How exactly are these definitions of "dimension" related and different?
Mathematics.
The least number of independent coordinates required to specify uniquely the points in a space.
Physics. A physical property, such as mass, length, time, or a combination thereof, regarded as a fundamental measure or as one of a set of fundamental measures of a physical quantity: Velocity has the dimensions of length divided by time.
http://www.answers.com/topic/dimension
A mathematical space is a bit of a malleable concept. Intuitively, you can see it as representing all the possible combinations of certain characteristics of objects.
The Euclidean space, for example, is a way of representing the three "dimensions" of height, length and width of certain objects. However, in mathematics, we say this space has dimension 3 because you need three independent coordinates to specify a point in that space. But if the objects have other characteristics that you want to represent and you can represent them by a number, then you can add additional coordinates. For example, you may want to represent colour so you might add another coordinate representing wavelength values. You'll then have a space of dimension 4 and so on if you want to add more information to the representation. A mathematical space is then something much more general than our conventional view of "space" and can be used to represent virtually anything measurable.
For example, you have 1000 stocks for which you have information about mean return and variance over 20 years. You can then represent these stocks on a mean-variance space which is then of dimension 2.
Sometimes "dimension" is used to mean the space defined by each one of the measured characteristics (i.e. "height is one dimension, length is another and width is the third"😉. This is used often, although technically incorrect from a purist mathematical sense. It seems this is close to how you describe dimension in physics, but since I'm not a physicist maybe someone else might comment.
Originally posted by PalynkaIs there something which differentiates between the type of dimension that are linear bases of a vector space (and thus orthogonal) and something like mass or volume?
A mathematical space is a bit of a malleable concept. Intuitively, you can see it as representing all the possible combinations of certain characteristics of objects.
The Euclidean space, for example, is a way of representing the three "dimensions" of height, length and width of certain objects. However, in mathematics, we say this space has dimension 3 b dimension in physics, but since I'm not a physicist maybe someone else might comment.
Originally posted by AThousandYoungI think they are very different. Mass and volume have an existence beyond mathematics, which is simply a language. One which is operationally incredibly efficient, but basically still just syntax.
Is there something which differentiates between the type of dimension that are linear bases of a vector space (and thus orthogonal) and something like mass or volume?
You can represent mass and volume in mathematical spaces, though. For example, using a pair of orthogonal vectors then any point in that space will represent an unique combination of mass and volume. But note that any 2 orthogonal vectors can be used for this representation. So you can have an infinity of mathematical spaces representing the same 2 dimensions (physics sense). You could also add another orthogonal vector representing another dimension (physics sense) and you would now have a mathematical space with dimension equal to 3 (math sense).
So they are very different concepts. It's hard to think what they have in common... 😕
Originally posted by PalynkaThe problem is that linear bases do not come to an end, while mass cannot go below 0.
I think they are very different. Mass and volume have an existence beyond mathematics, which is simply a language. One which is operationally incredibly efficient, but basically still just syntax.
You can represent mass and volume in mathematical spaces, though. For example, using a pair of orthogonal vectors then any point in that space will represent an ...[text shortened]... nse).
So they are very different concepts. It's hard to think what they have in common... 😕
In addition there is inertia along spacial dimensions, but not along mass.
EDIT - I'm putting this discussion on my blog.
http://athousandyoung.blogspot.com/2009/12/dimensions.html
Originally posted by AThousandYoungThat's not correct. The scalars need not be the real numbers and can be any set (the positive reals, for example). Also, nothing prevents you from representing something by a subspace in a more general space. For example, the image of a function represented in the real plane is a subspace of the real plane, yet it's easier for us to visualize it as a curve in the real plane, despite it being only a one dimensional object (although sometimes axes are useful to represent domain and codomain).
The problem is that linear bases do not come to an end, while mass cannot go below 0.
In addition there is inertia along spacial dimensions, but not along mass.
EDIT - I'm putting this discussion on my blog.
http://athousandyoung.blogspot.com/2009/12/dimensions.html
Representing an additional concept like inertia would require adding an additional coordinate. If you want to add time, add another dimension, etc. Like I said, mathematical spaces (not just vector spaces) are very malleable concepts and can be used to represent virtually any set. So mass, length, width, etc. have as much in common with mathematical spaces as GDP growth and inflation, in the sense that we can measure these concepts and map them into coordinates. That's all there is to it.
Originally posted by Palynkai think ultimately what you are saying (if i understand correctly) is that mathematical spaces and mathematical "dimensions" are much more adaptable concepts than the physics "physically-based" conception of a dimension (i.e. a physically measured quantity). in fact, i think that it is exactly that versatility and "malleable" nature, as you put it, that differentiates it from the definition of a dimension in physics. in a sense, the physics-version of a dimension is a more specific subset of the mathematical "coordinate" in a space that is describing physical quantities. coordinate systems and mathematical spaces can be used as the mathematical language with which to describe the more limiting physics conception of a dimension.
That's not correct. The scalars need not be the real numbers and can be any set (the positive reals, for example). Also, nothing prevents you from representing something by a subspace in a more general space. For example, the image of a function represented in the real plane is a subspace of the real plane, yet it's easier for us to visualize it as a curve i ...[text shortened]... that we can measure these concepts and map them into coordinates. That's all there is to it.
not sure if that helps clarify for ATY, or even if i am completely wrong in my understanding of your comment palynka! don't mean to speak for you just trying to join in a cool discussion 🙂
Originally posted by AetheraelI agree completely, thanks for helping to clarify it!
i think ultimately what you are saying (if i understand correctly) is that mathematical spaces and mathematical "dimensions" are much more adaptable concepts than the physics "physically-based" conception of a dimension (i.e. a physically measured quantity). in fact, i think that it is exactly that versatility and "malleable" nature, as you put it, that d ...[text shortened]... omment palynka! don't mean to speak for you just trying to join in a cool discussion 🙂