6 ms·

a personal pet peeve (brace yourselves, it's pedantry): Technically speaking, we're talking about four dimensional space. It doesn't really make sense to call

a personal pet peeve (brace yourselves, it's pedantry): Technically speaking, we're talking about four dimensional space. It doesn't really make sense to call such a space "The Fourth Dimension", any more than real life space is "The Third Dimension", or a tabletop is "The Second Dimension". This sometimes trips people up into arguing over whether The Fourth Dimension "is" time, or whatever. For that matter, maybe the first dimension is time, and the second, third, and fourth dimensions are space. These things aren't ordered, and in fact you can't really distinguish between the three familiar spatial dimensions: imagine trying to point along dimension one, whatever that means. The familiar three-dimensional space as we know it is three dimensional because you can put three straight lines to meet at right angles to each other, and no more. And you can label those lines x, y, and z if you like and pick their orientation. Four dimensional space allows you to cram another in. Two dimensional space only allows two, and one dimensional space is just a single line.

felipeccastro 3y ago
That's not pedantry, that's clarifying, haven't thought like that before. Thank you.
munchler 3y ago
Meh. If a circle was able to escape Flatland by rising into the third dimension, how would you prefer to describe it? Similarly, if we were able to escape 3-space by moving into a 4th spatial dimension, what would you call it? If this hypothetical 4-space is Euclidean, then it contains exactly one dimension that is perpendicular to our familiar 3-space, so we would be perfectly justified in calling it The Fourth Dimension.
- roywiggins 3y ago
  Fair, but two perpendicular Flatlands embedded in the same 3D space won't be able to agree on which is the fourth dimension. It's fine as a shorthand when Flatlanders talk to each other, but "the fourth dimension" still won't be an unambiguous direction, for Flatland A it's actually one of Flatland B's two dimensions, and vice versa. For us any dimensions perpendicular to the entire universe will be "special", but only for us. Native four dimensional critters won't see what's so different about the ana/kata axis, unless our universe happens to be their tabletop RPG.
  - munchler 3y ago
    I think you’re making my point. The Fourth Dimension is still uniquely defined for our 3-space (which is the entire universe, as far as we know). I think that’s pretty solid justification for giving it a distinct name. Hypothetically, if there are other (infinite) 3-spaces embedded in our 4D metaverse, then we either intersect them (which would cause all sorts of problems), or they are parallel and agree with our definition of The Fourth Dimension.
    - roywiggins 3y ago
      Our universe might be a much more complicated manifold than a plane in the space it's embedded in. Of course, we experience it as a flattish 3D space, but maybe gravity makes it four dimensionally lumpy. Or something else, who knows. Maybe it's actually closed and is the surface of a very, very large 4D sphere. This would make the ana/kata vectors pointing outwards away from the universe where I am and the ana/kata vectors where you are not line up, and there wouldn't be any way to decide which ana/kata vector is the special one, even between different points inside our universe. But really, my objection is mostly that "The Fourth Dimension" makes it sound like a "Dimension" is a kind of place, which is confusing.
      - lupire 3y ago
        The article is extremely clear in explaining the fourth dimension is a direction, not a place. You appear to be projecting your past experiences onto the article.
        roywiggins 3y ago
        Quite likely I am! And of course the article was fine. But I will note there was already a comment on it about how "the fourth dimension is time, actually." So this particular phrasing does seem to keep confusing people.
        lupire 3y ago
- bbor 3y ago
