A different but to my taste more mathematically conventional way to make this argument would be instead to say that a symbol is a function from the square to [0

A different but to my taste more mathematically conventional way to make this argument would be instead to say that a symbol is a function from the square to [0, 1], that is, a grayscale image. A symbol, because of limitations of either its writer/printer or its reader/viewer, should have some "regularity": as you move around the image, there should be some quantitative restrictions on how rapidly the darkness of the gray color should change. In this kind of setup, the space of symbols is compact by a version of the Arzela-Ascoli theorem, which I think was fairly well-known by the time Turing was working. This also has the advantage of being straightforward to generalize to things like colored images by just changing [0, 1] to, say, [0, 1]^3 to represent RGB space, or whatever you want, as some others are mentioning in the comments. As an aside, this is an interesting companion read: "A pedagogical history of compactness" (https://arxiv.org/abs/1006.4131 https://arxiv.org/abs/1006.4131) As a working mathematician, I can say that this kind of argument has become totally routine and, were I reading Turing's paper carefully, after seeing "epsilon" and "compact" I would think "makes sense" and move on. But, historically speaking, it's interesting to realize how recent the development of the abstract idea of compactness was when Turing was writing---the time between Frechet and Hausdorff's work on compactness (see the pedagogical history) and Turing is about the time between Google being founded and today.