The mathematical definition
of
dimension is not the one
used by sane human beings
Adapted for the Internet from:

Why God Doesn't Exist

    1.0   The definition of Mathematics has nothing to do with the definition of the rest of the world

    The average man thinks of a dimension as:

    “A measure of spatial extent, especially width, height, or length.” [1]  

    In everyday parlance, this is the context in which we normally use the word dimension. When we say that
    a box is three-dimensional, we are referring to the three mutually-perpendicular directions in which it faces
    or points. We are saying that the box has length, width, and height.

    We are surprised, however, when we find that the dictionaries and encyclopedias relegate this usage to
    ordinary speech.

    “ In common usage, a dimension (Latin, "measured out") is a parameter or measurement
      required to define the characteristics of an object—i.e. length, width, and height or size
      and shape. In mathematics, dimensions are the parameters required to describe the
      position and relevant characteristics of any object within a conceptual space…”   [2]

    “ in common parlance, the measure of the size of an object, such as a box, usually given
      as length, width, and height.”   [3]

    The three dimensions are commonly called length, width, and breadth”   [4]

    In Mathematics and Mathematical Physics, the word dimension means something else:

    “ 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 fundamen-
      tal measures of a physical quantity: Velocity has the dimensions of length divided by
      time.”  [5]

    “ The difference in the conditions of equilibrium of an ideal gas which is given by two
      independent variables, such as pressure and temperature, for a two-dimensional manifold,
      likewise the points on a sphere, or the system of pure tones (in terms of intensity and
      pitch).” (p. 84) [6]

    “ A physical property such as mass, length or time.” [7]

    The dimension speed, for example, is length divided by time [8]

    dimensions can also be other physical parameters such as the mass and electric
      charge of an object, or even, in a context where cost is relevant, an economic
      parameter such as its price. [9]

    And if the ‘physicists’ created their own dimensions, why couldn’t everyone else do the same…  

    “ Love: an important dimension in marital research and therapy”  [10]

    “ These are the (very smart) Mr. Smart's seven ‘dimensions’ along which a religion can be
      quantified.”  [11]

    It is not surprising that Hilbert ended up talking about 'infinite' dimensions. Suffice to say for now that
    none of these definitions pertains to Physics.

    The establishment's version is that the word dimension is a synonym of coordinate as well as any other
    characteristic you can think of (e.g., smell, sound, taste, mood, etc). The mathematical version has to do
    with locating a point within a physical object:

    “ dimension: Mathematics. The least number of independent coordinates required to
      specify uniquely the points in a space.”   [12]

    “ Dimension: is the number of coordinates needed to specify a point on the object.” [13]   

    Indeed, it is perplexing that the word dimension is not even explicitly defined in the Wolfram ‘Physics’ site.
    You will only find a definition in its Mathematics site.

    So there is absolutely no doubt. The mathematical idiot defines a dimension (architecture) in terms of
    coordinates (location, orientation). In mathematical jargon, an object is 3-D not because it has length, width,
    and height, but because it takes three numbers to locate a point somewhere in or on it. The mathematicians
    haven’t realized that they are using coordinates to define the physical characteristics of an object.

    So what is the problem? After all, a mathematician should be able to define a word as he sits fit or as needed
    for his dissertation. Certainly, he does not need to follow my prescription or advice.

    The obvious problem with the 'common' definition (length, width, and height) is that if it is not a part of
    Science, the mathematicians should not invoke it in their dissertations. The mathematicians give lip service
    to the 'street' definition of dimension and use it anyways. This way, when you pin them down on an
    inconsistency, they can always claim in retrospect that the formal definition is not length, width, and height.
    First, they take pains to differentiate between the 'length, width, and height' and the 'number of numbers'
    definitions. Then, these are obviously not the same.

    So I propose a solution to this serious problem. If I see anyone in the religion of Mathematical Physics using
    the length, width, and height version of dimension, I have a right to call him an idiot. Agreed? If Mathematical
    Physics has it correct, any person using the words length, width, or height in formal Math should, in this day
    and age, be deemed retarded. Maybe this method of mine will raise awareness that the mathematicians are
    bumbling fools.


