the independence of dimensions

March 2, 2009 11:59 AM

Does the curvature of space mean that the dimensions of height, breath and width are not independent of each other?

This is more of a conceptual question than a physics question. I am currently developing a dimensional model to explain the 'conceptual space' of emotions, and I insist that the dimensions (or variables) chosen have to be strictly independent of each other, or else the model is inefficient and one has used the concept of 'dimension' badly. I then got a criticism that because space is curved this shows that dimensions need not be independent, or orthogonal to each other.

I don't know enough physics to properly refute this point, but it seems to me that the curvature of space doesn't show that the dimensions themselves are non-orthogonal, but simply that the 3 dimensions do not fully describe physical space. Curvature should not be explained by the mutual constraint/influence of those 3 dimensions, but the situatedness of physical space in a higher dimensional structure. It's not like if you moved to the right or left for any amount of time you would wind up also moving a bit upwards or forwards (even if you would move in a great circle)? Can someone who knows more about these things settle this??

posted by leibniz to science & nature (45 comments total)

I would say this is largely an irrelevent question, as what you're doing is borrowing a three dimensional way of mapping that assumes all dimensions are independant (e.g., Euclidean, solid geometry) for the purposes of what you wish to achieve. The fact that non-euclidean geometries exist, and that curves can occur in realspace owing to gravity and wotnot, is rather beside the point.

To answer your actual question, though, check out the section on embedding in the curves link above. I'm not quite sure if that's what you're after but it may be germane. Sorry to only half answer but I am suddenly pressed for time owing to landlord arrival.
posted by Sparx at 12:16 PM on March 2

Curvature should not be explained by the mutual constraint/influence of those 3 dimensions, but the situatedness of physical space in a higher dimensional structure.

This is a common misconception.

The three-dimensional space we observe everyday objects in, and go about our business in, is so nearly Euclidean (i.e., not curved) as to be indistinguishable from entirely non-curved space. One result of that is that it is difficult for the human mind to comprehend curved space.

So, we use an analogy: imagine a two-dimensional space, and in it a two-dimensional being. However, unknown (at least through casual observation) to this two-dimensional being, the space he's living in is the surface of a sphere. And we draw all sorts of conclusions and observations from this.

This is a useful teaching tool, as the curved two-dimensional space is visualized as existing in non-curved three-dimensional space. We know what a sphere is and how it behaves, because we see spheres in our everyday, nearly-non-curved three-dimensional space.

What people forget at this point is that this is a model for curved two-dimensional space. We visualize curved two-dimensional space as existing within a non-curved three dimensional space because that's easy for us to imagine. But it's a mistake to conclude that if curved two-dimensonal space exists, there must be some third dimension, unseen to the two-dimensional being. Nothing in the mathematics of curved two-dimensional space requires a third dimension. Likewise, the fact that our universe is non-Euclidean—is "curved"—does not imply that some unseen fourth spatial dimension exists. (Other spatial dimensions may exist, but the fact that the universe is non-Euclidean is not evidence of that.)

However, I think the underlying question is really a mathematical and philosophical one, not a physics one. If you believe completely independent, orthogonal variables are necessary, I don't see why an analogy to physical dimensions is relevant. So what if spatial dimensions aren't independent (although they are)? Why would that mean your variables shouldn't be independent?
posted by DevilsAdvocate at 12:18 PM on March 2 [2 favorites has favorites]

Uh, the curvature of physical spacetime (near concentrations of mass-energy) has absolutely no bearing on creating an abstract mathematical model of n dimensions.

This objection is equivalent to objecting to a survey because two of the questions exhibit a Heisenberg-like minimum multiplied uncertainty in the answers; it's someone using unrelated knowledge in a completely different domain to intimidate you by inapplicable analogy and the use of similar terms that have different meanings in different contexts. Pure gibberish, isn't it?

With regards to curvature itself, eliminate the word "should" from your vocabulary; that word encourages preconceived notions to taint your thought processes.

Curvature is funny in physics. You can move left and somehow end up in a slightly different spot than you intended in a curved spacetime. Forget the nonsense about the escape velocity from a black hole exceeding the speed of light (that's a happy coincidence); you cannot escape from a black hole because, once inside the event horizon, the curvature of spacetime is such that, in a weird way, all directions head down to some degree, like some nightmare where every step you take away from the fanged clown leaves you mysteriously closer to its lipstick-smeared maw.
posted by adipocere at 12:20 PM on March 2 [1 favorite has favorites]

What's worth noting about DevilsAdvocate's model is that the 2D beings can tell (though it may be difficult) that their space is curved. They just won't necessarily be able to use a 3D model to think about it.
posted by dfan at 12:21 PM on March 2

I guess the question is: is the assumption that dimensions should be independence a fair one?

