Adapted for the Internet from:

Why God Doesn't Exist
A mathematician
doesn't know what
an angle is

    1.0   The angle of Math

    Euclid defined an angle as an inclination:

    “ A plane angle is the inclination to one another of two lines in a plane which
       meet one another and do not lie in a straight line. (Bk. I, Def. 8)…And when
       the lines containing the angle are straight, the angle is called rectilinear.”
       (Bk. I, Def. 9)  [1]

    Unfortunately, he forgot to tell us what an inclination is. Like all mathematicians, Euclid was careless
    with definitions and relied on synonyms, assuming that his audience already knew what he was
    talking about. We can be 'inclined' to forgive him because he was among the first to attempt to
    formalize geometric concepts, but what excuse do his disciples have?

    Nevertheless, under any interpretation, inclination is a static attribute and, despite allowing for
    angles comprised by curved lines, Euclid had the angle circumscribed to a plane. The mathematical
    physicists of subsequent generations altered both of these attributes. In Mathematical Physics, an
    angle is essentially the 'amount of rotation' needed to bring a line segment into correspondence
    with another.

    “ the amount of rotation needed to bring one line or plane into coincidence
       with another”  [2]

    “ Given two intersecting lines or line segments, the amount of rotation
       about the point of intersection (the vertex) required to bring one into
       correspondence with the other is called the angle θ between them.”  [3]

    The result is not a photograph, but a movie. The ET has to wait until the relativist brings one line in
    correspondence with the other to learn what an angle is. In other words, the contemporary definition
    of the word angle is another mathematical proof. It is not surprising that the mathematicians end up
    talking about angles lying on the surface of a sphere:

    “ The solid angle Ω subtended by a surface S is defined as the surface area
       Ω of a unit sphere covered by the surface's projection onto the sphere.”  [4]

    So what really is an angle?


    2.0   The angle of Physics

    Webster defines an angle as:

    “angle: the figure formed by two lines diverging from a common point.” [5]

    This pretty much summarizes what most people understand by the word angle. However, it leaves
    unresolved whether by figure, the authors meant that an angle is a physical object. The word figure
    is defined as:

    “ Mathematics. A geometric form consisting of any combination of points, lines,
       or planes: A triangle is a plane figure.” [6]

    “ The outline, form, or silhouette of a thing… An indistinct object or shape”  [7]

    So, is an angle a physical object? Are they referring to the shape we have before us or to the
    opening, to the space between the two sticks?

    If we were to concede that an angle is a figure, the angle is suspiciously much like an unfinished
    object. An angle is not enclosed all around like an ideal table or a circle. The value of the word angle
    is in the spacing between the sticks and not in the sticks themselves.

    Therefore, an angle, like an edge, is not an object in its own right. For example the letter ‘c’ can be
    conceptualized as a completed ‘c’, in which case we can regard it as a stand-alone object contoured
    by edges. Another perspective shows the ‘c’ as an incomplete ‘o’, in which case, despite being 2-D,
    it is actually proposed as the edge of an object whose construction is still in progress. I will use the
    word pattern to refer to incomplete geometries such as angles, parallels, and perpendiculars. From
    a conceptual point of view, the angle is not an object formed by two 2-D objects such as rectangular
    Euclidean lines, [8]  but an incomplete pattern formed by two edges of one or two objects which are
    irrelevant to our field of view (By the way, an edge, like a number, is also a pattern.)

    The next question we need to address is whether these edges touch. Many have intuitively
    developed the notion that, when two objects touch, the portion of their respective surfaces that
    come in contact somehow blend to become one. For instance, when the vice clamps down tight on
    a board, the marks left on the wood would appear to indicate that its surface came in direct contact
    with the vise metal. However, from a strictly conceptual point of view, this would imply that the
    wood and the metal had become one (i.e., made of a single piece). In order to remain discrete,
    space must contour the clamp and the wood at all times. For example, when the atoms of your
    fingers press down against the atoms of your palm they continue being discrete entities. Space
    must remain the boundary of each atom if we are to continue to talk about discrete atoms. The
    question, then, is whether two surfaces ever actually touch each other if there must be space
    between them at all times! Let me run that by one more time. When two cars crash, did they touch
    each other? You answer 'Of course!' Then why is each atom in the entire system wrapped
    individually in space at all times?

    Therefore, for the purposes of Physics, an angle is not constructed with two separate lines. An
    angle is, by definition, a single object whose straight edges meet at one end, but not at the other.
    When we look at an angle, we should not be distracted by the two lines. The word angle only refers
    to the spacing between them. If we fuse the edges where two objects meet, we have in effect
    amended the definition retroactively. We are no longer talking about TWO separate objects.

    I extrapolate the same notion to ideal edges. If two edges are independently continuous they
    cannot fuse to form a single entity,  more so when an edge, like a plane, is simply an attribute of
    a 3-D object. Space must at all times delineate the object to which the edge belongs. Hence, unlike
    the edge, a vertex cannot be a continuous pattern despite that we treat it as such. Under a
    sufficiently powerful microscope a vertex should exhibit a gap between each edge. Thus, the
    vertex is not a 'point,' but a 'pattern' formed by two coplanar edges that meet, but which don't
    touch. If these edges touch (i.e., superimpose), then we are again treating the entire pattern
    as a single, continuous object.

    This argument instantly eliminates 0º and 180º angles in Physics. In order for one edge to form an
    angle with another edge, they must conceptually touch only at one common end. This means that
    we are treating the entire pattern as a single, continuous object. In Physics, whether it is formed by
    one or two objects, the angle formed is treated as if it forms part of a single object:

    angle: A pattern formed by two straight edges fused at one end of an object lying
    on a plane facing the observer. For the purposes of the word angle, the only pattern
    which is relevant is the one formed by the two edges, whether they belong to one or
    to two objects. The objects themselves are irrelevant for the purposes of concep-
    tualizing an angle.

    Synonyms: vertex, node, corner, divergence, knee, bend, fork, incline, intersection.
Is that an
angle, Steve?
No. Not quite, Bill.
I haven't yet
rotated the
compass.


    Pages in this module:

    1.   This page:  A mathematician doesn't know what an angle is

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