  I’m just an amateur but it seems like the argument above yours is pretty airtight, just based on the difference between the mathematic definition of Dimension (“…is informally defined as the minimum number of coordinates needed to specify any point within [a space]”) and the colloquial definition (“a space”) which your last sentence seems to rely on. If I’m reading the first paragraph of Wikipedia right (surely an airtight source for a pedantic argument about advanced mathematics!) “dimensionality” is an adjective describing a space (a set?), so saying that we moved to “the fourth dimension” is about as meaningful as saying we moved to “The Euclidean” or “the big”. Rather than “a euclidean space” or “a big space”. That said you’re colloquially absolutely correct, of course. If I was giving advice to fiction writers or journalists I’d definitely endorse your common-sense argument.
- staplers 3y ago
  if we were able to escape 3-space by moving into a 4th spatial dimension We are perpetually suspended in this '4th dimension' given that we are orbiting a galaxy and star. Find a way out of the observable universe which doesn't move and you might escape this fourth dimension.
tshaddox 3y ago
> It doesn't really make sense to call such a space "The Fourth Dimension", any more than real life space is "The Third Dimension", It makes a lot more sense to refer to time as "the fourth dimension" than to refer to all three spatial dimensions together as "the third dimension." Time is indeed one of the four dimensions that we're discussing here! Of course you can quibble that the four dimensions are not in some fixed order, but in this context we're clearly referring to time as the fourth dimension because it's the one being introduced as an addition to the other three.
- datascienced 3y ago
  I think time is special in Einsteins equations. It therefore should be considered special. Been a while since I read about it.
  - roywiggins 3y ago
    In Minkowski spacetime, time is definitely special, the metric signature for it is reversed compared to the spatial dimensions.
- JdeBP 3y ago
  The headlined article was written in 1884, and it is not discussing time as "the fourth" dimension. Nor is roywiggins.
  - lupire 3y ago
    Geometrically (not physically), time is isomorphic to a Euclidian space dimension. This is how movies work.
kang 3y ago
Its not that simple. Time it seems IS the first (0th?) dimension. A point is space denotes existence in time of the object & observation by the subject. In other words, rate of change of existence is observed as time. Rate of change of a point is observed as a line. A moving point accepts line as its track, moving interval accepts square as its track and moving square accepts cube as its track. 2-eyed observer has 3D vision & 1-eyed observer as 2D vision (try touching your fingers with one-eyes closed exercise) has some role about role of observation as well. Further, dimensions being relative vs absolute makes more sense. In absolute sense, time is its own dimension T & point line cube are L, L^2 & L^3. A 3D object, a cube, has 2D object, plane, as its boundary & 1D object, lines, as its dimensional denotion. A square has 1D object as its boundary & n-2=0D objects, points, as its dimensions, relatively speaking. This is important because of the number of eyes? So basically, those 2D hypothetical characters in your physics are 1-eyed creatures, lol.
xanderlewis 3y ago
> space as we know it is three dimensional because you can put three straight lines to meet at right angles to each other, and no more. To continue in the spirit of your pedantry, it’s not really important that the lines meet at right angles; to say that space is three dimensional is just to say that a point in space can be uniquely expressed by specifying three numbers: the ‘coordinates’ of the point. What you say is true, but dimensionality in general doesn’t require the notion of ‘angle’ to make sense. In vector space/linear algebra terms it’s to say that we live in three dimensional space because 3 is the size of a maximal linearly independent set of vectors, or alternatively the size of a minimal spanning set or vectors. They need not be orthogonal, but orthogonality does imply linear independence (it’s stronger).
- lupire 3y ago
  Can you give an example of any n-dimensional system where you can't make a basis of n perpendicular vectors? "Right angle" isn't any kind of angle, it's the angle that means "perpendicular" . You don't necessarily need any other angles besides 0 (parallel) and right (perpendicular) in a system.
  - xanderlewis 3y ago
