Non-Turing Computers are the New Non-Euclidean Geometries

While it goes without saying that computing devices need space and time (spacetime), nothing in Turing’s theory of idealized computation prepares one for the rich nature of a computer’s dependence on the underlying spacetime geometry. Nor does that theory prepare one for the vast array of non-Turing computers (and one computer à la Turing) obtained by painting ordinary Turing machine hardware on a variety of different geometries. Different geometries yield radically different computers, and the range of computers extends way beyond Turing’s machine. Explicating and assessing this discovery is the object of this article.