Justin Riddle #10 - Foundation of Math in Crisis

In episode 10 of the Quantum Consciousness series, Justin Riddle explains that mathematics does not have a firm foundation in first order logic. While it seems intuitive that mathematics should be able to be described by simple principles, there is no simple way to explain how humans can comprehend mathematics. What happens when we try to use first order logic to explain how first order logic works? Well, we run into paradox: situations that are impossible to resolve and do not make any sense. For example, take the liar’s paradox, “this sentence is false.” There is no way to make sense of this sentence, because it leads to paradox. The content of the sentence is not supposed to refer to the truth value of the statement itself. Similarly, you cannot use first order logic to refer to the truth value of first order logic itself. Sets cannot describe sets (Bertrand Russell’s set paradox). And most dramatically, digital computers cannot even describe digital computers (Alan Turing’s halting problem). All of this leads Roger Penrose to conclude that “humans are not using a knowably sound algorithm to ascertain mathematical truth.” While we can understand things and learn mathematics, it’s really not clear how this is possible and something more than digital computers are required! Quantum mechanics offers a novel lens to this problem as the first order logic that appears in particles also has a supervening wavefunction that does not appear to follow the rules of first order logic. I don’t have the answer for you, but I sure have a lot of questions for you to consider!