    > Can you give an example of any n-dimensional system where you can't make a basis of n perpendicular vectors? Well, yes. In a totally general vector space (in the mathematical sense of a vector space over a field; it’s the same notion as used in physics but perhaps more general) there’s no such thing as ‘perpendicular’ (aka ‘orthogonal’) — you need an inner product space for that. Of course, (at least in finite dimensions) you can always define such an inner product and you can certainly create orthogonal bases (see: Gram-Schmidt process), but my point was that the notion of dimension doesn’t require it to make sense. Indeed it’s not part of the usual definition; we only need to be able to talk about sets of vectors and their spans.
- roywiggins 3y ago
  Yeah, I had a bit in there at first on linear independence but I couldn't fit it in without feeling like I was digressing!
sdwr 3y ago
Ahhh, perfect pedantry. Constructive, illuminating, accurate, and (most importantly) calling the OP wrong.
- aatd86 3y ago
  I haven't understood. If I take a 1D space, I just need an orthogonal vector to find a 2D space (the second dimension). If I take a 2D space, same...(3rd dimension). So if I take a 3D space...? Needs a 4th vector (4th dimension) to find a 4 dimensional space? ?_? (wut?) is the pedantry really accurate here or is it sarcasm or should I go sleep..
  - sdwr 3y ago
    The point is "a" fourth dimension vs "the" fourth dimension. Our 3D space uses all 3 dimensions interchangeably. It makes sense to talk about "the" 3 dimensions of space. If you look at 3D + time, time acts differently than the other 3D. - we only move through time in 1 direction, at a rate of pretty much 1 second per second. - things break more easily than they go back together (ex. shattering a glass) So time doesn't fit in with the other dimensions. Or does it? Einstein crammed it in there, and proved that time would be the same except we're being constantly pulled in the timewards direction by a really heavy object that we can't see. Or something like that.
layer8 3y ago
The article doesn’t call the space the fourth dimension, it (correctly) calls the additional direction the fourth dimension.
bmitc 3y ago
> These things aren't ordered, and in fact you can't really distinguish between the three familiar spatial dimensions: imagine trying to point along dimension one, whatever that means. I actually don't know what this means. You can certainly distinguish between the three spatial dimensions. Pointing along dimension one is irrelevant. Once you point in a direction, you can define the others given an orientation, and there are only two unique orientations on a three dimensional manifold.
- lupire 3y ago
  What they are saying is that you can't make a non-arbitrary identification of each dimension. You can call them 1,2,3, and I can call them 2,3,1, and that changes nothing except your arbitrary labels. A third party has no way to know whose choice of "1" is better for any reason related to the space itself, only by reference to some preferred object in the space, like the direction my pencil is pointing right now. Space itself has no such preferred object or direction.
  - bmitc 3y ago
    > What they are saying is that you can't make a non-arbitrary identification of each dimension. You can call them 1,2,3, and I can call them 2,3,1, and that changes nothing except your arbitrary labels. I don't understand the point. What is the supposed consequence of that? As far as I can tell, it's an empty statement. And as I said, there are only two orientations in three dimensions. For an orientation, which is an equivalence class of ordered bases, the order actually does matter. You can call them whatever you want, but once you've called them something, their order matters, which was my original retort.
    - lupire 3y ago
      > You can call them whatever you want, but once you've called them something, their order matters, which was my original retort. Right and different "local cultures" in Flatland call them different things, so the order matters. Orientation is a red herring. The same issue appears in 2-D manifolds inhabited by 1-D sub manifolds that one day "discover" the "other" dimension.
kazinator 3y ago
You can point to a dimension; you just have to pin down a coordinate system and then point to an axis. It does make sense to speak about a dimension. Say that some creatures in 2D space are ganging up from all sides on a creature which has access to 3D. As they close in, that creature can evade them by "disappearing into the third dimension". What it means is that it moves in such a way that its motion has a component that is orthogonal to the 2D plane, and only that component is relevant to the success of its escape. In the coordinate system in which that 2D plane makes up the first two dimensions, the orthogonal axis is "the third dimension".

