Stephen Hawking vs Richard Feynman

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. 

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

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).

Someone who will just have the insight to see something, some relatively simple principle which is just eluding everybody else because everybody is looking from the wrong perspective, even if his technical proficiency is not one of the best? (such as Einstein, he was not technically a top scientists, he just had the insight of realising things, basic principles, which the others did not see. He actually had to learn non-euclidean maths to develop mathematically what he had just realised by intuition, general relativity).

Or perhaps a super-skilled mathematician? who may come up with solutions to for example the unification of relativity and quantum mechanics from purely mathematical grounds?

Or perhaps someone more in the field of cosmology? (given that several of the biggest enigmas are in this field, dark energy, dark matter, black holes...).

As great as Feynman was, I am not sure that he would have been able to crack the current enigmas, for some reason we seem to have hit a barrier of a different level. It is not normal that with so many scientists and resources nobody is able to make significant fundamental progress in 45 years, this had not happened in the last few centuries. 

 

And marimba as I recall, and throat singing. 

Hawking provided vocals for Pink Floyd's Division Bell.

I can't know what would attract Feynman's interest, but there are quite some open questions in physics.

The first and most obvious is the attempt to reconcile Relativity with Quantum Mechanics, this is considered the Holy Grail of physics. There is no absolute reason why they should be possible to reconcile, but it seems nice that they should, we always strive for unification as it seems to infuse deeper meaning to the phenomena we observe in the universe.

The next major challenge is the nature of dark energy and dark matter, after all our best theories can only account for 4% of the universe surrounding us, so that is not very uplifting.

Life is another big challenge (and the phenomenon of complexity in general) which no current theory is capable to explain. Why and how do the constituent elements of the universe combine into ever more complex structures? and why and how do these structures reach a point when they become masters of their own destiny, apparently overtaking the very laws of nature?

On a more abstract plane, the subject of FreeWill is also one of the big questions, to which extent is the world pseudo-deterministic (we know at root, quantum level, it is not, it is random, but on larger scales the laws of nature seem to be pretty deterministic. Where is the border and why is there such a border?).

Related to this last point, the subject of quantum decoherence is also still a mystery (why the quantum rules and the macroscopic rules are so different). Where and how does the quantum world become the macroscopic world?