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knb 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.
The Magical economy generally doesn't seem to make a lot of sense. In canon it seems like every witch/wizard works for the Ministry, Hogwarts, or for a small restaurant or shop. I think wizards can pretty much conjure every basic material into existence except food and gold. That does leave an interesting question of where magical Britain is getting its food and gold from. There might be some witch/wizard farmers out there that we never hear about, or they could be just stealing food from muggles. Maybe the ministry sanctions some small scale trade between the magic/muggle world?
well... In cannon, the weaselys have a garden...and you can enlarge food with magic, though rowling never specifies what the multiplicative limit is. (it is STRONGLY implied there is one, though. seventh book.)
So a little bit of gardening by the women goes a long way, though it may only be poor families that actually resort to it.