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I will not dwell on "unreasonableness" because in my opinion, the word is being used in an emotional way to express appreciation and wonder, not to assert a factual claim that the precise amount of effectiveness quantitatively exceeds some rigorously defined threshold of reasonableness. And de gustibus non est disputandum.

However, I think that the part about "providing new intuitions that are not ordinarily accessible to practicing mathematicians via their traditional training" is fairly easy to explain. Many applications of logic to other areas of mathematics center around some kind of transfer principle. One way to think about transfer principles is as follows: We are studying some area of mathematics, and we are able to formalize not just the mathematical objects themselves, but everything we can say about the objects. That is, we are able to rigorously define a formal language that is able to capture (virtually) everything we want to say about the objects. Then by analyzing the formal language, we are able to draw conclusions about some other domain that is not quite the same as our original domain, but to which the formal language applies equally well.

This kind of argument does indeed involve a type of abstraction that is different from "usual" mathematical argumentation, because instead of examining the mathematical objects themselves, we examine the language that we are using to talk about the objects. Examining mathematical language is a natural thing to do when considering "meta-mathematical" questions such as consistency; after all, how else can you analyze the limits of mathematical reasoning other than by formalizing mathematical language? But the part that surprises some people is that the move from studying mathematical objects to studying the language used to talk about the objects can yield concrete results about the mathematical objects themselves, and not just abstract meta-mathematical results. Without detracting from the awe and joy that we feel when we contemplate mathematical beauty, I would submit that this should not really be any more surprising than the general fact that mathematical abstraction—at least, the right kind of abstraction—can yield concrete consequences.

As for the analogy to physics, I personally don't think it goes beyond the truism that a different perspective can yield new insights. For the parallel to be more than that, I think we would have to argue that the use of physical intuition amounts to an unusual process of abstraction, and this does not seem plausible to me.