A new & peculiar essay, on a (very!) unusual approach I've been using to deepen my understanding of mathematics: http://cognitivemedium.com/srs-mathematics
Lakatos' "Proofs and Refutations" may be of use to you as you refine your views on these matters. Oh, and by the way: a normal linear operator is one whose (self-adjoint) real and imaginary parts commute. This points the way to a rather different proof from the one you describe.
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"If you take an idea that seems outstandingly creative, it usually didn’t appear in someone’s head by magic. Some thought process led to it, and this can be broken into smaller ideas until you get down to the lowest level of [the] thought process, which is essentially routine."
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Thus spake
@wtgowers
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