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Thread: Proof: Physical Dimensions behaving as algebraic quantities

  1. #1 Proof: Physical Dimensions behaving as algebraic quantities 
    Forum Junior AndresKiani's Avatar
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    Is there a rigorous proof that explains why physical dimensions behave as algebraic quantities. This concept allows Physicists to incorporate dimensional analysis in their calculations and theories. I'm interested in a proof or rigorous argumentation to why this works. Any help would be great.


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    Forum Junior AndresKiani's Avatar
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    I acknowledge that there may not be a formal proof for this, but perhaps a satisfying explanation or any form of rigor would be great.


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    I wonder if one defines a sub dimensional space as one where one of the higher dimensions is frozen as a constant?

    So do you have to work downwards? First assume your n-dimensional space and then "carve out" your series of lesser numbered dimensions?

    I assume you are talking about dimensional spaces that actually describe physical and spatio-temporal scenarios?

    Or is it a purely mathematical process you are wanting?

    Edit: did you see my earlier thread from last month ?

    Dimensions
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    Quote Originally Posted by geordief View Post
    I wonder if one defines a sub dimensional space as one where one of the higher dimensions is frozen as a constant?

    So do you have to work downwards? First assume your n-dimensional space and then "carve out" your series of lesser numbered dimensions?

    I assume you are talking about dimensional spaces that actually describe physical and spatio-temporal scenarios?

    Or is it a purely mathematical process you are wanting?

    Edit: did you see my earlier thread from last month ?

    Dimensions
    I'm looking for a purely mathematical argumentation. I think I can prove just why algebraic operations work on these species by thinking about functions that map length to area for example. However, I think that one could imagine of a better argumentation involving more abstract mathematical structures like spaces and fields.
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    Quote Originally Posted by AndresKiani View Post

    I'm looking for a purely mathematical argumentation. I think I can prove just why algebraic operations work on these species by thinking about functions that map length to area for example. However, I think that one could imagine of a better argumentation involving more abstract mathematical structures like spaces and fields.
    Is there anything of interest in this other thread I started on another forum?
    Surfaces in higher dimensions - Mathematics - Science Forums
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    Quote Originally Posted by geordief View Post
    Quote Originally Posted by AndresKiani View Post

    I'm looking for a purely mathematical argumentation. I think I can prove just why algebraic operations work on these species by thinking about functions that map length to area for example. However, I think that one could imagine of a better argumentation involving more abstract mathematical structures like spaces and fields.
    Is there anything of interest in this other thread I started on another forum?
    Surfaces in higher dimensions - Mathematics - Science Forums
    No, that thread isn't helpful in this discussion. The definition of a surface in mathematics isn't relevant here, at least I don't see a relevancy... I'm also on that forum, my username is Symmetrica.
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    Quote Originally Posted by AndresKiani View Post
    Is there a rigorous proof that explains why physical dimensions behave as algebraic quantities. This concept allows Physicists to incorporate dimensional analysis in their calculations and theories. I'm interested in a proof or rigorous argumentation to why this works. Any help would be great.
    i am not sure if i fully understand your question, but i'll give it a try:
    in the SI system of measurement, you have basic dimensions, from those you can form derived dimensions, using those in a measurement gives a standard format quantifier x unit of measurement.eg:
    pressure=force acting on a surface: 1Pa=1N/m^2
    kinetic energy is force acting over a distance : E=F x s, J=N x m (a)
    force is acceleration acting on a mass: F=m x a, N=kg x m/s^2 (b)
    plug-in (b) in (a)
    J=kg x (m/s)^2, E=m x v^2
    replacing v (velocity) by c (celeritas,latin for velocity used for speed of light and speed of sound)
    E=mc^2
    however, applying mathematical methods, you get:
    E=1/2 mv^2
    "Of course today every rascal thinks he knows the answer, but he is deluding himself" (A.Einstein)
    guilty as charged
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    Quote Originally Posted by AndresKiani View Post
    Is there a rigorous proof that explains why physical dimensions behave as algebraic quantities. This concept allows Physicists to incorporate dimensional analysis in their calculations and theories. I'm interested in a proof or rigorous argumentation to why this works. Any help would be great.
    I assume that your question is about things like 1 kg + 2 kg = 3 kg. If so, ask yourself what the meaning of 1 kg + 2 uvulas might be.

    The equals sign requires commensurate quantities on both sides. That includes especially the units.
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  11. #10  
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    Quote Originally Posted by tk421 View Post
    Quote Originally Posted by AndresKiani View Post
    Is there a rigorous proof that explains why physical dimensions behave as algebraic quantities. This concept allows Physicists to incorporate dimensional analysis in their calculations and theories. I'm interested in a proof or rigorous argumentation to why this works. Any help would be great.
    I assume that your question is about things like 1 kg + 2 kg = 3 kg. If so, ask yourself what the meaning of 1 kg + 2 uvulas might be.

    The equals sign requires commensurate quantities on both sides. That includes especially the units.
    I think it's more about why a Newton is a kilogram × metre / second˛ rather than something like log(kilogram) × emetre / sin˛(second)
    There are no paradoxes in relativity, just people's misunderstandings of it.
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