This is a critique of certain aspects of the thesis (often referred to as the Church-Turing thesis) that effectively computable coincides with Turing machine computable. The author’s main arguments are: (1) Turing machine computation is actually a second-order concept, providing a framework for a particular idea of computation, but not actually realizing it in any physical (and therefore effective) manner; and (2) effective processes, understood to take place physically (in matter) do not fall under the Turing machine framework.

Indeed, point one seems to be clear enough. In fact, a careful look at the way in which, say, complexity theory results are formulated should convince one that computations within the Turing model are metaphorical, and that substantial effort is needed to convert them to schemes that can be instantiated in physical computing engines. The whole issue of correspondences between symbols and physical tokens, and so on, is still in a very confused state, as the author observes.

Point two is also clear, and rather more important. The author notes several mundane activities that reveal the necessity of a “creative” component supplied by matter. The author remarks that almost any effective process has the following two-fold structure: there is a recipe-like specification, and, then, the actualization of the process is carried out by physical processes, often a kind of phase transition in matter. These processes, though appealing to our present physics, are still somewhat mysterious. Combining these two points, the author’s main positive assertion is that any successful demonstration of an effective process that cannot be realized by a Turing machine will involve some material phenomena, as opposed to a purely mathematical construction.

There is one significant oversight in this paper. It seems to me that a major epistemological problem still attaches to the scope of the Turing model, in terms of, say, the knowability of concrete, countable well-orderings (for example, conceived as sets of ordered pairs of natural numbers) whose ordinals are at least that of the Church-Kleene ordinal. Here, the very existence of a knowable notation for these objects is open. Similarly, but even more unclear, is the issue of the knowability of a model of set theory (assuming consistency). It is not at all clear that these questions can be resolved by appeal to physics alone. In this respect, so far, Turing computability appears to present a limit to our cognizing powers in regard to mathematical objects.