I was thinking, if on the two dimensions of the earth's surface, me and my friend start one foot apart and both move in a 'straight line' west (if that could be determined without a frame of reference??), we should eventually cross each other's path. But in this case maybe neither of us has moved north or south, but we've both moved in the third dimension?
posted by leibniz at 12:24 PM on March 2

Read Flatland by Edwin Abbot. (note, it's on project gutenberg and other places. See the links at the bottom)

Bottom line, I think you are correct-ish in saying that in our 3D worlds, we can only express orthogonality in three directions. However, just like you can project a 3D image onto a 2D surface, you can also project higher order objects into a 3D or 2D representation (sorta like a shadow).

Basically, in a higher order dimension, you would need similar words to height, width, depth that would describe orthogonal directions to those 3.

Take a point. Create another point orthogonal to that point at distance X and you get a line of length X.

Take that line and copy/drag it orthogonal another distance X and you get a square with the sides = X

Take that square and extrude it orthogonal to itself another distance X and you get a cube with all sides = X.

Now, here's the fun part. Take that square and extrude(?) it orthogonal(?) to itself by distance of X and you get a hypercube with sides(?) = X.

We can't conceive of this object (without resorting to 2D/3D simplifications) but there IS a concept of orthogonality at higher dimensions.

Ok. Hope that helps. I hope someone who spent more than 6 hours one night in college reading up on this stuff comes in with REAL Science :)
posted by johnstein at 12:24 PM on March 2

By the way, if you want to look up the math behind this stuff, the magic word is "manifold". An n-dimensional manifold looks like normal n-dimensional space at a small enough scale but at a larger scale you can notice how it's distorted. For example, a hollow sphere is a 2-dimensional manifold.
posted by dfan at 12:28 PM on March 2

I guess the question is: is the assumption that dimensions should be independence a fair one?

1) Possibly, although I don't know enough about psychology to say. Is the assumption of independent variables common in psychological models?

2) This has absolutely nothing to do with physical, spatial dimensions. Perhaps the use of the term "dimension" is confusing you. Replace it with "variables" in your thought process if that helps.

Perhaps the person offering the criticism was trying to illustrate that it can be very hard, if not impossible, to find independent variables in psychology. How would you do that, exactly? Obviously, any two emotions are not going to be independent. Whether there's some more advanced mathematical functions based on emotions that are independent of each other, I couldn't say, but perhaps that's what this person was getting at. In which case the analogy is a bad one, but the underlying criticism might still be valid.
posted by DevilsAdvocate at 12:36 PM on March 2

Yes, this is a math problem, and not a physics problem.

Locally, as dfan says, for any n-manifold you can always choose a set of n coordinates that are orthogonal to each other. (You don't have to though, but you can. It's convenient if you do.)

But if you walk far enough-- e.g. on the globe if you walk to the north pole-- those coordinates might not be well defined anymore.


By the way, even if we were only two dimensional beings, if we measured the angles of triangles very very accurately... we'd find that triangles on the surface of the earth always have angles that add up to slightly more than 180 degrees. This is one example of how you can tell a surface is curved without reference to another dimension.

(Somewhat unrelated, but I have to mention, is that space, on a universe-sized scale, isn't actually curved. In fact it's very very close to flat, see flatness problem. Spacetime however has curvature even on a universe-sized scale).
posted by nat at 12:41 PM on March 2

I am currently developing a dimensional model to explain the 'conceptual space' of emotions, and I insist that the dimensions (or variables) chosen have to be strictly independent of each other, or else the model is inefficient and one has used the concept of 'dimension' badly.

So you have some variables tapping into emotional content, and you're saying that either:

(1) Each of these variables are perfectly uncorrelated with every other variable, or
(2) The model is complete crap

?

I then got a criticism that because space is curved this shows that dimensions need not be independent, or orthogonal to each other.

If intended sincerely as a statement of that concern, this is a dumb criticism. Whether space is or isn't curved has absolutely nothing whatsoever to do with the mutual orthogonality of abstract concepts.

