The Hypotenuse Problem (I am confused)

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• 8.7k
I think I get what you mean but you must surely have had the experience of not being able to say "no, this is impossible" to someone, someone you love dearly perhaps, and then quickly thought of something appropriate - the next best thing to say. Imagine you're in the same position with me (not that I'm a lovable character). What's, for you, the next best thing to say?
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What's, for you, the next best thing to say?
I'm not sure I get that. You riddle me too hard a riddle there, I'm afraid.
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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.

Oh my. I think that's hopelessly convoluted, where did you get it?

Here's what Euclidean space is. My reference here is for example Calculus on Manifolds by Spivak, page 1. [That's a pdf link].

Given the real numbers $\mathbb R$ and a positive integer $n$, Euclidean n-space is defined as the set of n-tuples $(x_i)$ with norm $\displaystyle |x| = \sqrt{\sum_{i=1}^n x_i^2}$. Spivak writes his indices upstairs ($x^i$ rather than $x_i$) in the manner of differential geometers, but we need not do that here.

The rest of what you wrote is overloaded with what software developers would call cruft. There is no underlying point space, the n-tuples ARE the points. There is no underlying Euclidean space, it's the norm defined on the n-tuples that characterizes Euclidean space. And an inner product is just an abstraction of the dot product, there's no isomorphism going on. It's true that one could in theory define different inner products on Euclidean n-space but I believe (if I recall and I didn't take the trouble to look this up) that they're all related by a linear scaling factor. Or at worst they all induce the same topology via the metric $d(x,y) = |x - y|$ so there's no important difference. I could be wrong, maybe there's some weird inner product you could put on the n-tuples but I don't see how that's important here.

I don't think I should comment on the rest of what you wrote because you have a lot of extra baggage in that one paragraph that's leading to a lot of conceptual confusion. So let's stay here and work this part out.
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I'm not sure I get that. You riddle me too hard a riddle there, I'm afraid

I feel the same way you're feeling, maybe we're in identical situations. All I can say is you're coming at the issue from a rather conventional point of view. 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
.
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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.

That's like saying I'm going to the store for oranges but I need to buy fruit as well. Oranges are the fruit I need to buy. An inner product is an abstraction of the dot product. You can call the dot product the inner product if you like and I usually do. But you are making a distinction that's not really there and introducing confusion. Is this something you got from a book? Maybe this is something I don't know about. You don't have a dot product AND an inner product. You have a dot product which can also be CALLED an inner product. They're the same thing, namely $\displaystyle x \cdot y = \sum_{i = i}^n x_i y_i$ where the $x_i$'s are the coordinates of $x$ and likewise for the $y$'s.

But I was saying that there is one more hop (probably) in my mind to how this intuition translates to Cartesian coordinates.

That hop is indeed in your mind and you're confusing the issue. The n-tuples ARE the points and the Euclidean norm is ALL the structure you need to define the distance and the dot (or inner) product, which gives you all the structure of Euclidean space.

You're thinking that the n-tuples are imposed on top of an existing space, and perhaps for some purposes that might be a useful point of view, but I don't think it's helpful here.
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Here's what Euclidean space is. My reference here is for example Calculus on Manifolds by Spivak, page 1
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.
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
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.

The rest of what you wrote is overloaded with what software developers would call cruft.
I personally do not object to extra formalism, if it makes the assumptions more explicit. But I see how you may feel this way.

There is no underlying point space, the n-tuples ARE the points.
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.

You can see how the OP assigns the n-tuples to the dollar-apple combinations. There is nothing irregular in that. But they define the dot product and metric, just as you do, and henceforth, they use the Pythagorean theorem and all associated geometrical intuitions from Euclidean geometry to their real-world interpretation. The idea of using intermediary mathematical structure is, that the affine space and its associated inner product space have no mechanical definition of inner product. It comes from the problem domain. It is supposed to exist "natively", not to be defined mechanically, like the dot product. Its properties must be verified before it can be used. So, instead of asking what does the dot product imply in reality, you have to ask what does the reality imply for the inner product. (Reminds of the famous Kennedy quote: " ask not what your country can do for you, ask what you can do for your country.") This is a shield against semantic errors. Nothing more, nothing less. You can bypass it, but then you have made one step in your mind implicitly that you could formally explicate.
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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
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.
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I don't think I can help

Sorry to hear that. You come across as more than capable of coming up with a good response. Too bad. Thanks.
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Sorry to hear that.
It's not a big deal. You will live.
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I should also say, that if I was having a practical problem, I would probably never deal with underlying vector spaces explicitly. I would only think about them. But if I was discussing semantics, in a question like this one, my investigation would be meta-mathematical, so I would talk about inner product spaces and fluid coordinate choices directly.
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I just realized that the hypotenuse isn't just its length, there's also slope to consider and the slope of the hypotenuse does have meaning - the going rate for 3 apples. I'm still not satisfied with the answers other posters have been kind enough to offer.

