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Where did they use the term "N-dimensional"? When you say N-dimensional are you talking about what the article refers to as "finite dimensional"? If so, I thi

Where did they use the term "N-dimensional"? When you say N-dimensional are you talking about what the article refers to as "finite dimensional"? If so, I think you're touching on something about the distinction between finite-dimensional and infinite-dimensional spaces. For any finite n, the cardinality of R^n and R are the same. I'm guessing that those bijections break down once you step into infinite dimensional spaces, but I haven't found a simple reference for this. Can someone verify this?

ajuc 8y ago
> Where did they use the term "N-dimensional"? From the article: > Researchers talk about a state space as having a certain number of dimensions, reflecting the number of independent characteristics you can adjust in the underlying system. > For example, even a sock drawer has a state space. Any sock might be described by its color, its length, its material, and how raggedy and worn it is. In this case, the dimension of the sock drawer’s state space is four. That's meaningless IMHO, 4 is the same as 1.
- philipov 8y ago
  Dimensionality is a type of structure that can be applied to quantities. Dimensionality refers to being able to find independent bases within a vector space. The study of vector spaces is a richer language than set theory. By allowing only cardinality to be meaningful, you are denying the value of dimensionality as an organizing principle. Therefore, its meaninglessness is tautological. Your mistake is to think that 4 is the same as 1. It is not. You are putting a hand up to one eye and complaining that stereoscopy is meaningless. You are pretending that only set theory exists, and claiming that therefore linear algebra is meaningless. It is rhetorical nonsense!
  - ajuc 8y ago
    > It is not. Why? If I can describe a system with 1 number why invent 4 parameters to describe it? And why not 100? The choice seems arbitrary.
    - philipov 8y ago
      Because you can't describe any system with just 1 number. You are arbitrarily choosing to only pay attention to the cardinality of a set. If you are discussing a vector space, there is more structure than that. This is what the study of linear algebra is all about. The amount of structure you require depends on the question you are asking. Not all questions equate to finding the size of a set. You are like a blind man feeling the tail of an elephant and claiming the elephant's nature is the same as a rope!
    - UnFleshedOne 8y ago
      Say you convert color of the sock in the drawer to a range [0,1] on your one and only coordinate. Then you convert dirtiness of the sock to a similar range and just add them together. Can you differentiate between a sock that is 0.4 on color and 0.6 on dirtiness from another sock which is 0.6 on color and 0.4 on dirtiness? I suppose you could encode it all to one number so that structure is preserved, but dimensions themselves are not going anywhere, they are in your choice of encoding that must reside outside of your one coordinate.
    - NotAnEconomist 8y ago
      Because dimension is about the connectivity structure of a set of objects, not the size of the set of objects. Sets with the same cardinality can have wildly different topologies, which describe the possible paths through the space, which in turn describes the nature of functions into and out of that space, as well as transformations and relationships within that space. A line through the center of a sphere doesn't necessarily touch the sphere in a "4 dimensional" connectivity structure, must touch the sphere in a "3 dimensional" connectivity structure, while a sphere doesn't even exist in a "1 dimensional" connectivity structure. The main reason that we use "4 dimensional" spacetimes for particle physics is that there are three independent components of motion with respect to time, and so it makes the math simpler to allow for independent variation of those parameters to functions, instead of compressing them into a single encoded number, eg, derivatives wouldn't function correctly (or likely even be defined) on a single parameter model. An additional reason that we don't use a "1 dimensional" spacetime is because the product of a "1 dimensional" time and a "1 dimensional" space is a "2 dimensional" spacetime -- and we don't have any models of reality that lack either space or time. (Which is good: any model needs to explain both, if only as an emergent feature.) Topology is about connectivity, not scale; dimension is a topological fact.

ajuc 8y ago

> Where did they use the term "N-dimensional"? From the article: > Researchers talk about a state space as having a certain number of dimensions, reflecting the number of independent characteristics you can adjust in the underlying system. > For example, even a sock drawer has a state space. Any sock might be described by its color, its length, its material, and how raggedy and worn it is. In this case, the dimension of the sock drawer’s state space is four. That's meaningless IMHO, 4 is the same as 1.

philipov 8y ago
Dimensionality is a type of structure that can be applied to quantities. Dimensionality refers to being able to find independent bases within a vector space. The study of vector spaces is a richer language than set theory. By allowing only cardinality to be meaningful, you are denying the value of dimensionality as an organizing principle. Therefore, its meaninglessness is tautological. Your mistake is to think that 4 is the same as 1. It is not. You are putting a hand up to one eye and complaining that stereoscopy is meaningless. You are pretending that only set theory exists, and claiming that therefore linear algebra is meaningless. It is rhetorical nonsense!
- ajuc 8y ago
  > It is not. Why? If I can describe a system with 1 number why invent 4 parameters to describe it? And why not 100? The choice seems arbitrary.
  - philipov 8y ago
    Because you can't describe any system with just 1 number. You are arbitrarily choosing to only pay attention to the cardinality of a set. If you are discussing a vector space, there is more structure than that. This is what the study of linear algebra is all about. The amount of structure you require depends on the question you are asking. Not all questions equate to finding the size of a set. You are like a blind man feeling the tail of an elephant and claiming the elephant's nature is the same as a rope!
  - UnFleshedOne 8y ago
    Say you convert color of the sock in the drawer to a range [0,1] on your one and only coordinate. Then you convert dirtiness of the sock to a similar range and just add them together. Can you differentiate between a sock that is 0.4 on color and 0.6 on dirtiness from another sock which is 0.6 on color and 0.4 on dirtiness? I suppose you could encode it all to one number so that structure is preserved, but dimensions themselves are not going anywhere, they are in your choice of encoding that must reside outside of your one coordinate.
  - NotAnEconomist 8y ago
    Because dimension is about the connectivity structure of a set of objects, not the size of the set of objects. Sets with the same cardinality can have wildly different topologies, which describe the possible paths through the space, which in turn describes the nature of functions into and out of that space, as well as transformations and relationships within that space. A line through the center of a sphere doesn't necessarily touch the sphere in a "4 dimensional" connectivity structure, must touch the sphere in a "3 dimensional" connectivity structure, while a sphere doesn't even exist in a "1 dimensional" connectivity structure. The main reason that we use "4 dimensional" spacetimes for particle physics is that there are three independent components of motion with respect to time, and so it makes the math simpler to allow for independent variation of those parameters to functions, instead of compressing them into a single encoded number, eg, derivatives wouldn't function correctly (or likely even be defined) on a single parameter model. An additional reason that we don't use a "1 dimensional" spacetime is because the product of a "1 dimensional" time and a "1 dimensional" space is a "2 dimensional" spacetime -- and we don't have any models of reality that lack either space or time. (Which is good: any model needs to explain both, if only as an emergent feature.) Topology is about connectivity, not scale; dimension is a topological fact.