs how rockets move. Even if they cannot communicate their method in a way we can understand. Economists sometimes use the phrase "as if", e.g. the alien acts "as if" they have a utility function, regardless of whether that's how they conceptualize their own action. Similarly. they would certainly act "as if" they had math and physics. But the question of whether that would imply they "do", I'm less sure of.