In Minkowski space-time (I hope I got this right), the hypotenuse, if the x axis is time and the y axis space, does have a meaning - it's the worldline of the object passing through 4D space-time and here too, like my apple-dollar scenario, the two axes are measuring totally different quantities. Whether that actually means something or whether it leads to an answer to my original question is not clear to me.

As it seems to me space-time is equivalent to apple-dollars.
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I would probably never deal with underlying vector spaces explicitly.

There is no "underlying" vector space. The n-tuples ARE the vector space.
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How are you detaching from the use of preferred origin and axial orientation then? Since, obviously, you are defining some point to be (0, 0), and some vectors to be (0, 1), (1, 0). For the universe, (0, 0) would be its center of gravity, or some other choice that someone deems excellent, for example. 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.

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

The emphasis is mine. And this last bolded statement is not really accurate. A lot of places, including many passages on wikipedia, use the old n-tuple defintion.
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How are you detaching from the use of preferred origin and axial orientation then?

Detaching from the use of preferred origin and axial orientation? I just can't parse that at all.

Since, obviously, you are defining some point to be (0, 0),

Yes, the ordered pair of real numbers (0,0).

and some vectors to be (0, 1), (1, 0).

Yes, the ordered pairs of real numbers (0,1) and (1,0), respectively. The ordered pairs ARE Euclidean space, they're not imposed on some underlying space.

I get that you must be making some point about coordinate systems, but your exposition is not adding clarity.

For the universe, (0, 0) would be its center of gravity,

I don't know anything about the universe. Other than what Einstein pointed out, that there is no preferred frame of reference. You can put the origin of a coordinate system anywhere you like.

or some other choice that someone deems excellent, for example.

Yes. We agree. You can put the origin anywhere that's convenient in any given context.

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.

Ok. I no longer know why I'm even in this thread. May I withdraw gracefully now? I have nothing new to add. Except that I read the OP's initial post (the OP's OP) and I don't think the hypotenuse means anything at all in the graph of apples versus dollars. It's like noticing that your thermometer reads 70 degrees Fahrenheit and that the mercury has reached a height of three inches above the base of the old-fashioned mercury-filled wall thermometer. The three inches is a true measurement, but it has no meaning in the context of measuring heat.

Wikipedia

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.

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.

Jeez, not that Michael Spivak needs the likes of me to defend his reputation. but this remark is a little off target. A lot off target in fact. Calculus on Manifolds is essentially a proof of the generalized Stokes' theorem from the viewpoint of modern differential geometry. It's a very modern book. despite its 1965 publication date. I wonder if you are thinking of something else.

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.

I can see that you do. It's just that you haven't explained it to me. I'll agree with you that we can impose a coordinate system on an arbitrary space, and that the space isn't the coordinate system. 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. That doesn't mean that your ideas about coordinate systems are wrong, it just means that after all this I still can't figure out why we're having this conversation. I should have quit while I was behind a long time ago.

As far as the OP, the length of the hypotenuse doesn't mean anything at all in the context of the graph of the prices of apples.
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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.

I agree. It was lame of me not to offer link, but it was from the same page that I referred to previously and it is the main article for Euclidean spaces in Wikipedia. My bad. The first paragraph was from the history of the definition, and the second was from the motivation of the definition.

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.

Many places define it this way. This is a hands-down concrete computations centric definition which bypasses affine spaces altogether, or considers affine spaces to be just useful for other types of computations (signal analysis and control engineering as you pointed out). It is how physicists and engineers usually think. They would similarly argue that real numbers are a particular set (such as Dedekind cuts in rational numbers), not just a type of mathematical structure - such as complete ordered field.
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Many places define it this way.

I'm going to gracefully bow out, or turn tail and run, as the case may be. I find myself passionately defending my side of an argument without even knowing what the argument's about.

Let me just refer you to the Talk page for the article in question, https://en.wikipedia.org/wiki/Talk:Euclidean_space . It has many passionate and bitter responses to the article that mirror some of the concerns expressed in this thread. IMO the article itself is a mess. But even so, after waving their hands and confusing the issue massively, and clearly inducing many of the confusions that you've been expressing, they finally give a technical definition:

A Euclidean vector space is a finite-dimensional inner product space over the real numbers.

