We turn the physical Church-Turing Hypothesis from an ambiguous source of sensational speculations into a (collection of) sound and well-defined scientific problem(s):
Examining recent controversies and causes for misunderstanding concerning the state of the Church-Turing Hypothesis (CTH), it is suggested to study the CTH 'sharpened' relative to an arbitrary but specific physical theory - rather than vaguely referring to "nature" in general. For this purpose we apply, and emphasize the utility of, concepts from philosophy: physical structuralism, ontological commitment, and constructivism. This general approach is then illustrated with some exemplary results on computability and complexity theory in computational physics. (C) 2009 Elsevier Inc. All rights reserved.