The question adds up to the following questions and I hope it’s not considered duplicate:
Where does the Pythagorean theorem "fit" within modern mathematics?
Why does the Pythagorean Theorem have its simple form only in Euclidean geometry?
Let $(E, d)$ be a metric space.
If $d$ is the Euclidean metric, then the Pythagorean Theorem is essentially the definition of $d$, or a trivial corollary.
If $d$ is not the Euclidean metric, then the Pythagorean Theorem does not hold (is this true?).
So: what is the underlying truth of the Pythagorean Theorem, in modern mathematical terms?
Note: since all the proofs are implicitly based on the invariance of the metric with respect to translations and rotations, I thought that the underlying “theorem” was that the Euclidean metric is the only one whose group of isometries is the Euclidean Group. This is in line with some of the answers to the questions, but this is not true (for a counterexample see
Does a group of isometries uniquely characterize a metric?)
I would be satisfied with an answer of the type: the Euclidean metric is the only metric that comes from an inner product (and therefore the set $E$ can be made a vector space), that is invariant with respect to rotations, translations and reflections, [+ more conditions].
Or please give a reference, I feel that this is pretty basic but I can't figure it out! Thanks.
Best Answer
The basic reference for the significance of the Pythagorian theorem is the illuminating article by Givental:
Givental, Alexander. "The Pythagorean theorem: What is it about?" Amer. Math. Monthly 113 (2006), no. 3, 261–265.