The Hypotenuse Problem (I am confused)
Suppose a 2 D space with a y-axis and an x-axis at the usual 90 degree angle to each other. You're at the origin (0, 0) and travel in this space, in a straight line, to the point (3, 4) which basically means you travelled 3 units along the x-axis and 4 units along the y-axis. We could form a right triangle with 3 as its base and 4 as its height. What distance did you actually travel? The actual distance you traveled is the hypotenuse given by square root of [(3^2) + (4^2)] = square root of 25 = 5 units.
Now imagine you're a shopkeeper and you sell apples at the rate: 3 apples for 4 dollars. In this case, every 3 units along the x-axis (you give 3 apples), there are 4 units along the y-axis (you receive 4 dollars). These values too will form a right triangle. My question is what does the hypotenuse mean? We understand the base is 3 apples sold, the height is 4 dollars received but what, if any, meaning can be assigned to the hypotenuse?
Since 3 is apples and 4 is dollars, the hypotenuse = square root of [(3^2) apples^2 + (4^2) dollars^2] doesn't make sense since we can't add apples^2 to dollars^2.
A way to go about making sense of the hypotenuse would be to replace the 4 dollars with 3 apples (since 4 dollars = 3 apples in terms of value) and we get hypotenuse = square root of [(3^2) apples^2 + (3^2) apples^2] = square root of [(2)×(3^2) apples^2] = (1.414)×(3 apples) = 4.242 apples. Since in a 2D space the hypotenuse is the actual distance traveled, it follows then that, in actuality, 4.242 apples were sold.
Likewise, replacing 3 apples with 4 dollars (since 3 apples = 4 dollars in value) we get the hypotenuse = square root of [(4^2) dollars^2 +'(4^2) dollars^2] = square root of [(2)×(4^2) dollars^2] = (1.414)×(4 dollars) = 5.656 dollars. Since, sorry for the repetition, in 2D space, the hypotenuse is the actual distance traveled, it again follows that the actual amount of money made is 5.656 dollars.
The original rate is 3 apple for 4 dollars = 0.75 apples per dollar and the rate calculated after replacing dollars with apples and vice versa is (4.242 apples)/(5.656 dollars) = 0.75 apples per dollar.
How is it possible that when you sold 3 apples, you actually sold 4.242 apples and when you made 4 dollars, you actually made 5.656 dollars?
[note: square root of 2 = 1.414]
I know I'm making a mistake. I just don't know where. Help!!!
Now imagine you're a shopkeeper and you sell apples at the rate: 3 apples for 4 dollars. In this case, every 3 units along the x-axis (you give 3 apples), there are 4 units along the y-axis (you receive 4 dollars). These values too will form a right triangle. My question is what does the hypotenuse mean? We understand the base is 3 apples sold, the height is 4 dollars received but what, if any, meaning can be assigned to the hypotenuse?
Since 3 is apples and 4 is dollars, the hypotenuse = square root of [(3^2) apples^2 + (4^2) dollars^2] doesn't make sense since we can't add apples^2 to dollars^2.
A way to go about making sense of the hypotenuse would be to replace the 4 dollars with 3 apples (since 4 dollars = 3 apples in terms of value) and we get hypotenuse = square root of [(3^2) apples^2 + (3^2) apples^2] = square root of [(2)×(3^2) apples^2] = (1.414)×(3 apples) = 4.242 apples. Since in a 2D space the hypotenuse is the actual distance traveled, it follows then that, in actuality, 4.242 apples were sold.
Likewise, replacing 3 apples with 4 dollars (since 3 apples = 4 dollars in value) we get the hypotenuse = square root of [(4^2) dollars^2 +'(4^2) dollars^2] = square root of [(2)×(4^2) dollars^2] = (1.414)×(4 dollars) = 5.656 dollars. Since, sorry for the repetition, in 2D space, the hypotenuse is the actual distance traveled, it again follows that the actual amount of money made is 5.656 dollars.
The original rate is 3 apple for 4 dollars = 0.75 apples per dollar and the rate calculated after replacing dollars with apples and vice versa is (4.242 apples)/(5.656 dollars) = 0.75 apples per dollar.
How is it possible that when you sold 3 apples, you actually sold 4.242 apples and when you made 4 dollars, you actually made 5.656 dollars?
