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kilobug comments on Harry Potter and the Methods of Rationality discussion thread, part 26, chapter 97 - Less Wrong Discussion
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Do we know this for a fact?
Objections:
Going to Hogwarts is prestigious, meaning there must be lower-status options available.
Hogwarts regularly hires apparently British replacement teachers, most of them with at least the appearance of educational experience. It is improbable that said experience comes exclusively from abroad or from being a private tutor.
There are too few pupils at Hogwarts to account for the entire underage wizarding population, given the size of the overall wizarding population and assuming the majority of wizards' children are also wizards (not to mention having to factor in Muggleborns).
It seems improbable that the booming school equipment business of Diagon Alley survives on one school's worth of customers, especially if most of them only shop once a year.
If most of the population of magical Britain have been through the same school, we would expect an extremely high degree of social interconnectedness, with most people knowing everyone of the same age at least by sight. There's no evidence of this.
On the other hand,
It is implied that letters coming on one's 11th birthday can only come from Hogwarts.
If one is expelled from Hogwarts, one is forbidden from practising further magic altogether.
No other British schools, or pupils or graduates thereof, are ever mentioned in canon that I can remember.
It didn't seem to me there were so much school equipment business. If you look at the shops Harry went to the first year, it's a dress shop, a book shop, a wand shop, a potion ingredient shop, a broomstick shop, a pet shop, ... none of them sound dedicated to school equipment. They probably have a lot of school equipment in stock/display during august, like supermarkets do here too, but continue selling their goods to adult wizards the rest of the time.
The rest of your objections do hold, but I just fold it into "canon HP isn't very consistent" suspension of disbelief... of course, it makes things tougher for EY when he's making a consistent version of it. But that's part of what makes HPMOR funny/interesting.