That said, that seems a strange thing to be so insistent on. If you're modeling emotional space as a hyperspace of mutually orthogonal dimensions, okay. Obviously I don't work in your field, but why do you need to insist so strongly that these variables really are mutually orthogonal? Like DevilsAdvocate, I suspect that the person who offered this criticism didn't mean it literally.
posted by ROU_Xenophobe at 12:50 PM on March 2

I am currently developing a dimensional model to explain the 'conceptual space' of emotions, and I insist that the dimensions (or variables) chosen have to be strictly independent of each other,

Simple solution: Don't do that. There's no reason to think emotions are orthogonal.

And anyway, there are all different types of mathematical spaces. Euclidean and non-euclidean spaces are different types of metric spaces. But either way, there are plenty of orthogonal spaces to think about in math, such as the nullspace of a matrix.
posted by delmoi at 12:55 PM on March 2

Yes, the assumption that dimensions should be independence a fair one. If you want to make a model where one measurement is weight of a person and the other measurement is month of birth it is likely that one has no relation to the other. On the other hand weight and number of ice cream sandwiches eaten a week might not be independent, they might correlated.

If you make a model where you assume that dimensions are independent but, when analyzing your observations, you find that the dimensions are not independent, then your model is faulty.

If you are creating a model to support a theory and have already decided that it will flawlessly account for all observations, then these finer points can be ignored.
posted by bdc34 at 12:57 PM on March 2

Just for background- the current dimensional model popular in emotion psychology uses two dimensions: arousal and valence (positive-negative). But these dimensions are extremely non-independent. You can't be in either an extremely negative or positive emotional state without being pretty aroused. I then go to some lengths to suggest an alternate model which has, as far as I can tell, completely independent dimensions, and which I suggest is a better model partly for that reason. This is potentially controversial because the two dimensional model is extremely widely used in psychology (and influences personality research as well).

(So we aren't saying that emotions are independent-not sure what that would mean anyway- we say the variables underlying the emotions are independent.)
posted by leibniz at 1:10 PM on March 2

You might want to rethink making your model analogous to a physical phenomenon. There's just no proveable relationship between the two, and it's doubly dangerous since you only have a passing familiarity with the science. There's a tendency to relate abstract concepts to physical phenomenon to make them more "true", but it just isn't the case. You're heading down the road of the Sokal Affair.
posted by electroboy at 1:12 PM on March 2

I don't think that a mathematical model defining a euclidean subspace needs to have anything to do with the complexities of general relativity, but your requirement that your space needs to have mutually orthogonal axes is superfluous. If your axes are linearly independent, you can construct a space that has the same power as your original space and that has orthogonal axes.

In other words, the power of your "dimensional model" has to do with the linear independence of the axes, not their orthogonality. If you find that your axes x and y are correlated in some manner, you can transform your space into non-correlated axes x' and y'.

But it seems like your question is: How do you justify the idea that less correlated axes are better than more highly correlated ones?

I would say that less correlated axes are easier to have an intuitive understanding about, but I'm not sure that's a convincing argument.
posted by demiurge at 1:22 PM on March 2

Does the curvature of space mean that the dimensions of height, breath and width are not independent of each other?

Yes.

If they were independent variables, then we're just talking about plain old affine space, with coordinates x_1, x_2, ..., x_n, which presumably has curvature zero. That is bordering on tautological, but it's not quite; because if curvature is defined locally, then it might take a step or two to show the curvature for affine space is everywhere zero.
posted by metastability at 1:23 PM on March 2

Also, if your model has more than two dimensions and you can show that they are independent, then your model has power that is impossible to completely reproduce in a two dimensional model.
posted by demiurge at 1:34 PM on March 2

I then go to some lengths to suggest an alternate model which has, as far as I can tell, completely independent dimensions, and which I suggest is a better model partly for that reason.

I'm not sure why a model with independent dimensions would be better, unless it made empirically superior predictions. I'm pretty sure this is what the analogy to physics was really intended to convey -- the reason a non-independent model of physical dimensions is superior is because it makes better predictions, and that there is no a priori reason to chose a model with independent dimensions. (I.e. in the history of physics, that choice turned out to be wrong.)
posted by advil at 1:34 PM on March 2

Is the consensus that the physics of spacetime shows that the spatial dimensions are not independent?

If so, one cannot assume that a dimension model must necessarily employ independent dimensions.