Which frankly is, on the one hand, at least consistent with my definition as a set of ordered n-tuples with the Euclidean norm; but on the other, is a little messy, because an inner product space is a far more complicated thing than a Euclidean space. Suppose we look up what's an inner product space? We find that, "In mathematics, an inner product space or a Hausdorff pre-Hilbert space[1][2] is a vector space with a binary operation called an inner product." Well that's helpful. If you've studied Hilbert spaces or functional analysis or quantum physics, or know what a Hausdorff space is, and know the difference between a Hilbert space and a pre-Hilbert space, you can maybe figure out what they mean by a Euclidean space.

I pronounce this article hopeless. The Wiki article is trying to blend too many disparate concepts from history and modern practice, trying to be both technical and beginner-friendly, and in the end obscures more than it clarifies. I wonder if you got your ideas just from reading this disaster of an exposition. The Talk page is unusually passionate, as Wiki Talk pages go, in their objection to the content of the main article. You should give it a read. The first paragraph is titled, "Wrong, wrong, wrong," and the rest of the Talk page goes on from there.

Can you at least tell me, did you come by your ideas solely from reading this article?

Let me suggest this. Ignore the Wiki article entirely. A Euclidean space of dimension n is the set of ordered n-tuples of real numbers with the Euclidean norm |x| as I defined it earlier. That definition requires only that you know what an ordered n-tuple of real numbers is. It's accessible to high school students. And from it, you can derive ALL of the properties of Euclidean space including the metric, the inner product, and the vector space and Hilbert space structure. That's the right definition.
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Can you at least tell me, did you come by your ideas solely from reading this article?
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.

I think that many people are opposed to the categorical style of thinking, that we are not defining mathematical structures to get the computations off the ground, but to abstractly define the conditions in which those computations are possible.
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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.

Ok. I'm out of ammo. Maybe you're right.
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Ok. I'm out of ammo. Maybe you're right.

I can see that you are being polite. Thanks for not sending me out with a curse.
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I can see that you are being polite. Thanks for not sending me out with a curse.

Well, like I say, I'm not entirely sure what we're disagreeing about. And you did actually make me think that I could be missing some subtleties. I know that we can impose a coordinate system on an already-existing object. And I know that Euclidean space is defined (at least by Spivak) as the set of n-tuples itself. So there's some subtle philosophical difference between a coordinate system imposed on an object, versus the coordinate system itself being the object. So I don't think you're entirely wrong. In any case I find the Talk page to that Wiki article interesting, as some of these points are brought out; for example the distinction, or lack of distinction, between $E^n$ and $\mathbb R^n$. And I'm only rude on this site to people who really really deserve it, and not that often.
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As it seems to me space-time is equivalent to apple-dollars.
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.

In your case, if you say that an apple costs one dollar as a fixed market price, that would give some semblance of similarity. So, you have a company, you have active assets - in fruit inventory and in currency, you know how they translate to each other in value, and you are trying to guesstimate both of them by least squares regression. The distance measures by how much you are off. And the vectors are orthogonal, when they exert influence that is separable in some sense - no projection on each other.

Either way, it is difficult for me to imagine the idea of right angles. May be we are talking about noise vectors, and we are considering perpendicularly acting sources of noise. I know I might be talking with hand-wavy terms and be inarticulate here, but the point is to give you just a general direction. I doubt that I could invent a completely sensible real example.
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So there's some subtle philosophical difference between a coordinate system imposed on an object, versus the coordinate system being the object.
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
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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.
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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.

I read the entire Talk thread and have concluded that I no longer have any idea what a Euclidean space is. LOL. However someone in that thread did reference Spivak's monumental Comprehensive Introduction to Differential Geometry, which on page 1 defines Euclidean space as the set of n-tuples of reals with the usual inner product. So again. clearly this is the modern analytic definition, but it apparently sidesteps the subtleties of classical and affine geometry. But my own preference is for the analytic treatment; just as I view an angle as being defined analytically as an arccosine after the cosine has been defined as the real part of the complex exponential, which itself is defined by a differential equation or a power series. There's no longer any geometry involved in the modern definition of angles; although of course one is free to use one's intuition, as we all do.

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

Ok. Spivak is a differential geometer. He wants to associate, or attach, a little copy of $\mathbb R^n$ to every point of a differentiable manifold. This is the viewpoint of modern geometry and in particular general relativity in physics. Or as Einstein said, once he got his theory back from the mathematicians he no longer understood it.

But in this point of view, there's no underlying space at each point that we coordinatize. Rather, there is a copy of Euclidian n-space at every point of some manifold, meaning a set of n-tuples. There's no secret underlying space under the coordinate space.

But you know, you did elevate me to a higher state of confusion. I'd be the first to agree that if we have a plane, it makes no difference where the origin is. But then the coordinate system isn't the plane and never was the plane. The plane is logically prior to the coordinate system. But from the modern point of view, the coordinate system IS the plane. Or at least it's the Euclidean space. So I have definitely become more confused but at a higher level.
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