I was considering asking this question on math stack exchange but decided not to because "first principles" seems like more of a physics thing.
I'm looking for a conceptual proof of the pythagorean theorem from first principles. Actually, it might be better to say the distance formula rather than the pythagorean theorem because I'm thinking about distances in three dimensions.
I don't find any of the usual proofs of the distance formula satisfying. There are a number of proofs from first principles in euclidean geometry but then I feel like I have to move triangles and squares around or break out proportions every time I use the distance formula. On the other hand there are a lot of conceptual proofs of the pythagorean theorem, e.g. using the dot product or the law of cosines, but each of these just pushes the question around. If I use the dot product to "prove" the pythagorean theorem I need to know why we use the dot product from first principles. If I use the law of cosines then I need to know why vectors that are $90^{\circ}$ to each other are perpendicular. I'm sure this sounds perverse so let me try to make what I'm looking for a bit more precise.
For me "understanding" a thing is always understanding it within a particular conceptual system. Within that system we have admissible intuitions and understanding is when those intuitions are matched with rigorous math. Those intuitions could be of the formal relations between various quantities and how we interpret that physically. So there are all these different ways to understand this stuff, but what I'm lacking is a way to understand them simultaneously. (--ok, I really tried here, and I know it is super confused but its not getting any better than this)
The distance formula is such an elementary and fundamental fact about our everyday experience that I feel like there must be some better way to understand it than "we can prove it and its mutually compatible with other concepts like inner products and angles and the math that relates them".
I've been wondering about this on and off ever since I learned about the dot product proof of the pythagorean theorem. At first I was more wondering why the the dot product lets us compute angles via $u \cdot v = \|u\| \|v\| \cos(\theta)$, and this still seems extraordinary to me, so I would want a way to understand the relation between inner products and angles from first principles too.
Right now my thinking is that from first principles (whatever first principles is) distance is a thing, rigid motions is a thing, distance is preserved by rigid motions, rigid motions includes translations, rotations, and reflections, so distance is translation invariant and respects scaling, so distance comes from a norm, then because the parallelogram law is so obviously true (not obvious to me at all), this norm comes from an inner product given by the polarization identity.
If I could understand why the parallelogram law is true from first principles then by the rest of the above I would have one of these nice conceptual systems from first principles through inner products, and that would include the pythagorean theorem.
But understanding why the parallelogram law is true seems at least as hard as the pythagorean theorem itself, since its essentially a more general form, so I'm going to settle with understanding how it can fail. I see two avenues for this. First there are various examples of norms where the parallelogram fails to hold, for example all of the $L^p$ spaces. Second, the pythagorean theorem, and presumably the parallelogram law, fail to hold in geometries with nonzero curvature.
I think this clarifies what I mean by "conceptual proof of the pythagorean theorem from first principles" so I think its reasonable to ask for approaches found in textbooks or expository papers to the pythagorean theorem / inner products / angles following a similar outline to what I gave above.