Hello Spherical ...
I'm no mathematician ... but it seems from what I've read that what I described is one of the ramifications of Godels Theorum ..
http://www.miskatonic.org/godel.html
"Boyer, History of Mathematics
Gödel showed that within a rigidly logical system such as Russell and Whitehead had developed for arithmetic, propositions can be formulated that are undecidable or undemonstrable within the axioms of the system. That is, within the system, there exist certain clear-cut statements that can neither be proved or disproved. Hence one cannot, using the usual methods, be certain that the axioms of arithmetic will not lead to contradictions ... It appears to foredoom hope of mathematical certitude through use of the obvious methods. Perhaps doomed also, as a result, is the ideal of science - to devise a set of axioms from which all phenomena of the external world can be deduced."
Actually, is this not kind of what you said anyway? ... if you consider your "closed logical system" as the universe which must be observed. Obviously if you are in the system such as we are .. it can't be observed properly .. that must take place from somewhere "outside of the system" so to speak.
I think that this in conjunction with Heisenbergs Uncertainty Principle shows that a truly accurate description of our reality by us is probably impossible.
I would like to return to the concept of invisible proofs. If in the material world we infer the existence of things by their effects so can we see the effect of the spiritual forces in our world.
And how do you recognize these spiritual forces? Just like science ... you must investigate ... but with both your heart and mind.