But meanwhile, maybe I can argue that our common sense of dimensions is as independent, and so using blatantly non-independent dimensions betrays this understanding?
posted by leibniz at 2:12 PM on March 2

But these dimensions are extremely non-independent. You can't be in either an extremely negative or positive emotional state without being pretty aroused.

...

maybe I can argue that our common sense of dimensions is as independent, and so using blatantly non-independent dimensions betrays this understanding?


You mean they aren't even theoretically separable?

The two dimensions can be theoretically orthogonal, and the concepts orthogonal, even if the values that we actually observe are correlated. Similarly from my world, party identification and ideology can be theoretically or conceptually orthogonal even though they are obviously rather highly correlated.

I don't work in your field. But I expect you're going to get a lot of resistance from psychometrics types who think you don't understand the difference between concepts being theoretically orthogonal versus their operationalized variables taking correlated values. I expect as well that, for them, the proof of your pudding will be whether your n-dimensional model has noticeably more explanatory power than the standard model. Not whether the dimensions in your model are such that values actually observed tend to be uncorrelated.

Again, this has nothing to do with physical space being curved.

Why do you think strong valence / low arousal is impossible? I get that when I read about Nazi atrocities -- a strong negative valence, but lower arousal than from things that aren't confined to a history book.
posted by ROU_Xenophobe at 2:40 PM on March 2

Don't confuse orthogonal and linear independence. A dimensional model must have linearly independent dimensions to not be redundant, but they don't need to be orthogonal.

I agree that generally orthogonal axes are easier to understand than non-orthogonal ones.
posted by demiurge at 2:47 PM on March 2

Demiurge: Don't confuse orthogonal and linear independence.

Sorry, can you explain linear independence more please?
posted by leibniz at 2:52 PM on March 2

I'm not sure I can do much better than wikipedia, but if you have a vector space, you have axes. All of those axes need to be linearly independent from each other or you can't get to everywhere in the space.

Imagine if you had skewed axes for the Cartesian plane. You could still get to everywhere on the plane by combination of those axes. Your axes p and q just have to have that property that there isn't any m such that you can't find scalars a and b such that a*p + b*q = m.
posted by demiurge at 3:15 PM on March 2

leibniz (oh the irony of these names), you are jumping all over several very different concepts and entire disciplines here.

Perhaps it would be best to drop any analogies to physics or math from your work and stick to writing what you know?

On the other hand, a quick grounding in basic math is valuable and should only take a few months of intense study to get under your belt. Couldn't hurt. At least read the above article about the Sokal Affair.

I cannot stress enough that, possibly in stark contrast to the fields of study you are used to, these terms you are throwing around ("dimension", "linear dependence", "orthogonal") all have precise, rigorous definitions. You don't get to change their definitions to suit your needs or play loose with what they mean.

The article on wikipedia above about linear independence is good. But I suspect will read like a bunch of jargon. That's because you are trying to start with what usually is a 3rd year course at the University level at the earliest.
posted by Riemann at 3:34 PM on March 2 [2 favorites has favorites]

- You're heading down the road of the Sokal Affair.

- At least read the above article about the Sokal Affair.

Please, please get the facts about this incident straight before quoting it every time someone in the humanities wants to incorporate mathematical or scientific concepts into their work. You are doing a huge disservice to both the liberal arts and to the sciences when you refer to this incident thoughtlessly.

Social Text was not a peer reviewed journal.

Let me say that again with the emphasis on what matters:

Social Text was not a peer reviewed journal.

Moreover, people referring to it in the context that you both just did frequently ignore the facts that Wikipedia mentions halfway through the article:

[Social Text was] collecting papers for an upcoming issue dedicated to the science wars, and [Sokal's] was the only article submitted by a "real scientist". The editors had a number of concerns about the quality of the writing, and requested changes which Sokal refused.

So what we have are the following facts. (1) The article was not submitted to the process of peer review, which means that it should not be given any more academic weight than an article published in Scientific American; (2) the article was accepted mostly because they wanted to be inclusive of academics working in the sciences; (3) the article was not approved of even by the editors, who requested changes that were refused! In other words, this doesn't reveal anything about critical theory or postmodernism. All it tells us is that the editors of Social Text were desperate for content and Sokal was out to get some press.

The "affair" should be understood as nothing more than this: Sokal tried his hand at postmodern critical theory and failed. That's it.
posted by voltairemodern at 8:20 PM on March 2

I have quite a specific question about the indepedence of the spatial dimensions. I have given the background for my desire to know the answer, but I am also interested in the issue by itself. I am quite able to draw my own conclusions.

