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.. of the theory of calculating machines, indicating it is possible in principle to frame questions that cannot be resolved: the problem is that determining exactly which questions "fall between the cracks" is itself a nontrivial problem.
The approach is this: one carefully describes certain finite programmable calculating devices, shows that some machine behaviors can be re-interpreted as statements about the machines' behaviors, and then pulls a trick rather like the one involved in the "liar paradox." The upshot is that certain questions about machine behavior cannot be answered by the machines themselves; and if one replaces the machines by better machines, the same problem reproduces itself for the new machines. This was first done in the 1930's by several people, including Alan Turing, and has been very carefully studied ever since.
A reasonable theory of consciousness, based on calculable physical phenomena, is likely to suffer the same defect: one will assume (because the philosophy of science absolutely requires it) that the human mind is essentially a programmable calculating device (say, the calculation is performed according to certain electrochemical laws) which provides certain outputs (perhaps in a probabilistic manner) in response to certain inputs; then, once the whole theory had been worked out in full detail, one should copy the basic technique of Turing -- with the conclusion that certain well-framed questions will remain unanswerable. But one then has the problem that an unanswerable question is not a scientific question -- and thus any belief that all such questions can be addressed in a logical and scientific manner "commits suicide."
The appropriate scientific stance, of course, is still to formulate whatever questions one thinks one knows how to answer and to try to answer them -- but in some sense, as one approaches the unresolvable problems, the logical terrain becomes erratic and increasingly difficult to predict, requiring specialized methods that shed light on limited areas
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