[note: square root of 2 = 1.414]
I know I'm making a mistake. I just don't know where. Help!!!
Comments (54)
For spatial coordinates, the "unit" along each axis is an arbitrary interval of space and applies in any direction whatsoever, which is why it can also be used to measure the hypotenuse. Moreover, the axis orientations are likewise arbitrary, so you could rotate them and say that you simply moved 5 units along the revised x-axis (or y-axis).
Apples and dollars do not work that way. Apples are truly discrete units that we can count individually, and dollars are units of exchange that measure value. When you assign apples to one axis and dollars to the other, you are simply representing the relationship between these two different units as a line corresponding to an equation: dollars = 4/3 x apples; the hypotenuse has no meaning at all. The same is true even if you assign both axes to apples or dollars; then the equation is apples = apples and dollars = dollars, while the hypotenuse still has no meaning at all.
Firstly, how can that be? Consider the much bandied about term, worldline in the theory of relativity. If time were the x-axis and space the y-axis then, the hypotenuse would be the worldline of an object that began existing at the origin. It amounts to the actual experience of that object in spacetime. So, even if the two axes in a coordinate system are different it is possible for the hypotenuse to have meaning.
Secondly, suppose that one simply wishes, on a whim, to give the hypotenuse a meaning. What meaning would you give it? 5 apple-dollars? Just like 5 space-time (worldline)? If apple-dollars has no meaning then how can space-time wordlines have meaning?
Time and space are continuous in themselves, so any units assigned to them are completely arbitrary, and you cannot measure a worldline except along the two different axes. Apples are discrete and dollars are defined, so comparing them to time and space is like comparing ... apples to oranges.
Right! Taking x to mean apples, y to mean dollars, we get the equation for the hypotenuse (5 units) to be (3x)^2 + (4y)^2 = 5^2 which becomes 9x^2 + 16y^2 = 25. This, it turns out, is the equation of an ellipse. The major axis of this ellipse is 5/3 and the minor axis is 5/4. I don't know where to go with this. :chin:
How about nowhere, since it is nonsense.
:rofl: Are you a mathematician?
Any ideas? :chin:
To begin with, I don't understand why you think it's a nonsensical question.
If the x-axis and the y-axis were distances then 5 represents the actual distance between positions (0, 0) and (3, 4).
Similarly, if a moving object, p, were to begin at (0, 0) and travel 4 units of distance in 3 units of time, it would be at point (3, 4). In this case 5 would represent the worldline of the object p in spacetime.
I would like to give a little clarification on the answer provided by @tim wood. The application of the Pythagorean theorem is intended for spaces equipped with distances and angles. More precisely, for any affine space with associated inner product space or even a module. The theorem there follows simply from the properties required of the inner product, which then makes the statement a rather unhelpful fact. Essentially, the theorem is axiomatic.
The introduction of this abstraction in the first place aims to eliminate the need for making any point the fixed origin, and any orientation automatically vertical, horizontal, etc. However, because angles and distances are defined, you can make use of Cartesian coordinate systems by choosing one point arbitrarily and one orientation (the abscissa) as reference, allowing you to express the other points and directions using numbers produced by the metric and norm in the affine space. (In fact, for the Pythagorean theorem, you don't need to use numbers, but say booleans with conjunction being multiplication and logical exclusion being the addition. You don't need to fix the unit of your quantities at all for non-unitary module either. No dollars, no euro, just abstract monetary value. But that just complicates the intuition here.)
This trick is sensible only because the dot product between the n-tuples matches the inner product between the vectors. This can proven using the axioms of the affine spaces and the fact that the two basis vectors are orthonormal - perpendicular and unitary (for modules they can be chosen to be of identical magnitude, and the two products are not equal, but homomorphic). The other properties are also consequently preserved in the translation process - the Pythagorean property (I don't call it a theorem, because it stems axiomatically), distances computed using the Pythagorean property (ironically, but it is already preserved), the tangent (being the ratio in two dimensions) between the coordinate vectors and the abscissa, etc.
What I am getting at is that we are first taught coordinates at school. We start with the description of numeric pairs and move to their correspondence to points in space. Unfortunately, this introduces angles and distances generically, by working out some arithmetic with the coordinate numbers. Affine spaces reinforce the notion that you can't derive such concepts mechanically. They are ascribed to the space semantically, and only then they become numerically expressed.