I have known about the Sokal paper for years. It is disheartening that people should even compare this case to that. The concept of dimension is a very common analogy, and I'm trying to sharpen up its use a bit. To slap down the humanites guy for trying to do this shows the exact kind of 2-cultures contempt that I am struggling against here.

There are certain concepts which can be centred in mathematics and logic but which are prime candidates for interdisciplinary comparisons. These include 'dimension' 'relation' 'mechanism' and even 'evolution'. No applied discipline has the authority to exclude other disciplines from using these terms. Analogies are central to human thinking, and an extremely useful way to approach problems by comparison to solutions in other fields. Moreover, it is possible that abstract structures are a fundamental level of reality appearing again and again in wildly different phenomenon (e.g. things like chaotic systems).

It is no coincidence that someone who uses the id 'leibniz' is interested in the nature of space and mathematics.
posted by leibniz at 1:01 AM on March 3

before quoting it every time someone in the humanities wants to incorporate mathematical or scientific concepts into their work.

Definitely...and also, (my impression is that) the OP isn't working in humanities, but in social sciences, which involves precise and rigorous mathematical modeling in many domains. (Though to be fair I have no idea what theories of emotion look like.) All Riemann's comment did was demonstrate ignorance, I thought it was really inappropriate.
posted by advil at 6:43 AM on March 3

Social Text was not a peer reviewed journal.

So lack of peer review means lack of fact checking too?
posted by electroboy at 6:59 AM on March 3

This Sokal paper is as much a red herring as the original criticism about curved spacetime.

I see nothing wrong with your premise that you can analyze emotions using a different model than those typically used. However, simply having orthogonal axes does not give the model more power, since you could transform the space to have orthogonal axes. You may want to brush up on your linear algebra so you can defend yourself better, and maybe talk to a statistician. You don't really need to talk to a physicist.
posted by demiurge at 8:22 AM on March 3

Definitely...and also, (my impression is that) the OP isn't working in humanities, but in social sciences, which involves precise and rigorous mathematical modeling in many domains

Precise and rigorous modeling, based on real world measurements, which then can be tested and refined. There doesn't seem to be anything here other than using mathematics to lend a veneer of credibility to something in an unrelated field. Might as well use numerology.
posted by electroboy at 9:02 AM on March 3

Psychological and other social science studies use statistics and mathematical modeling without being "numerology". But this is far off-topic from the problem that leibniz was trying to solve.
posted by demiurge at 11:03 AM on March 3

It is worth remarking that bringing up Sokal in this context is actually a reasonable point to raise, at least as a caution.

The original reason that Sokal wrote his paper was because of many instances of misuse of scientific terminology, in partiuclar mathematical and physical ones, in many well-regarded papers in a variety of disciplines. Whether or not you think that his paper is a scathing critique of post-modernism aside, he had a valid point in that regard.

So don't use these terms simply because they sound more science-y. Use them if you understand them, and the analogy actually makes sense. From what you've written above, I would suggest at least being careful. There might be some validity to your approach, but you haven't successfully demonstrated it here.

Go back. Read up on Wikipedia about curvature, orthogonality (not the poster!) and linear independance. Then be careful and don't overstate your case.

Even people who have some training make mistakes: For example, metastability's comment

If they were independent variables, then we're just talking about plain old affine space, with coordinates x_1, x_2, ..., x_n, which presumably has curvature zero. That is bordering on tautological, but it's not quite; because if curvature is defined locally, then it might take a step or two to show the curvature for affine space is everywhere zero.

is unfortunately quite false, despite their showing evidence of knowing some of the subject material. These are complicated, subtle ideas. Tread carefully.
posted by vernondalhart at 4:13 AM on March 4 [1 favorite has favorites]

Ok, sorry bout that. What I should have said was, If we're talking about plain old affine space equipped with the standard Euclidean metric, (Euclidean space would have been better), then it's curvature is zero, assuming you don't define curvature perversely. The curvature of Euclidean space is zero- that's all.
posted by metastability at 6:10 AM on March 4

and so one possible way of rephrasing the poster's question is "Are there affine varieties, not isomorphic to C^n, possessing a metric, such that the curvature of that metric is zero?" maybe someone who knows differential geometry can answer that, or correct me if it doesn't make sense.
posted by metastability at 6:35 AM on March 4