felipeccastro 3y ago

That's not pedantry, that's clarifying, haven't thought like that before. Thank you.

munchler 3y ago

Meh. If a circle was able to escape Flatland by rising into the third dimension, how would you prefer to describe it? Similarly, if we were able to escape 3-space by moving into a 4th spatial dimension, what would you call it? If this hypothetical 4-space is Euclidean, then it contains exactly one dimension that is perpendicular to our familiar 3-space, so we would be perfectly justified in calling it The Fourth Dimension.

roywiggins 3y ago
Fair, but two perpendicular Flatlands embedded in the same 3D space won't be able to agree on which is the fourth dimension. It's fine as a shorthand when Flatlanders talk to each other, but "the fourth dimension" still won't be an unambiguous direction, for Flatland A it's actually one of Flatland B's two dimensions, and vice versa. For us any dimensions perpendicular to the entire universe will be "special", but only for us. Native four dimensional critters won't see what's so different about the ana/kata axis, unless our universe happens to be their tabletop RPG.
- munchler 3y ago
  I think you’re making my point. The Fourth Dimension is still uniquely defined for our 3-space (which is the entire universe, as far as we know). I think that’s pretty solid justification for giving it a distinct name. Hypothetically, if there are other (infinite) 3-spaces embedded in our 4D metaverse, then we either intersect them (which would cause all sorts of problems), or they are parallel and agree with our definition of The Fourth Dimension.
  - roywiggins 3y ago
    Our universe might be a much more complicated manifold than a plane in the space it's embedded in. Of course, we experience it as a flattish 3D space, but maybe gravity makes it four dimensionally lumpy. Or something else, who knows. Maybe it's actually closed and is the surface of a very, very large 4D sphere. This would make the ana/kata vectors pointing outwards away from the universe where I am and the ana/kata vectors where you are not line up, and there wouldn't be any way to decide which ana/kata vector is the special one, even between different points inside our universe. But really, my objection is mostly that "The Fourth Dimension" makes it sound like a "Dimension" is a kind of place, which is confusing.
    - lupire 3y ago
      The article is extremely clear in explaining the fourth dimension is a direction, not a place. You appear to be projecting your past experiences onto the article.
      - roywiggins 3y ago
        Quite likely I am! And of course the article was fine. But I will note there was already a comment on it about how "the fourth dimension is time, actually." So this particular phrasing does seem to keep confusing people.
        lupire 3y ago
bbor 3y ago
I’m just an amateur but it seems like the argument above yours is pretty airtight, just based on the difference between the mathematic definition of Dimension (“…is informally defined as the minimum number of coordinates needed to specify any point within [a space]”) and the colloquial definition (“a space”) which your last sentence seems to rely on. If I’m reading the first paragraph of Wikipedia right (surely an airtight source for a pedantic argument about advanced mathematics!) “dimensionality” is an adjective describing a space (a set?), so saying that we moved to “the fourth dimension” is about as meaningful as saying we moved to “The Euclidean” or “the big”. Rather than “a euclidean space” or “a big space”. That said you’re colloquially absolutely correct, of course. If I was giving advice to fiction writers or journalists I’d definitely endorse your common-sense argument.
staplers 3y ago
if we were able to escape 3-space by moving into a 4th spatial dimension We are perpetually suspended in this '4th dimension' given that we are orbiting a galaxy and star. Find a way out of the observable universe which doesn't move and you might escape this fourth dimension.

tshaddox 3y ago

> It doesn't really make sense to call such a space "The Fourth Dimension", any more than real life space is "The Third Dimension", It makes a lot more sense to refer to time as "the fourth dimension" than to refer to all three spatial dimensions together as "the third dimension." Time is indeed one of the four dimensions that we're discussing here! Of course you can quibble that the four dimensions are not in some fixed order, but in this context we're clearly referring to time as the fourth dimension because it's the one being introduced as an addition to the other three.

datascienced 3y ago
I think time is special in Einsteins equations. It therefore should be considered special. Been a while since I read about it.
- roywiggins 3y ago
  In Minkowski spacetime, time is definitely special, the metric signature for it is reversed compared to the spatial dimensions.
JdeBP 3y ago
The headlined article was written in 1884, and it is not discussing time as "the fourth" dimension. Nor is roywiggins.
- lupire 3y ago
  Geometrically (not physically), time is isomorphic to a Euclidian space dimension. This is how movies work.

kang 3y ago

Its not that simple. Time it seems IS the first (0th?) dimension. A point is space denotes existence in time of the object & observation by the subject. In other words, rate of change of existence is observed as time. Rate of change of a point is observed as a line. A moving point accepts line as its track, moving interval accepts square as its track and moving square accepts cube as its track. 2-eyed observer has 3D vision & 1-eyed observer as 2D vision (try touching your fingers with one-eyes closed exercise) has some role about role of observation as well. Further, dimensions being relative vs absolute makes more sense. In absolute sense, time is its own dimension T & point line cube are L, L^2 & L^3. A 3D object, a cube, has 2D object, plane, as its boundary & 1D object, lines, as its dimensional denotion. A square has 1D object as its boundary & n-2=0D objects, points, as its dimensions, relatively speaking. This is important because of the number of eyes? So basically, those 2D hypothetical characters in your physics are 1-eyed creatures, lol.

xanderlewis 3y ago

> space as we know it is three dimensional because you can put three straight lines to meet at right angles to each other, and no more. To continue in the spirit of your pedantry, it’s not really important that the lines meet at right angles; to say that space is three dimensional is just to say that a point in space can be uniquely expressed by specifying three numbers: the ‘coordinates’ of the point. What you say is true, but dimensionality in general doesn’t require the notion of ‘angle’ to make sense. In vector space/linear algebra terms it’s to say that we live in three dimensional space because 3 is the size of a maximal linearly independent set of vectors, or alternatively the size of a minimal spanning set or vectors. They need not be orthogonal, but orthogonality does imply linear independence (it’s stronger).

