Feynman's question on the mathematical machinery underlying nature Physicist Richard Feynman said in his lectures

"It always bothers me that according to the laws as we understand them
  today, it takes a computing machine an infinite number of logical
  operations to figure out what goes on in no matter how tiny a region
  of space, and no matter how tiny a region of time. How can all that be
  going on in that tiny space? Why should it take an infinite amount of
  logic to figure out what one tiny piece of space/time is going to do?
  So I have often made the hypothesis ultimately physics will not
  require a mathematical statement, that in the end the machinery will
  be revealed and the laws will turn out to be simple, like the chequer
  board with all its apparent complexities. But this is just
  speculation." (The Character of Physical Law - Ch. 2)

what does he mean by "in the end, the machinery will be revealed and the laws will turn out to be simple". what machinery is he referring to? Is he asking for example what machinery "solves" the endless mathematics? or perhaps how the mathematics unfolds seemingly by itself.
also, these were given in 1965. have we come anywhere closer to answering his question since then?
 A: There is this talk by Stephen Wolfram that I think has some key insights into what Feynman meant. The idea is that simple rules can induce very complex behaviour, a nice example of this is cellular automata.
If one just looks at an animation of a cellular automaton and tries to describe the global behaviour, this description might be very complicated indeed. However, if you zoom in and slow down and meticulously catalogue all the steps you might be able to deduce the simple rules which make the thing tick.
Feynman's intuition was that our current understanding of physics is a very rough one, to push the cellular automaton analogy further, it is like we can only see what happens to $3\times 3$ blocks or bigger, but to really understand what happens we should look at the individual $1 \times 1$ blocks.
You can fully specify what happens in a cellular automaton by listing out all possible configureations in a $3 \times 3$ block (if you allow diagonals). If you want to describe it at the level of $3 \times 3$ blocks you need to list out all possible configurations of a $5 \times 5$ block, surely a much more complicated way to do this.
I should add a disclaimer that even though this is what I understand what Feynman meant, there is no way to be absolutely sure.
On your second question, I do not feel qualified to answer such a broad question about the state of physics as a whole. But if I would have to hazard a guess, I would say that even though physics has definitely progressed since then, it is not in the direction that Feynman had in mind when he said this.
