  Turing's (1936) paper on computable numbers has played its role in
underpinning different perspectives on the world of information. On the one
hand, it encourages a digital ontology, with a perceived flatness of
computational structure comprehensively hosting causality at the physical level
and beyond. On the other (the main point of Turing's paper), it can give an
insight into the way in which higher order information arises and leads to loss
of computational control - while demonstrating how the control can be
re-established, in special circumstances, via suitable type reductions. We
examine the classical computational framework more closely than is usual,
drawing out lessons for the wider application of information-theoretical
approaches to characterizing the real world. The problem which arises across a
range of contexts is the characterizing of the balance of power between the
complexity of informational structure (with emergence, chaos, randomness and
'big data' prominently on the scene) and the means available (simulation,
codes, statistical sampling, human intuition, semantic constructs) to bring
this information back into the computational fold. We proceed via appropriate
mathematical modelling to a more coherent view of the computational structure
of information, relevant to a wide spectrum of areas of investigation.