Oookay. sorry for cluttering up this thread. But here is an example of an affine space with a zero-curvature metric: GL_n(C). As an (affine) open subset of C^{n^2}, it inherits the euclidean metric, yet it is not isomorphic to C^N. In other words, the coordinates on GL_n(C) actually do satisfy some relation, yet the curvature is zero. So the converse, "non-zero curvature implies isomorphic to C^N", which is how I'm interpreting the OP's question, is false. Cool.
posted by metastability at 7:16 AM on March 4

one thing though: GL_n(C) is not closed. if we insist on closed, then maybe we could take a closed subset of it. \end{derail}
posted by metastability at 7:32 AM on March 4

Ah Stupid! by that logic, any closed subvariety of C^n will inherit the euclidean metric. i give up for now, i'll ask someone about it. (thanks for asking this question though! )
posted by metastability at 7:40 AM on March 4

Admittedly, the question confuses "independence" and "orthogonality", so it is a bit unclear whether or not that is a correct rephrasing of the question.

Anyhow, any affine subset (or analytic, or simply closed at all) inherits the euclidean metric from C^n, but that doesn't mean that the inherited metric is flat. At least in the real case, that is. Consider a hyperboloid in R^3. It has strictly negative curvature, and it is smoothly embedded in a flat space.

I'll hedge a bit on complex affine space, but I would be shocked if the case were any different. Be that as it may, the question above suggests that we would be dealing with real coordinates anyhow.
posted by vernondalhart at 8:21 AM on March 4

'Orthogonal' is clearly a metaphor in the case I am ultimately interested in. 'Independent' is the more important property. You don't need to get into the complexities of 'affine space'. Please just say, (and hopefully explain in English) whether in curved space, the 3 spatial dimensions are independent of each other, such that moving along one doesn't entail moving along another, even if it's just a tiny bit. Or in other words, that occupying a certain position on one dimension does not limit the range of positions one could occupy on other dimensions.
posted by leibniz at 2:24 PM on March 4

They are. Consider the two-dimensional analogy: use the surface of a sphere as a model for a positively curved two dimensional space. You can think of the two dimensions as latitude and longitude. You can change your latitude without changing your longitude, and vice versa. Having a given longitude does not restrict your choice of latitude, or vice versa, except at the two poles where longitude becomes meaningless. (And for a negatively curved space, you don't even have any poles as an exception).
posted by DevilsAdvocate at 2:59 PM on March 4

hmm. Interestingly though, as one approaches the pole, one's longitudes while still all there (and infinitely refinable) become more bunched up.

And is this sort of independence true of all (maximally efficient) sets of dimensions?
posted by leibniz at 3:41 PM on March 4

Which sort of independence? The bunching up of longitudinal lines?

What that reflects is that there is a singularity in the chosen system of coordinates. Not of the surface, mind you, but simply of the representation of points on the surface via a certain system of coordinates.

A more simple example (although pretty much identical, really) is to look at coordinates on the plane R². We can use standard x, y coordinates, and there is nothing funny. However, if we choose polar coordinates (where r represents the distance from the origin, and θ represents the angle between our point, the origin, and the x-axis), then we have a singularity at the origin again. The coordinate r is well define everywhere, but θ is not.

R² is certainly not singular, simply our choice of coordinates on it.

What this comes down to is that we may or may not be able to choose a single system of coordinates which covers the entire object in question that is well defined and non-singular. We instead have to choose coordinates locally (i.e. in "small enough" areas) and glue these together. It was suggested upthread that you look at the wikipedia definition of a "Manifold". That's really the idea here.
posted by vernondalhart at 1:42 AM on March 5

Not the bunching up (that's just interesting). All I would need for my analogy is that you can be at one position on one dimension while at any position on the other dimensions (within the defined boundaries of those dimensions). Do you agree with this vernondalhart?

And if it is true of particular locations, would it also be true of vectors? (that you can move in one dimension without constraining your movement on another dimension). Maybe not right? Some spaces may permit only certain sorts of movements.
posted by leibniz at 2:42 AM on March 5

If I understand what you're saying correctly, then yes. That is arguably the definition of dimension.

I say arguably, because there are many different definitions of dimension based on context, but in the case of the surface of the earth, or manifolds in general, then yes. And it's certainly true for vectors, the collection of which are a local model for manifolds.
posted by vernondalhart at 3:23 AM on March 5

Thanks everyone, I will try to bear all these distinctions in mind.
posted by leibniz at 3:59 AM on March 5