Popper and Turing's Challenge

In Computing Machinery and Intelligence, Alan M. Turing posed the question "Can machines think?". Popper's answer in The Self and Its Brain is "without hesitation that they cannot", and "Turing's challenge should not be taken up". Why does Popper refuse Turing's challenge? First, we look at what Popper and Turing actually said, in order to put some aspects of the problem in perspective. Second, because Popper's view about mechanical procedures might to some extent match that of Gödel, a glance at Gödel's interpretation of Turing's work may help to find a way through Popper's thought. By this route, we reach the conclusion that Popper fails to see that what Turing was really proposing can be viewed as a genuine application of his own method of conjectures and refutations.