A Note on the Physical Possibility of Transfinite Computation

On Epistemically Useful Physical Computation

“…Using Turing machines in classical universes whose spatial dimensions are described by hyperreal, rather than real, lines, Aitken and Barrett (2009, 2010) show that what is computable using a Turing machine depends on the physical theory in which it resides. They conclude that, generally, what is physically computable depends on the physical theory.…”

“…Any process can be considered to be a computation of almost anything if one is liberal enough with what one considers a computation, as the pancomputationalism literature demonstrates. 2 We will want to consider models of physical computation that follow Aitken and Barrett’s (2009, 2010) prescription in light of Piccinini’s epistemological concerns. To give us a tractable starting point, I will define a model of computation as follows.…”

“…We will see that Piccinini’s executability subconstraint, owing to ambiguities in how to determine what is buildable, can leave open a wider range of possibilities for computational operations than seems desirable and that seems to have been acknowledged in the literature. By investigating ways one can describe what counts as “physically constructible” (what we call “physically buildable” to avoid confusion with other ideas tied to the word constructible ), we will see that Piccinini implicitly supplements this concept with a notion of what operations finite observers can use which is based in classical computation, rather than on the physical theory itself, in contrast with the prescription Aitken and Barrett (2009, 2010) describe. Nonetheless, I will argue that adapting the implicit classical notion underpinning Piccinini’s discussion of his usability constraint to a particular physical theory is a strong starting point for investigating physical computation in that theory, though we should be explicitly aware that we are making an assumption.…”

“…As Aitken and Barrett (2009, 2010) note, what is physically computable depends on the selection of a model of computation relative to a particular physical theory, so we may hope that our understanding of what is buildable, and hence potentially usable for finite observers, in a physical theory will emerge from the physical theory and its descriptions of finite systems. However, attempting to derive a notion of buildability directly from an account of finite systems in a physical theory succumbs to a bootstrapping problem, in which our notion of what is “buildable” is meant to allow us to determine which physical processes finite observers can reliably implement (as part of a computation), but determining what counts as buildable requires that we already know which physical processes finite observers can reliably implement (as part of building the components of computing systems).…”

The Nature of Physical Computation

“…Hogarth also pointed out the non-recursive computational powers of such devices, and suggested that the class of computable functions (in the broad sense) depends on the properties of the spacetime. 56 More recently, Etesi and Németi (2002), Hogarth (2004), Welch (2008), Button (2009), and Barrett and Aitken (2010) further explore the computational powers of these devices, within and beyond the arithmetical hierarchy.…”