Some BackgroundTowards the end of the nineteenth century there was a general feeling among scientists that explaining the universe was on the verge of being done and dusted: there were still some minor issues to be resolved, but essentially all the elements of a complete explanation were in place and all that was left for the future was further refinement. So much for science, but in the field of mathematics things were not quite as rosy - the branch known as set theory was undergoing a major crisis as a result of the discovery that it permitted logical contradictions.
The idea of sets is so simple that even I can comprehend it but the nature of the particular problem requires a little thought to understand (at least for me it does, anyway). The crux of the issue is this: set theory allows for the creation of a set of all sets that are not members of themselves, resulting in sets that are members of themselves only if they are NOT members of themselves. Paradox! Disaster! I can only just about get my head around this (if you're looking for a more complete analysis go here) but the next link in the chain that is leading us (I promise) to Hawking - Gödel's incompleteness theorem - leaves my comfort zone as a distant spot dwindling on the horizon (so any requests for clarification would be better addressed to a slab of concrete than to me).
Here we go, anyway: both the set theory problem and the incompleteness theorem(s) are variants of the liar's paradox, an old philosophical conundrum. What Gödel succeeded in doing (by means that are completely beyond my ability to grasp) was to encode 'this statement is unprovable' and embed it in arithmetic. The implication of this remarkable feat is that mathematics is incomplete. In other words, there will always be statements which are true but which can never be proved to be true.
Meanwhile back at the homestead ...Along came Einstein and abolished some common sense notions about the universe, but his theories subsumed Newtonian mechanics so - albeit with a little grumbling - they were accepted without too much of a problem. And then there was Heisenberg who demonstrated that, at least at the subatomic level, there could be no such thing as certainty, and that was a heavy blow but the desire to arrive at a theory of everything persisted, and thus was born string theory, superstring theory, M theory, and wherever-the-fuck-we-are-now theory.
There are good philosophical reasons (see Hume and Popper et al) to suppose that even if a Theory of Everything is possible we can never be sure that we've arrived at it, but Hawking's point, insofar as I understand it (and he's not the first or only person to make this argument, just the most famous), is that because mathematics is incomplete physics, too, must be incomplete and because mathematics never can be complete neither can physics. Bye-bye Theory of Everything. Obscurely, that makes me happy!