lupire 3y ago
Can you give an example of any n-dimensional system where you can't make a basis of n perpendicular vectors? "Right angle" isn't any kind of angle, it's the angle that means "perpendicular" . You don't necessarily need any other angles besides 0 (parallel) and right (perpendicular) in a system.
- xanderlewis 3y ago
  > Can you give an example of any n-dimensional system where you can't make a basis of n perpendicular vectors? Well, yes. In a totally general vector space (in the mathematical sense of a vector space over a field; it’s the same notion as used in physics but perhaps more general) there’s no such thing as ‘perpendicular’ (aka ‘orthogonal’) — you need an inner product space for that. Of course, (at least in finite dimensions) you can always define such an inner product and you can certainly create orthogonal bases (see: Gram-Schmidt process), but my point was that the notion of dimension doesn’t require it to make sense. Indeed it’s not part of the usual definition; we only need to be able to talk about sets of vectors and their spans.
roywiggins 3y ago
Yeah, I had a bit in there at first on linear independence but I couldn't fit it in without feeling like I was digressing!

sdwr 3y ago

Ahhh, perfect pedantry. Constructive, illuminating, accurate, and (most importantly) calling the OP wrong.

aatd86 3y ago
I haven't understood. If I take a 1D space, I just need an orthogonal vector to find a 2D space (the second dimension). If I take a 2D space, same...(3rd dimension). So if I take a 3D space...? Needs a 4th vector (4th dimension) to find a 4 dimensional space? ?_? (wut?) is the pedantry really accurate here or is it sarcasm or should I go sleep..
- sdwr 3y ago
  The point is "a" fourth dimension vs "the" fourth dimension. Our 3D space uses all 3 dimensions interchangeably. It makes sense to talk about "the" 3 dimensions of space. If you look at 3D + time, time acts differently than the other 3D. - we only move through time in 1 direction, at a rate of pretty much 1 second per second. - things break more easily than they go back together (ex. shattering a glass) So time doesn't fit in with the other dimensions. Or does it? Einstein crammed it in there, and proved that time would be the same except we're being constantly pulled in the timewards direction by a really heavy object that we can't see. Or something like that.

layer8 3y ago

The article doesn’t call the space the fourth dimension, it (correctly) calls the additional direction the fourth dimension.

bmitc 3y ago

> These things aren't ordered, and in fact you can't really distinguish between the three familiar spatial dimensions: imagine trying to point along dimension one, whatever that means. I actually don't know what this means. You can certainly distinguish between the three spatial dimensions. Pointing along dimension one is irrelevant. Once you point in a direction, you can define the others given an orientation, and there are only two unique orientations on a three dimensional manifold.

lupire 3y ago
What they are saying is that you can't make a non-arbitrary identification of each dimension. You can call them 1,2,3, and I can call them 2,3,1, and that changes nothing except your arbitrary labels. A third party has no way to know whose choice of "1" is better for any reason related to the space itself, only by reference to some preferred object in the space, like the direction my pencil is pointing right now. Space itself has no such preferred object or direction.
- bmitc 3y ago
  > What they are saying is that you can't make a non-arbitrary identification of each dimension. You can call them 1,2,3, and I can call them 2,3,1, and that changes nothing except your arbitrary labels. I don't understand the point. What is the supposed consequence of that? As far as I can tell, it's an empty statement. And as I said, there are only two orientations in three dimensions. For an orientation, which is an equivalence class of ordered bases, the order actually does matter. You can call them whatever you want, but once you've called them something, their order matters, which was my original retort.
  - lupire 3y ago
    > You can call them whatever you want, but once you've called them something, their order matters, which was my original retort. Right and different "local cultures" in Flatland call them different things, so the order matters. Orientation is a red herring. The same issue appears in 2-D manifolds inhabited by 1-D sub manifolds that one day "discover" the "other" dimension.

kazinator 3y ago

You can point to a dimension; you just have to pin down a coordinate system and then point to an axis. It does make sense to speak about a dimension. Say that some creatures in 2D space are ganging up from all sides on a creature which has access to 3D. As they close in, that creature can evade them by "disappearing into the third dimension". What it means is that it moves in such a way that its motion has a component that is orthogonal to the 2D plane, and only that component is relevant to the success of its escape. In the coordinate system in which that 2D plane makes up the first two dimensions, the orthogonal axis is "the third dimension".