In your particular case, you have two axes and you assume that they are perpendicular, i.e. that there is meaning to the angle between them. What does this angle express? You will find out that the main requirement of the Pythagorean theorem, the rightness of one angle, is untenable in your case, because the appearance of orthogonality between your basis vectors is only the remnant of some pedagogic intuition. As tim wood pointed out, the distance is not specified either, aside from strides along the coaxial directions. In summary, we can create a pair of numbers from virtually anything, but that does not confer the necessary structure for a Cartesian coordinate system corresponding to a choice of origin and orthonormal basis in some kind of affine space.
If memory serves, Pythagora's theorem works only for right triangles and yes the axes that I used are perpendicular and yes there's a right triangle (3, 4, 5) formed. There are 3 apples that come at a total cost of 4 dollars. What's the hypotenuse in terms of apples and dollars? That's all I'm asking.
The square root of the square of the number of apples plus sixteen. An amazing breakthrough in marketing!
I don't get your joke. We have two items in our list: apples and dollars, each of them forming a side of a right triangle. What's the hypotenuse in terms of apples and dollars?
If I ask a similar question with distance, the answer is quite obvious: the hypotenuse is the shortest distance between the two points that form the ends of the hypotenuse; the hypotenuse is a distance.
How did you come to the realization that the angle is perpendicular when constructing your mathematical model of the problem domain? What property in the actual domain of application of your model prompted the idea? If you were not concerned with any domain when you constructed the model, how do you come to need to ask semantic questions now?
Essentially, you are trying to bypass semantic questions when constructing your geometry, but you want to coerce semantics at a later date, through some synthetico-analytic magic.
P.S. The above came off a bit tacky. Not trying to be stern or anything...
A number representing the square root of the square of the number of apples plus sixteen
Here goes...
1. There's the usual 2D Cartesian coordinate system (x and y axes are perpendicular). I'm not willing to debate on the matter of whether the x and y axes are perpendicular or not because that's a given. Imagine now a person travels 3 m along the x axis and 4 m along the y axis. The person started from point (0, 0) and is now at the point (3, 4). If I know calculate the length of the hypotenuse it's 5 m and this 5 m is a distance just like the 3 m traversed along the x axis and the 4 m along the y axis. In other words, the hypotenuse makes sense to me - it's a distance, just like the adjacent and the opposite sides
2. There's, again, the usual 2D Cartesian coordinate system (x and y axes are perpendicular). This time, however, the x axis is labeled as apples and the y axis is labeled dollars. 3 apples cost 4 dollars. Here too there are 2 points: (0, 0) which represents 0 dollars for 0 apples and the other point (3, 4) which represents 3 apples for 4 dollars. This time too we can construct a hypotenuse which is 5 "something". 5 isn't dollars, neither is it apples. We have nothing else to choose from since we haven't anything else to choose from apart from apples and dollars.
:chin:
The whole point is that the 2D Cartesian coordinate system is not a picture. It is ascription of coordinates to some plane of points, which points correspond in pairs to vectors, which vectors individually correspond to lengths and in pairs correspond to angles. Ok, the points are dollars-apples, but the remaining properties are not automatic. They are not provided by the Cartesian coordinate system for you, mathematically. They are provided by you, originally, so that you can justify the use of Cartesian coordinate system. Otherwise, what you have are just pairs of numbers corresponding to points, and the rest is as real as Tolkien's world.
All I can say is that I'm talking about Descartes and you're talking about Descartes' father. We're talking past each other.
Sure, but I'm not following your point. The Pythagorean theorem goes back over a couple of thousand years, and Cartesian coordinates go back only to Descartes. We don't need an orthogonal coordinate system to have the theorem. Everyone agrees with that I'm sure. I am not understanding the point you're making. A right angle (if I remember my high school geometry) is when you have a line intersecting another line and making equal angles on each side. No coordinates or numbers needed.
My point is that you need to have the concepts of "angles" (so that they can be equal), "directions" (so that you can make the points on your lines aligned, i.e. colinear), "distances" apriori, before resorting to analytic geometry. (And formally, we would call that an affine space today. Although the terminological designation would not be present historically, the ideas would be the same.) It would be backwards thinking if we started with pairs of numbers, declared them to be the Cartesian coordinate system for implicit space of entities and finally tried to infer a sensible explanation of the nature of the metrics of those entities.
