“The mathematical proofs that Hilbert's tenth problem and the tiling problem are not soluble by computational means are difficult, and I shall certainly not attempt to give the argument here. The central point of each argument is to show, in effect, how any Turing-machine action can be coded into a Diophantine or tiling problem. This reduces the issue to one that Turing actually addressed in his original discussion: the computational insolubility of the halting problem-the problem of deciding those situations in which a Turing-machine action fails ever to come to a halt. In 2.3, various explicit computations that do not ever halt will be given; and in 2.5 a relatively simple argument will be presented-based essentially on Turing's original one-that shows, amongst other things, that the halting problem is indeed computationally insoluble.”