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there's this really bizarre thing that happens to you when you first start to learn rigorous mathematics, which is that geometry is taken away from you and you have to conduct a series of arcane rituals to get it back. to get back the *pythagorean theorem*, you need to:
3. construct the rationals Q as equivalence classes of ordered pairs (a, b) of integers where b is not zero (representing the quotient a / b) 4. construct the real numbers R as either equivalence classes of cauchy sequences of rationals or as dedekind cuts
5. construct the euclidean plane as the cartesian product R^2 = R x R, define the euclidean inner product and distance / length on it, define orthogonality in terms of the inner product, and then *finally* prove: if v, w are orthogonal then |v + w|^2 = |v|^2 + |w|^2
this is sort of a hazing ritual you have to go through to properly enter the domain of "actual mathematics" and as far as i can tell it would have looked completely perverse to any practicing mathematician working before the year 1900 or so
the reason we have to do this is to reassure ourselves that the euclidean plane "actually exists," and the reason we have to do *this* is rooted in the foundational crisis in mathematics: russell's paradox, the incompleteness theorems, all that stuff
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