I'm not sure I get that. You riddle me too hard a riddle there, I'm afraid.What's, for you, the next best thing to say? — TheMadFool
A Cartesian coordinate system is an assignment of n-tuples to the points in a point space underlying Euclidean space, such that the dot product between the n-tuples is isomorphic to the inner product between the displacement vectors of the points from the origin. — simeonz
I'm not sure I get that. You riddle me too hard a riddle there, I'm afraid — simeonz
.no actual geometric model — simeonz
I agree with the fact that we can define the dot product as you specify, but we need inner product as well, or we are just manipulating unitless numbers that don't correspond to anything. — simeonz
But I was saying that there is one more hop (probably) in my mind to how this intuition translates to Cartesian coordinates. — simeonz
I don't doubt that such sources are authoritative in their own right. I think that such treatment is a little outdated in style, because the mathematics skip a little modern abstraction, in pre-Russellian (pre-Frege) manner of thought. I do not oppose the dot product and metrization you provide. It indeed fits the axiomatic requirements. Affine spaces can be defined over n-tuples (as both point and product spaces) and that Cartesian coordinate systems can simply be rigid transformations over some preferred innate coordinates. However, I have something else in mind. Something akin to the Wikipedia definition. I agree that it comes from a much less reliable source, but it is what I mean.Here's what Euclidean space is. My reference here is for example Calculus on Manifolds by Spivak, page 1 — fishfry
Inner product space over the real numbers does not imply a vector space of n-tuples, but simply that the scalar field of the vector space are the reals. That allows us to use real numbers in the Cartesian coordinate system, without worrying of the underlying units of measurement that correspond to the unit distance stride in our point space.A Euclidean vector space is a finite-dimensional inner product space over the real numbers.
A Euclidean space is an affine space over the reals such that the associated vector space is a Euclidean vector space. — Wikipedia
I personally do not object to extra formalism, if it makes the assumptions more explicit. But I see how you may feel this way.The rest of what you wrote is overloaded with what software developers would call cruft. — fishfry
Then the distance in your point space is inherently unitless, and the angles are synthetic. In the OP's case, the answer then would be that the hypotenuse length is also unitless. The units do not come from the metric, which is just a number, but from the nature of the unit vectors, which in the case of an n-tuple are just numerical. But for many applications, there are underlying units. Sure, you can informally make the association, but if you were willing to formalize it, it would be stated explicitly in the point and inner product spaces. For example, your vectors would be currency or distances, your points would be capitals or locations, etc.There is no underlying point space, the n-tuples ARE the points. — fishfry
I'm all for contrarian thought, but I don't think I can help. Distances in the Euclidean geometry, where the theorem applies, are supposed to have units that persist in all directions. Your units change with direction, pure apples and pure dollars coaxially, and some shade in between in all other directions. The angles are also arbitrary. If you rescale your dollars to cents, you will skew the space (shear map it) and thus change all angles, yet nothing in the problem domain changes. You have no native semantics for your calculations and that makes our interpretations kind of, sort of, futile.I'm asking you to relax, bend, ignore, contradict the rules/principles/whathaveyou that's making you think that there's
no actual geometric model — TheMadFool
I don't think I can help — simeonz
Despite the wide use of Descartes' approach, which was called analytic geometry, the definition of Euclidean space remained unchanged until the end of 19th century. The introduction of abstract vector spaces allowed their use in defining Euclidean spaces with a purely algebraic definition. This new definition has been shown to be equivalent to the classical definition in terms of geometric axioms. It is this algebraic definition that is now most often used for introducing Euclidean spaces.
...
A reason for introducing such an abstract definition of Euclidean spaces, and for working with it instead of R n {\displaystyle \mathbb {R} ^{n}} \mathbb {R} ^{n} is that it is often preferable to work in a coordinate-free and origin-free manner (that is, without choosing a preferred basis and a preferred origin). Another reason is that there is no origin nor any basis in the physical world. — Wikipedia
How are you detaching from the use of preferred origin and axial orientation then? — simeonz
Since, obviously, you are defining some point to be (0, 0), — simeonz
and some vectors to be (0, 1), (1, 0). — simeonz
For the universe, (0, 0) would be its center of gravity, — simeonz
or some other choice that someone deems excellent, for example. — simeonz
But in my variant, the choice is made by the use of Cartesian coordinate system which uses orthornormal basis and origin after the fact. The underlying space has no (0, 0) in it, just abstract locations, and there are no special orientations or planes, just abstract vector directions. — simeonz
Wikipedia — simeonz
I don't doubt that such sources are authoritative in their own right. I think that such treatment is a little outdated in style, because the mathematics skip a little modern abstraction, in pre-Russellian (pre-Frege) manner of thought. — simeonz
I do not oppose the dot product and metrization you provide. It indeed fits the axiomatic requirements. Affine spaces can be defined over n-tuples (as both point and product spaces) and that Cartesian coordinate systems can simply be rigid transformations over some preferred innate coordinates. However, I have something else in mind. — simeonz
When you copy Wiki paragraphs could you please give the full link? I can't search every Wiki article on coordinate systems, vector calculus, Euclidean space, inner product spaces, and so forth in order to see what the context is. — fishfry
But Euclidean space is in fact the coordinate system. Euclidean space is exactly the set of n-tuples with the usual norm, distance, and inner product. — fishfry
Many places define it this way. — simeonz
No. In fact, it is one of the few places which concurs with the manner in which I was taught to think of analytic geometry. Not as working with numbers directly, but with coordinate systems that use vector bases to define numeric representations of the underlying coordinate free space.Can you at least tell me, did you come by your ideas solely from reading this article? — fishfry
I can see that you are being polite. Thanks for not sending me out with a curse. — simeonz
Even if the Minkowski space were Euclidean, because there is room for disagreement on definitions (edit: its inner product is not suitable, so it shouldn't be), there is at least some pertinent relationship between time and space in it. There is specific distance that field interactions can traverse in a given amount of time, assuming no significant gravity.As it seems to me space-time is equivalent to apple-dollars. — TheMadFool
Exactly, nomenclature or not. Not all philosophical differences translate to definitions and definitions are merely conventions. That is, there is always going to be some contention and heat on the issue, of who establishes the right linguistic terms for mathematics. I am contented to use either, as long as people understand the philosophical distinction and we can talk about thatSo there's some subtle philosophical difference between a coordinate system imposed on an object, versus the coordinate system being the object. — fishfry
I don't want to drag you back into a dispute. We can agree to disagree. I'll be fine with any compatible definition, proviso the ideas for its proper application are the same. — simeonz
Exactly, nomenclature or not. Not all philosophical differences translate to definitions and definitions are merely conventions. That is, there is always going to be some contention and heat on the issue, of who establishes the right linguistic terms for mathematics. I am contented to use either, as long as people understand the philosophical distinction and we can talk about that — simeonz
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