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I am interested in how today's professional mathematicians view the Pythagorean theorem, in terms of how the theorem fits within the axiomatic framework of mathematics. I often come across textbooks that define length by the Pythagorean theorem, so that the theorem is in essence a definition or axiom. In more modern mathematics such as linear algebra, is the Pythagorean theorem generally just used as the definition of length? Is it more conventional today to treat the Pythagorean theorem as a definition (or axiom) rather than a theorem? Are there any modern proofs of the Pythagorean theorem that don't rely on Euclidean geometry (like a proof that utilizes linear algebra/the dot product, etc.)?

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    What is modern mathematics for you? Mathematics after 1980?2010-08-25
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    I have posed the following question, now a community wiki, What is Modern Mathematics? Is this an exact concept with a clear meaning? [http://math.stackexchange.com/questions/3447/what-is-modern-mathematics-is-this-an-exact-concept-with-a-clear-meaning2010-08-27

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