Seeing as you’re a physicist who has thought so deeply about Gödel’s theorem, do you think the absence of a theory of everything in mathematics suggests there might be no theory of everything in physics?
I totally think about that. Why should we think, since physics is so rooted in mathematics, that there is going to be a physical theory of everything? The way we usually think about the Big Bang is: The universe is born, and it’s born with initial data. There are laws of physics, and somehow the initial data is just… something else. We really are dishonest about where that comes from. What if the law of physics that describes the origin of the universe is something that has to make a claim about itself, which is a classic self-referential Gödelian setup for a tangle. [A Gödelian tangle is an unprovable, self-referential mathematical statement, such as, “This statement is unprovable.”] What if the laws of physics have to make a claim about themselves in such a way that they themselves become somehow uncomputable?
I’m also super interested in the idea that the initial data of the universe could contain irrational or uncomputable numbers. Then the universe could never finish computing the consequences of the initial conditions. Maybe we can’t predict what’s coming next because every digit of the initial data is a toss of a coin.
But it’s not enough if I only have words, and I’ve never found something to write down in math, so I’ve just kind of waffled. I think a smart thing to do would be to look at a specific Gödelian tangle that exists in mathematics and try to map that to fictitious laws of physics. Then you would have a universe in which there was a Gödelian tangle. There are constructive things to try.