    2.0   The length, width, and height definition is the one that the mathematicians use!

    If length, width, and height is not the formal definition of dimension, then there is something terribly wrong in
    Mathematical Physics because everyone and his mother invokes this definition in practically everything he
    writes:

    “ The way you describe it, it would be an equilateral triangle, with height equal to
      sqrt(3)/2 times the base, but in this case either the square would have to be a
      rectangle or the circle would be an ellipse. They couldn't both have the same
      height and width.”  [14]

    “ For a rectangle, we usually refer to the length and width… For a parallelogram,
      the important thing to remember is that the "height" is no longer a side of the
      shape, but the "base" is… The height is the distance between the top and bottom,
      measured perpendicular to them. The base is the length of the bottom. If we had
      said "width", you might think it meant the greatest side-to-side distance, which is
      not the base” [15]

    The concept of a fourth dimension is one that is often described in considering its
      physical implications; that is, we know that in three dimensions, we have dimensions
      of length (or depth), width, and height. The fourth dimension is orthogonal to the other
      three spatial dimensions…Usually, the fourth dimension is identified with time. In this
      case, the concept of an additional spatial dimension would be referred to as the fifth
      dimension.”  [16]

    “ Is there a formula that relates the length, height and width of a cuboid, without the
      need to know the volume or the cuboid? This is really just the same problem in three
      dimensions that you already solved in two dimensions.  [17]

    “ the container is box-shaped, so the volume is the product of length and width and
      height”  [18]

    “ All cubes are three dimensional because they differ with each other in size by all of
      the three measurements that we know of - width, length, and height.”  [19]

    “ three-dimensional shapes take up space in three different directions: length, width
      and height.” [20]

    “ A rectangular solid has three important dimensions: length (l), width (w), and height (h).
      If you know these three measurements, you can find the solid’s surface area, volume,
      and diagonal length.”  [21]

    “ In physics, spacetime is a model that combines three-dimensional space and one-
      dimensional time into a single construct called the space-time continuum, in which
      time plays the role of the 4th dimension.  [22]

    So we have our first contradiction in the ‘official’ version of dimension of Mathematical Physics. We
    have to conclude that the mathematicians are a bunch of idiots or that they are contradicting their
    definition. Which is it? Is a dimension related to structure (length, width, and height of an object), to
    location (coordinates needed to locate a point on an object), or to the directions in which we can move      
    (i.e., vectors)?

    We have to conclude that the length, width,and height definition the mathematicians so disdain is alive
    and well. This IS the most used version in Mathematics. This and not the idiotic 'number of numbers' is
    the true definition of dimension. This is also what mentally sane human beings understand by the wod
    dimension.


    3.0   The mathematical definition of dimension is fatally flawed

    The mathematical 'number of numbers' definition nevertheless fails anyways. It suffers from irrecoverable
    problems:

    1.   The mathematicians confuse dimensions, coordinates and vectors with each other.
         They go from architecture to location, from orientation to motion, and from motion to
         architecture or to location back again in the same presentation without noticing. To
         them it’s all the same. Indeed, a mathematician uses coordinates to define a dimension
         and explains dimensions in terms of coordinates

    2.   The mathematical definition of the word dimension cannot be used consistently and,
         therefore, is unscientific. The formal definition of Mathematics is the true ‘ordinary’
         definition and the true scientific definition of dimension is the length, width, and height
         of Physics.

    3.   The mathematicians are not even talking about dimensions, coordinates or vectors.
         They are talking about number lines. There are no such things as dimensions, coor-
         dinates, or vectors in Mathematics. The mathematicians would have no use for them.

    Let's address each objection individually.
I wonder if
research funds
qualify as a
dimension?
Bill, the Universe is
unknowable because
there are so many
dimensions.


    Pages in this module:

    1.    This page:  The mathematical definition of dimension is not the one used by sane human beings









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