So the situation is perhaps about as discouraging as it could possibly be. We know we're wrong, yet we lack access to phenomena capable of forcing us to correct our errors (and of guiding our guesswork as we struggle to correct them.)
The great thing about a mathematical uniqueness theorem is that it promises to remove all historical contingency. Things must be a certain way, not in obedience to tradition, but because otherwise one or more of these utterly transparent foundational principles would be violated.
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The yearning for this sort of understanding of quantum field theory drove Arthur Wightman and his various colleagues and disciples to seek axioms for the discipline that might simultaneously free it of all dependence on historical accident and expose its absolute essentials.
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The program of the axiomatic quantum field theorists remains incomplete (and indeed one can win a million dollars merely by proving that it can be completed), but that doesn't mean it was entirely without successes. We do now know why some things must be so, and not otherwise.
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