Adapted for the Internet from:

Why God Doesn't Exist

    1.0   Humble origins

    Euclid defined a plane as a category of figures:

    " A surface is that which has length and breadth only. (Bk. I, Def. 5) [1]... The edges
       of a surface are lines (Bk. I, Def. 6) [2] …A plane surface is a surface which lies
       evenly with the straight lines on itself." (Bk. I, Def. 7)  [3]

    In other words, he visualized planes as ideally flat, static figures with boundaries. This is what Geometry was
    supposed to be all about: objects and the relations within and between objects.

    His successors, however, have misconstrued Euclid’s intentions and heavily mathematized his figures, especially
    during the 19th and 20th Centuries. Hoping to axiomatize his rather loose definitions and put them on sounder
    foundations, the mathematicians ended up amending his definitions to the point that they are not even similar to
    Euclid’s. In fact, the geometric figures of modern day Geometry have nothing to do with Geometry. The
    mathematicians revamped Geometry so much that it is no longer a static science, which is what this discipline was
    supposed to be from the start. All of the geometric 'figures' of today are dynamically 'constructed'. They are
    sculptures in the making. They have to do with limits and volumes and not at all with ideal planes and solids. When
    the mathematician invokes the word square or sphere, he is not referring to a figure you can see in a photograph.
    He is talking about a movie of the making of a square or a sphere. The idiots of Mathematics have converted
    geometry into a dynamic 'science' in order to make it more practical and useful for their purposes. What they
    ended up doing is blending conceptually static and dynamic versions into a surrealistic version of Geometry.


    2.0   The contemporary definitions of the word plane

    Modern dictionaries typically define a plane from a structural perspective:  

    " a surface in which a straight line joining any two of its points will lie wholly in
      the surface"  [4]

    " a surface containing all the straight lines connecting any two points on it." [5]

    The mathematicians prefer to define planes by ‘constructing’ them dynamically, sliding one straight line along
    another.

    " A plane is a two-dimensional doubly ruled surface spanned by two linearly
       independent vectors" [6]

    However, some of them prefer static geometric figures and clarify that planes are made of lines lying side by side
    which in turn are made of points.

    " In mathematics, a plane is a fundamental two-dimensional object. Intuitively, it
       may be visualized as a flat infinite sheet of paper." [7]

    " Lines are used to construct all other figures in plane geometry, including angles,
       triangles, squares, trapezoids, circles, and so on. Since a line has no beginning
       or end, most of the ‘lines’ one deals with in geometry are actually line segments
       —portions of a line that do have a limited length." [8]

    " all figures in a plane, as well as the plane itself, may be thought of as a set of
       points." [9]

    Of course, the definitions of point and line invoked in these definitions of plane are misconceived to begin with.
    They are predicated on erroneous notions. Therefore, the foregoing definitions of plane relativists live with today
    are on flimsy ground from the start. It leads to incongruous notions because now the plane is a static image rather
    than a movie of a scanning line. The static plane is constructed not with dots, but with locations.

    Like a line, a plane is alleged to be infinite:

    " Definition: An imaginary flat surface that is infinitely large and with zero thickness

       Clearly, when you read the above definition, such a thing cannot possibly really
       exist. Imagine a flat sheet of metal. Now make it infinitely large in both directions.
       This means that no matter how far you go, you never reach its edges. Now imagine
       that it is so thin that it actually has no thickness at all. In spite of this, it remains
       completely rigid and flat. This is the 'plane' in geometry. It fits into a scheme that
       starts with a point, which has no dimensions and goes up through solids which
       have three dimensions:"  [10]

    With so many versions, we ask in exasperation: What in tarnation is a plane?

    Clearly, the modern mathematicians have emphasized ‘construction’ over architecture and structure. The difference
    is that structure and architecture refer to a stand alone statue. A statue only has location (i.e., occupies a single
    frame in the universal movie). A construction may be one of two things. It is either a description of what the static
    plane is made of or it is a recipe indicating how to make a figure. A geometric figure is supposed to be a still image.
    The mathematicians ask you to watch a movie of a line sliding along another. If both notions are incorporated into
    the definition of the word plane, we can see how useful such a malleable contraption can be to a theorist. The
    ambiguous plane becomes a powerful tool to explain anything. It doesn’t follow, however, that the juror understood
    the theory.  How can a triangle be infinite? How would an infinite triangle differ from an infinite square or an infinite
    circle if we cannot see the boundaries of the figure? Obviously, the mathematicians are using the word plane in
    inconsistent ways. This is unscientific. There should only be one definition for the purposes of Science! A plane
    cannot be several different notions or incorporate all of them because then it would be impossible to make sense
    of the theory. That is the status today.


    3.0   Are planes 2-D, or do we require two numbers to locate a point on a plane?

    The biggest sources of confusion with planes in Mathematical Physics are related to the number of dimensions.
    Either a plane is a geometric figure that has two dimensions (width and height) or it is a location specified with
    two numbers (x, y) referenced to a point of origin. It can't be both because these notions are irreconcilable. So
    which of the following definitions is correct?

    " A plane is a geometric figure with only two dimensions: width and length. It has
        no thickness."  [11]

    " Planes are two-dimensional. A plane has length and width, but no height, and
       extends infinitely on all sides. Planes are thought of as flat surfaces, like a tabletop.
       A plane is made up of an infinite amount of lines. Two-dimensional figures are
       called plane figures." [12]

    " locating a point on a plane (e.g. a city on a map of the Earth) requires two parameters
       — latitude and longitude. The corresponding space has therefore two dimensions,
       its dimension is two, and this space is said to be 2-dimensional (2D)." [13]

    Width and height are dimensions. This means that they are conceptually made of a single piece and therefore
    indivisible. The mathematical number line, instead, is a segmented concept. Dimensions have direction but no
    magnitude. They are entirely qualitative concepts. Number lines, in contrast, have magnitude, but not direction.
    And so on.  So does a plane requires 2 numbers or coordinates to be 2-D? Is it 2-D because we can locate a point
    on it using two number lines? Or is a plane 2-D because it has width and height?
After smoking
one of these
stogies, I am
convinced that a
plane is
something that
moves.
So this is how they
come up with their
definitions of
geometric figures!  
Now I get it!

    4.0   The Equation plane

    And of course, we always find the ubiquitous and most incongruous 'equation' definition of a geometric figure:

    " A plane is the set of all points whose coordinates (x,y,z) satisfy a linear equation
       in x, y, and z." [14]

    " Plane: A two-dimensional surface defined by linear equations."  [15]

    " Complex plane: complex numbers can be thought of as points in a plane"  [16]

    The mathematicians call this being 'rigorous.' They go to extremes to 'formalize' and make their definitions more
    'objective', meaning more scientific. In their world, a plane is not something you point to so that even a baby can
    understand. A plane is something you construct point by point with a computer and define with an equation or a
    set of numbers. So the idiots, in their immense ignorance, end up converting a geometric figure into a movie of
    how they constructed an ideal sculpture. An equation is a shorthand for how something moves one step at a time.
    You plug one value into the variable and plot the point. Since the mathematical statue is always made of an 'infinite
    number of points', it is always in production. The motion pictures being filmed at the Mathematical Studio have no
    chance of ever making it to the Big Screen.
The definition of the
word plane
A plane is made of
points like the drops
of whiskey pouring
out of this bottle
A plane is infinite like the trip
this weed is giving me.

A plane is rough
like this floor.

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        Copyright © by Nila Gaede 2008