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pedanterrific comments on Harry Potter and the Methods of Rationality discussion thread, part 17, chapter 86 - Less Wrong
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OK. I'm thinking of this in terms of Harry being able to see Bellatrix because it's his cloak. Harry should then be able to see the other Harrys because they're also wearing his cloak, unless the Cloak distinguishes between "master" and "time-travelled master", or the "loan" part is significant enough that Harry wouldn't be able to see someone under the cloak if they just pick it up without him expressly loaning it to them. If that counts as "stealing" and transfers ownership then you could "loan" the Cloak to everyone and they'd never be able to take it from you.
There's something unsettling about a cloak that hides you from everyone, except its Master, unless the Master is also you.
It seems reasonable that Cloak belongs to Harry. Then successive timetravels instatiate Cloak1 with owner Harry1, Cloak2 with owner Harry2, etc... What's unsettling is that Mad Eye can see through the true cloak of invisibility!
I think EY phrased it as "more narrowly-focused artifacts can defeat other artifacts in the area of their specialty", or something like that (does anyone have the reference?).
Of course, that still seems odd, because the Cloak's specialty is "being invisible" and the Eye's is "seeing in every direction at once, seeing through solid objects, and seeing magic including hiding or invisibility spells". I must be missing something. Maybe the Cloak's specialty actually has to do with Death, and the mundane invisibility is a side effect?
Maybe it's something to do with order of construction. Either the Eye is more ancient and thus more powerful in general, or the Eye was made after the Cloak and therefore its creator knew about and wanted to defeat the cloak.
Oh yeah, obviously it would be more powerful since it was made before or after the cloak.
All of those Eye functions can be summed up as "seeing bloody everything" (presumably within some range of the user), which at least seems more narrowly focused than your description, so perhaps its creator thought so too, which seems to count for something here, what with the broom physics and all.
That's a plausible thought on the Cloak, though.
Yeah, thinking about it a bit more, it might be that the Eye is actually specialized to "detect all hidden things" or something- you can't hide by being behind a wall, you can't hide by being behind the user, you can't hide by being invisible.