Don't you see what I mean when I say that you have only dollars and apples corresponding to pairs of numbers and no actual geometric model?
I guess I don't follow your point. The historical evolution is well known, from Euclid to Descartes. And in modern math we start with a 2-dimensional coordinate system and define the Euclidean distance. Either way works. What exactly is the question or issue?
I say that the Pythagorean theorem applies to affine spaces over inner product spaces, because in affine spaces any two points from the underlying point space map to a vector, and we have the requirement of distributivity for inner products, hence:
Thus, for any triangle, the square of the norm of the third side equals the sum of the squares of the norms of the other two sides plus their doubled inner product . If the sides are orthogonal to each other, the inner is zero, hence the theorem. The trick then is to guarantee that we preserve the inner product when we move to n-tuples, which we do with a coordinate system. A Cartesian coordinate system is assignment of n-tuples to points, such that the implicit basis corresponding to the strides in unit distances along the coordinate axes is orthonormal. It cannot be orthonormal, if we haven't ascribed angles and distances.
Quoting fishfry
I don't see it that way really. We still come from the geometric perspective, to define angles and distances in one way or another, and only then we have the privilege of calling an n-tuple of points being from a Cartesian coordinate system. Cartesian coordinate systems come with semantics that need to be defined apriori. They are not just mechanical assignment of pairs of numbers to some arbitrary point space.
There is only one sense, in fact, in which I am not correct. And it is that a Cartesian coordinate system might be a applied to the very n-tuples, with vectors being n-tuples, distances and angles computed in the usual way, etc. But then, we couldn't talk about apples and dollars, because since the underlying point space is just a mechanical bonding of numbers, it is unitless.
Edit: That is my perspective anyhow. That is how I was taught analytic geometry at uni. But I am a software guy, so you may wish to exercise some reasonable caution and not take my word for it.
I know that the Pythagorean theorem can be proven with constructive geometry, with only areas of aligned triangles. This is however not about Cartesian coordinates as far as I am concerned, because it is not about analytic geometry. In fact, the irony is, that you would most likely use such constructive proofs to validate the sensibility of the assumptions of affine space to your application domain before moving to analytic geometry. You would use the constructive proof to guarantee that the inner product for orthogonally directed vectors is zero (involving also the definition of inner product through cosine of angles and distances), and then you would also get the Pythagorean theorem to your orthogonally directed vectors in the affine space, making the exercise a little vacuous. But the constructive proof requires domain-level intuition - moving triangles, aligning their sides, etc. Those are not analytic. They are intuitional.
That sounds right. I don't know much about affine spaces. But basically an affine space is a vector space that's "forgotten its origin" and you don't need any privileged origin to have the Pythagorean theorem be true.
But you are saying this as if someone is denying it. I don't think anyone is denying that the Pythagorean theorem is false in affine spaces. Help me understand what is the point of the thread. I don't think anyone disagrees with what you said here.
Quoting simeonz
No that's not true. We define [math]\mathbb R^2[/math] as the set of ordered pairs of real numbers. Then we define the usual Euclidean Euclidean distance, and we define the usual dot product. Then the angle between two vectors is the arccosine of the dot product of their normalized versions. That is,
[math]x \cdot y = |x| |y| \cos \theta[/math] so that [math]\theta[/math] can be defined as [math]\arccos(\frac{x \cdot y}{|x| |y|})[/math]. I assume you agree. And we can even formalize the arccosine by defining the cosine as the real part of the complex exponential function, and the arccos as its inverse. All this can be done without reference to geometry and we can even define angles without geometry. I'm guess you know this but disagree for some reason?
Quoting simeonz
No they don't and yes they are. You just define Euclidean n-space or in general you can define an abstract inner product space and everything works out fine without geometric semantics. For example if instead of Euclidean n-space we can work in generalize inner product spaces and all the theorems carry over directly. I can't see the point of objecting to this but maybe I'm misunderstanding you.
https://en.wikipedia.org/wiki/Inner_product_space
Quoting simeonz
There's no Cartesian coordinate system in an inner product space but there is a notion of an orthonormal basis. That's Fourier series, functional analysis, and quantum physics based on Hilbert space. All this is standard. I don't follow your point.
