I had the opposite reaction: I lost faith in the book's analytical process only ten pages in, when Hofstadter claimed that Bach's canon was a "strange loop" simply because it modulates to a higher key. Sure, if you keep doing this you can get to the same key, an octave higher, but that's only because pitch itself is periodic, repeating the same notes every octave. There's nothing unique or mysterious about Bach's canon in this regard; anything that modulates up a step has the same property (for example, the chorus to "Mandy" by Barry Manilow).

I think the far more interesting observation is that octaves in music work this way, where we as humans somehow perceive frequencies in a 2:1 ratio as "the same, but different." A 440 and A 880 aren't the same pitch, and we can clearly perceive the difference between them, and yet the seem "the same" to the point that we give them the same letter and think of them as the same. I find that very interesting and slightly mysterious.

Anyway, since music is my area of expertise (as opposed to the other things described in the book) I lost faith that his explanations of the other subject material wouldn't have similarly sloppy thinking. The impression I got was that, in an effort to weave an interesting and engaging narrative, he was too quick to see things as instances of these fanciful concepts like "strange loops."