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Velorien comments on Harry Potter and the Methods of Rationality discussion thread, part 15, chapter 84 - Less Wrong
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You're right, no hint of an explanation why wands are necessary has ever been given. Spontaneous underage magic, as well as high-level wandless magic, would be evidence in favour of the "conceptual limitation" theory.
On the other hand, this wouldn't explain why Bacon, who apparently lacked this conceptual limitation, was severely held back in his research by lack of a wand. Also, it doesn't explain why high-level wizards continue to use wands for the overwhelming majority of their spellcasting.
Maybe it's not that wands are needed to cast spells, but that they amplify magical power (and perhaps adds focus to a target). While the magically powerful are able to cast high level wandless magic, most are unable to. Hence, they have to use wands to make their spells powerful enough to have an effect. Children have spontaneous magic but they can't cast as much as adults normally can with wands.
Perhaps Roger Bacon just wasn't magically powerful. -shrugs- Not all great thinkers have to have tons of strength. Er, wasn't he Muggleborn? If the "Muggleborns-are-weaker" theory is true, then it makes sense.
My hypothesis for the reason why high-level wizards continue to use wands is that they've simply grown dependent. If they've been using magic-amplifying wands ever since they were eleven, then they would be used to being assisted by the wand. I think this matches my mental model of Quirrell, who is seen doing a lot of wandless magic (stopping spells midair, spontaneously combusting inkwells). He seems like the kind of person that would train himself to use his wand as little as possible. And even if he can't duel without, his magical ability is certainly very impressive.
However, the spells they do cast are fully as powerful as those of adults with wands.
Pretty sure this theory has been unambiguously dismissed both in canon and in MoR.
Otherwise your hypothesis is credible, though I still don't accept it as I can't see all the high-level wizards we know being dependent on wands when there are so many advantages to wandless magic (and when high-level wizards tend to be ones with strong, independent personalities).
I think both have been silent on the question of whether there is any notion of inherent "power levels" at all, let alone whether it is heritable or whether it is correlated to being a "muggleborn".
EDIT: It's clear in MoR that - if Harry's hypothesis on magic heritability is true (a big if), then other non-binary factors seem unlikely to be correlated to being a "muggleborn". However, I felt that Harry very strongly anchored on that hypothesis, which was one of my reasons for being annoyed with him and eventually stopping reading (to pick it back up later on, obviously)
In fact more powerful than most adults; there's a line in Chapter 78 that "If [Mr and Mrs Davis]'d been children young enough for accidental magic they probably would've spontaneously Disillusioned themselves.", which we know requires significantly above-average power in MoR. (Assuming that line from the narrator isn't exaggeration.)
If you accept the hypothesis, wanded magic has the not-insignificant advantage of being more powerful. What's the advantage of wandless magic?
I thought it was obvious. What if you're without a wand?
If you're in battle without a wand, it seems to me that either 1) you've been ambushed, or 2) you've been disarmed. I don't really see that the ability to cast understrength spells helps all that much in either situation.
That depends on how creative you get. Even understrength spells, especially if unexpected, could tip the balance in your direction - especially if all you're looking for is, say, an opportunity to escape. Even if you lose your gun, a rock can still be useful.
Yeah, thinking about it a little more, even just wandless Apparation would be pretty useful.
Even just wandless Apparation? Wandless Apparation! Of all the defensive magic options available that don't involve time travel that's quite possibly the first pick. I'd take it over the ability to cast Avada Kedavra (at all). I'd consider taking it even if it meant sacrificing my ability to cast any offensive dueling spell ever.
Once that is in place it is time to research as many alertness and general paranoia spells as possible.
Do we have any indication how difficult it is to cast Anti-Disapparation Jinxes, in canon or MoR?
The fact that Quirrell ends his spiel about how the correct tactic is usually "Just Apparate away!" with the fact that Dark Wizards can still reliably threaten even people who can do that indicates it's at least possible in combat time (i.e. doesn't require a day to cast it on a house, say).
Edit: But yeah, that's obviously the best single choice (though I think you're selling the AK a little low). Number two would be Accio, I guess.
Non-reliance on wands is a big one, since watching the movements of an opponent's wand, or disarming them, are combat fundamentals. Being able to cast spells unnoticed is another one (consider the powerful effect of the mid-interrogation Memory Charm in the Order of the Phoenix). Also, it's presumably better training in terms of building up power and skill to cast spells without a crutch.
Besides, many spells don't really need extra power to work, as they have a binary effect (like the Quietus Charm) or typically target objects that can't resist (such as the Vanishing Charm).
Nitpick, but Quirrell cast a Quieting Charm on the rocket in the Azkaban escape, but Harry's ears were still ringing enough afterward that he couldn't hear Bellatrix shouting. So it's not a binary effect; there could be Quieting Charms that are capable of silencing louder noises than others.