> But the reals? Well they are invented purely so we can take the square root of two.
I agree with what you say, but I also think that we overly conflate "importance" with "the order we found/defined them in". It's a big leap (conceptually, arithmetically, etc.) to go from the rationals to the reals; sure that's what the ancients actually did, but we don't need to follow the same path when e.g. generalising some result, teaching mathematics in school, defining a computer mathematics tool, etc.
There's no reason to "skip past" algebraic numbers, computable numbers, etc. especially since they're most often the sets we're dealing with; e.g. in high school most problems were something like 'find the real number x such that something-involving-x = 0'.
PS: You've inspired me to have another go at reading the road to reality; it's been gathering dust on my shelf for years!
I agree with what you say, but I also think that we overly conflate "importance" with "the order we found/defined them in". It's a big leap (conceptually, arithmetically, etc.) to go from the rationals to the reals; sure that's what the ancients actually did, but we don't need to follow the same path when e.g. generalising some result, teaching mathematics in school, defining a computer mathematics tool, etc.
There's no reason to "skip past" algebraic numbers, computable numbers, etc. especially since they're most often the sets we're dealing with; e.g. in high school most problems were something like 'find the real number x such that something-involving-x = 0'.
PS: You've inspired me to have another go at reading the road to reality; it's been gathering dust on my shelf for years!