The Orbit Method
Over at Not Even Wrong, there’s a discussion of David Vogan’s talks at Columbia about the “orbit method” or “orbit philosophy”. This is the view that there is — or at least there should be — a correspondence between unitary irreps of a Lie group 
This is described as a “method” or “philosophy” rather than a theorem because it doesn’t always work, and remains poorly understood in some cases, while at the same time having shown itself to be a powerful source of inspiration in representation theory.
What he doesn’t say in so many words (but which I’m just rude enough to) is that the same statement applies to a lot of theoretical physics. Path integrals are, as they currently stand, prima facie nonsense. In some cases we’ve figured out how to make sense of them, and to give real meaning to the conceptual framework of what should happen. And this isn’t a bad thing. Path integrals have proven to be a powerful source of inspiration, and a lot of actual, solid mathematics and physics has come out of trying to determine what the hell they’re supposed to mean.
Where this becomes a problem is when people take the conceptual framework as literal truth rather than as the inspirational jumping-off point it properly is.
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Every “method” or “philosophy” that doesn’t work all the time stands as a challenge: we should figure out precisely how much truth it contains, and how much delusion, until we get something that works all the time. I’m a firm believer that mathematics is perfect: in math, whenever things don’t work “as well as expected”, whenever we’re tempted say “unfortunately…”, it means we’re suffering some confusion – and we need to find that confusion and root it out. When we understanding things correctly, there are no flaws.
Of course, this is a never-ending task.