  The mathematical notion of incompleteness (eg of rational numbers,
Turing-computable functions, and arithmetic proof) does not play a key role in
conventional physics. Here, a reformulation of the kinematics of quantum theory
is attempted, based on an inherently granular and discontinuous state space, in
which the quantum wavefunction is associated with a finite set of finite bit
strings, and the unitary transformations of complex Hilbert space are
reformulated as finite permutation and related operators incorporating complex
and hyper-complex structure. Such a reformulation, consistent with Wheeler's
`It from Bit' programme, provides the basis for a novel interpretation of the
Bell theorem: that the experimental violation of the Bell inequalities reveals
the inevitable incompleteness of the causal structure of physical theory. The
kinematic reformulation of quantum theory so developed, provides a new
perspective on the age-old dichotomy of free will versus determinism.
