Normally you define the dot product (or inner product more generally) with a few conditions that it has to satisfy, and then define two vectors to be orthogonal if their dot product equals 0: <u,v> = 0.
Then Pythagoras's theorem falls out of that - if we have two vectors u, v that are orthogonal, we can use the conditions in the definition of the dot product to prove that ||u + v||^2 = ||u||^2 + ||v||^2 (where ||u|| is the norm of u, defined as sqrt(<u,u>)).
(It's not a hard proof, because the definition of dot product says it's additive in the first slot, meaning <u+v, w> = <u,w> + <v, w>. So it's easy to prove things about sqrt(<u+v, u+v>) by splitting out the u's and v's. It's a bit hard to write this on HN because no mathjax though.)
If you mean you'd start with the axiomatic definition of an inner product space (bilinear, symmetric, positive definite) then I agree that proving that formula is trivial. The problem with a really abstract approach like this is that you haven't actually proved Pythagoras's theorem in Euclidian space. How do you know that the usual definition of dot product satisfies these axioms (or indeed anything does)? How do you show that the "distance" you've defined from the inner product is what we'd expect to be distance (e.g. that it's rotation invariant) or that what you've defined as "orthogonal" from your inner product is related to angle in Euclidean space?
It's not hard to show all that, but by the time you've done it you'll have accidentally proved Pythagoras's theorem along the way. It's like you've gripped on a tube of toothpaste and said "see, there's nothing there" but really you just squeezed it all to the other end.
As a fun illustration of this, note that <x,y> := 2x₁y₁ + x₂y₂ satisfies the axioms but doesn't give you the normal distance measure (e.g. it's not rotationally invariant: |(1,0)| = 2 while |(0,1)| = 1) and has a different notion of "orthogonal".
If you want to prove Pythagoras's theorem on Euclidean space, aren't there about a thousand proofs from that? Including the semi-original from Euclid? I assume it was proved there. And yes of course, you have to start with a bunch of axioms and earlier proofs about Euclidean space, but that's always true isn't it?
> It's not hard to show all that, but by the time you've done it you'll have accidentally proved Pythagoras's theorem along the way. It's like you've gripped on a tube of toothpaste and said "see, there's nothing there" but really you just squeezed it all to the other end.
Funny metaphor. Yes, I don't think you can "simply" prove Pythagoras's theorem without a bunch of background assumptions, it's just that usually these are all assumptions we've already learned (explicitly or implicitly). And if you want to start without assumptions, like in the case of defining inner product space from scratch, then you are by definition starting abstractly and therefore left with the problem of showing this maps onto Euclidean space, somehow.
I agree that there's are lots of proofs of Pythagoras's theorem on Euclidean space. The comment I replied to said that it follows "directly" from the definition of the dot product. That's all I was disagreeing with. They had missed that they were using some or other property of the dot product that was actually proved from Pythagoras in the first place, or some other non-trivial fact about Euclidean geometry.
And I certainly don't mean to imply anything about the importance of abstract inner product spaces. In fact my masters thesis was about Hilbert spaces. And I find it pretty interesting that you can prove something like Pythagoras on the inner product form I mentioned at the end of my last comment.
Ah yes, you're right of course, and I haven't really thought about it in this way before - that either you're based on "real world" geometry, in which case things are a bit harder to prove but make sense, or you're more abstract, in which case you can define things to be easier to prove e.g. Pythagoras, but the complexity is in the mapping between your definitions and the "real world".
> In fact my masters thesis was about Hilbert spaces. And I find it pretty interesting that you can prove something like Pythagoras on the inner product form I mentioned at the end of my last comment.
That's pretty cool, you're definitely more knowledgeable than I am, I'm just a math amateur :)
Then Pythagoras's theorem falls out of that - if we have two vectors u, v that are orthogonal, we can use the conditions in the definition of the dot product to prove that ||u + v||^2 = ||u||^2 + ||v||^2 (where ||u|| is the norm of u, defined as sqrt(<u,u>)).
(It's not a hard proof, because the definition of dot product says it's additive in the first slot, meaning <u+v, w> = <u,w> + <v, w>. So it's easy to prove things about sqrt(<u+v, u+v>) by splitting out the u's and v's. It's a bit hard to write this on HN because no mathjax though.)