3 ms·

there are hundreds of proofs for the pythagorean theorem, surely some of them using the axioms of geometry

• mkl 7y ago
  This isn't just using the axioms of geometry, though, it's trying to prove the Pythagorean theorem using the Pythagorean theorem as an axiom. Hence circularity.
  • posterboy 7y ago
    no, not exactly. the euclidean norm can be defined to fulfill a couple basic properties or equations (like composibility, or schwarz' inequality, but I don't recall exactly), and it's pure coincidence, if you will, that the norm is equal to the root mean square. That's not circular reasoning. The name euclidean norm implies that the geometric angle (no pun intended) was the motivation, but what's really central is a question of epistemology. There's not much of a point to prescribe a certain approach over another, without a good argument.