Stephen Hawking vs Richard Feynman

I guess that one could say that psychology is physics taken to its ultimate complexity level. But along the way up the complexity ladder there seems to be some point were the physics is not anymore describable by our usual understanding of laws of physics, precise mathematical equations, at some point it seems likely that other things such as "principles", "trends", "patterns", "attractors" etc take over definite mathematical laws. There has been some progress in the study of complexity and mostly it has result in identification of some general principles, pattern trends etc rather than any mathematical equations in the traditional physics sense (that's why many hardcore physicists still hesitate to recognise complexity as a real branch of physics in the same rank as the traditional areas).

No but you can explain them that music and harmony were considered a pure science by Pythagoras and Co.

Several very clever people have said that at some moment, but most more modern reviews of the subject by very clever people conclude that one thing has nothing to do with the other. Godel's theorem basically says that any sufficiently complex formal system (and which complies with a couple of requirements such as arithmetic does) will have undecidable propositions. Translated into a physics theory this would mean that even if we knew the "Theory Of Everything", the theory would predict (or allow) the possibility of things which we could never know if they do actually happen in the physical world or they don't, even if we never observe them we could never prove that they can not or do not happen, the theory would remain necessarily undecissive about it.

An analogy is the decidability of the randomness of a sequences of numbers. We can never prove that a sequence of numbers is truly random, but we can prove that it is not random by finding a more compressed form. Not finding any more compressed form is not a proof that it is uncompressible (truly random), it can be that it truly is or it can be that we have not yet found the compression algorithm.

A different angle to the question is that knowing a theory does not equal to knowing or even being in principle able to know its manifestations. Quantum theory has the famous Heisenberg Uncertainty Principle, we know the mathematical theory but it tells us that we can never know precisely certain features of the world. Similarly, there are theories we know mathematically (such as general relativity) for which computing solutions is in practice is often impossible. And yet another take is chaos physics, where knowing the mathematical theory does not help in being able to know what the system will do after a short time.

A Theory Of Everything, even if it exists and it could ever be found, does not absolutely mean that we could know, deduce or compute much about the physical world.