MathMage comments on Harry Potter and the Methods of Rationality discussion thread, February 2015, chapter 112 - Less Wrong
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I think that since this story is Harry Potter and the Methods of rationality, its going to be Harry who saves the day. But if this wasn't a story...
Harry is in a totally impossible position (baring some experimental transfiguration). The best hope lies with McGonagall. She's been seen going off scrip before and if she realises the magnitude of the threat Volde now poses (perhaps Harry could impress this upon her, and point out that Volde has recently resolved to stop wasting time and just take over the world) she might realise that the only way left to stop Volde is to go totally off-script, and seek help from the Muggles.
While a gun can't hurt a shielded wizard, the shielding only has so much strength. Bring in the Muggle army with overwhelming numbers, heavy weapons, helicopter gunships and so forth. Wizards have access to apperation, dissilusionment etc which might still allow them to win against a pure muggle army, but a combine force of Muggles and Wizards would also have access to anti-apperation jinxes and so forth.
Admittedly, this ends centuries of secrecy. Therefore, the second option she has is to go to the Wizard UN or whatever their equivalent is, and explain that without reinforcements she will be forced to go to the Muggles for help.
I am dubious that "seeking help from the Muggles" is as straightforward as you seem to think it is.
I didn't say it would be easy, but what other option does she have?
At present, McGonagall has no options because she doesn't know what's going on. In order for your question "What can McGonagall do?" to even make sense, we must first postulate that Harry/Hermione get out of this situation and that one of them is able to contact McGonagall while evading control or surveillance. Even if that is resolved without obviating your reason for considering McGonagall (that Harry is in an impossible situation), it's not clear that McGonagall could:
Persuade enough Muggles to take her seriously
Anticipate Voldemort's intentions and avenues of attack
Explain these to the Muggles in sufficient detail that the Muggles could devise countermeasures (McGonagall certainly can't devise Muggle countermeasures for them)
Do all this in the time frame before Voldemort could take over magical Britain, not to mention whichever Muggles McGonagall has gone to for help
Avoid Voldemort while doing all of this
Perhaps Harry could will a patronus to take an extremely small form?
Doable, given enough time. If she is morally flexible enough, she could simply memory charm those at the top of the command chain, but... this isn't the sort of thing I'd imagine her doing.
This is a problem. Unless Harry explains it to her, I don't think she will understand just how bad the situation is.
I'd seek advice from Muggleborns. But actually, in canon the muggle prime minister at least knows about the existence of wizards. I wonder if the muggles already have a (top secret) plan in place for what to do if wizards attack them?
This is the big problem.
Can't you Imperious a few heads of state?
(Though probably not, or someone would have done it, though McGonagall may be aware of the details of the restrictions.)
It is of course only quite resonantly that Muggles have gained enough firepower to be a threat to shielded wizards.
It is mentioned that Grindelwald confunded Neville Chamberlain, and that the Nazis were his "muggle pawns". So I think its probable that "Imperious a few heads of state" has already happened and the result was WWII.
McGonagall would probably want to use a different method that didn't involve unforgivable curses though.