Well those are not mutually exclusive. Of course we use geometric intuition to get the analytic approach off the ground, but that's true of everything. Is that what you're saying, that we need the ancient geometric intuition to ground the modern analytic approach? Perfectly well agreed. But again, what of it? Nobody's disagreeing.
ps -- I see that you're not the OP. I should quit while I'm behind here. What I know about all this is that inner product spaces are a vast abstraction of ancient Euclidean geometry. But who would disagree? The law of cosines was known to Euclid and is the same concept as the dot product.
I still want to be certain that you concur with me on the definition of Cartesian coordinate systems.
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. And Euclidean space is indeed a special case of affine space (something I blurred over a little here), such that the vector space is inner product space and the field of the inner product space is ordered and complete (or which would be the same - the real numbers). Other then that, the point space and vector spaces are arbitrary. That means that we have already defined an inner product (not just dot product for the n-tuples) somewhere. And we can't merely define the dot product over the n-tuples and map the n-tuples to points (mind you, by points I don't even mean locations, but objects that conform to the requirements), or we would have what appears to be considered "generalized" version of the Cartesian coordinate system. There is no geometric sense in it. It is more or less, assignment of n-tuples to points, which can be manipulated with arithmetic.
As I said, we can also use n-tuples as the underlying inner product space and n-tuples as the underlying point space, but then we cannot talk about dollars and apples, or sensible angles and distances, because those are mechanical constructions now.
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.
Quoting fishfry
To some extent. But I was saying that there is one more hop (probably) in my mind to how this intuition translates to Cartesian coordinates. We first justify the requirements of the affine spaces with the constructive proofs, such as the properties of the inner product in the inner product space. Then we assign n-tuples to the points in the point space, proving that we preserve the inner product with the dot product. Since, in the OP's question there is no inner product, just dot product, there is nothing to preserve and no Pythagorean semantics to be had. We either have arbitrary assignment of numbers to points somewhere, in some semantic domain, or we work with numbers as our semantic domain, and those numbers have no units.
I'm not sure I get that. You riddle me too hard a riddle there, I'm afraid.
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 [math]\mathbb R[/math] and a positive integer [math]n[/math], Euclidean n-space is defined as the set of n-tuples [math](x_i)[/math] with norm [math]\displaystyle |x| = \sqrt{\sum_{i=1}^n x_i^2}[/math]. Spivak writes his indices upstairs ([math]x^i[/math] rather than [math]x_i[/math]) 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 [math]d(x,y) = |x - y|[/math] 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.
Quoting simeonz
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
Quoting simeonz.
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 [math]\displaystyle x \cdot y = \sum_{i = i}^n x_i y_i[/math] where the [math]x_i[/math]'s are the coordinates of [math]x[/math] and likewise for the [math]y[/math]'s.
Quoting simeonz
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.
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.
[quote=Wikipedia]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.[/quote]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.
Quoting fishfry
I personally do not object to extra formalism, if it makes the assumptions more explicit. But I see how you may feel this way.
Quoting 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.
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.
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.
Sorry to hear that. You come across as more than capable of coming up with a good response. Too bad. Thanks.
It's not a big deal. You will live.
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.
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.
There is no "underlying" vector space. The n-tuples ARE the vector space.
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.
[quote=Wikipedia]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.[/quote]
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.
Detaching from the use of preferred origin and axial orientation? I just can't parse that at all.
Quoting simeonz
Yes, the ordered pair of real numbers (0,0).
Quoting simeonz
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.
Quoting simeonz
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.
Quoting simeonz
Yes. We agree. You can put the origin anywhere that's convenient in any given context.
Quoting simeonz
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.
Quoting 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.
Quoting simeonz
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.
Quoting simeonz
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.
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.
Quoting fishfry
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.
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:
[i]
A Euclidean vector space is a finite-dimensional inner product space over the real numbers.[/i]
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.
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.
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.
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 [math]E^n[/math] and [math]\mathbb R^n[/math]. And I'm only rude on this site to people who really really deserve it, and not that often.
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.
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
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.
Quoting simeonz
Ok. Spivak is a differential geometer. He wants to associate, or attach, a little copy of [math]\mathbb R^n[/math] 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.