1.0   Mathematical dualities come in handy to explain anything

Euclid defined the word point as ‘that which has no part’ (Bk. I, Def. 1)  [1]  He appears to be saying
that a point is a zero-dimensional (0-D) figure. This is not ‘difficult’ or ‘hard’ to visualize as the
mathematicians are fond of saying. It is impossible to visualize! We may have difficulty visualizing
an infinitesimal point. We absolutely cannot visualize a  0-D point. A mathematician is an individual
who is more concerned about being politically correct  than about being precise in his statements.

Therefore, before we can continue with any other aspect of Euclid's book, we must resolve what he
had in the back of his mind with this 'no part' point of his. We must assume that he intended to use
his point in the context of architecture. You would think that this was his goal because the subject
matter he was discussing was Geometry. Throughout his book, Euclid uses his 0-D point to
construct and contain his line segments and higher dimensional figures:

“ The ends of a line are points.” (Bk. I, Def. 3)

“ The edges of a surface are lines.” (Bk. I, Def. 6)

“ Rectilinear figures are those which are contained by straight lines (Bk. I, Def. 19)

“ A pyramid is a solid figure contained by planes which is constructed from one plane
to one point.” (Bk. XI, Def. 12)

Evidently, Euclid was concerned with constructing a system of definitions of physical entities. He
could have intended it in no other way. Geometry is a static, visual discipline. Without a shape to
visualize, there is no such thing as Geometry.

“ Geometry is the study of shape and size. Geometry was probably first developed to
measure the earth and its objects.” [2]

“ Geometry: The branch of mathematics whose primary subject is spatial relationships
and shapes of bodies. Geometry studies spatial relationships and shapes, while
ignoring other properties of real bodies (density, weight, colour, etc.).”  [3]

Yet, throughout Elements, Euclid incongruously uses his 0-D dot to designate positions and
locations:

“ And the point is called the center of the circle.”  (Book I, Def. 16)

“ If on the diameter of a circle a point is taken which is not the center of the circle…”
(Bk.III, Prop. 7)

Now there is a significant qualitative difference between a dot and a location. The former is an
object: a geometric figure. The latter is an abstract concept. You cannot visualize a location without
an object.

So let's review what Euclid is attempting to do to see if he can get away with it:

1. Euclid introduces the simplest of figures and wants you to believe that he is talking about
a dot. He is alluding to something that is tiny (i.e., infinitesimal) but that still has shape. He
is referring to an object that, as a minimum, has width and height (Fig. 1).
 Fig. 1   DotA mathematician marks a point on a piece ofpaper or tells you that it looks like the one atthe end of this line. When magnified, youverify that these dots have two dimensions:width and height.

2. Afterwards, he uses this infinitesimal, yet 2-D dot throughout his presentation to construct
and contain geometric figures such as planes (triangles, circles) and solids (cubes, spheres).
All of these figures have shape. Obviously, this mathematician is not talking about abstract
concepts such as love, beauty, or justice.

3. However, now he asks you to make believe that this 2-D dot is really 0-D:

“ Point: A dot that indicates a definite position or location. A point has no width,
depth, or length.” [4]

or non-dimensional...

A point shows an exact position or location on a plane surface. It is important
to understand that a point is not a thing, but a place. We draw a point by
placing a dot with a pencil. This dot may have a diameter of 0.2mm, but a point
has no size. No matter how far you zoomed in, it would still have no width.” [5]

[Draw a location? I wonder how the mathematicians do that!]
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4. Then, he uses this 0-D dot to designate positions and locations on his geometric figures.

I believe that this is quite unethical of him. Euclid self-servingly uses this ‘entity’ in two irreconcilable
manners throughout his dissertation. When you ask, ‘How do you construct your solids with
locations?', He answers, ‘They are not locations. They are dots. See?’ And he makes a mark on a
paper with his crayon. When you ask, ‘How can you fit an infinite number of these dots between
any two dots on a line?’ or ‘How can you make a smooth circumference with dots?’ He answers,
‘They are not dots,stupid! They are locations. Don’t you have any imagination?’

Unfortunately, Euclid and his successors cannot have it both ways. If the jurors were to grant Euclid
his malleable dot, they would be unable to follow his theories.

Allow me to put you to the test with the following statements. Are the authors referring to a dot or to
a location?

“ The ends of a line are points… A straight line is a line which lies evenly  with the points
on itself.” (Bk. I, Defs. 3 and 4 respectively)

[Dot or location?]

“ Adding points to a line results in an object that violates one or more of the following
axioms…" (Hilbert's 20th Axiom)  [5]

[What is Hilbert adding to his line: dots, ordered pairs, or locations?]

“ tangent - a straight line or plane that touches a curve or curved surface at a point but
does not intersect it at that point”  [6]

[How does a tangent ‘touch’ a location or an ordered pair?]

Between any two points on a line there may be an infinite number of intermediate
points.”  [7]

[If there is always a point between any two points on a line, are we talking about dots
or locations? If you opt for dots, it becomes an issue of size. We determine the
diameter of the dot and afterwards can only fit a finite number of such dots in any
given length of line. On the other hand, it can be argued that there are countless
locations between any two locations. The problem is that we cannot construct a
physical line with locations. Geometry absolutely requires shapes. A location is
nothing without 'something' in it!]

In science, it is illegal to use conflicting definitions of a strategic word during the presentation. A
strategic word is one that makes or breaks your theory. For the purposes of Physics, ‘that which
has no part’ or 'no extension' is known as nothing. Structure-less is a synonym of void, for in what
respects would nothing differ from Euclid’s ‘no part’ point otherwise? Under his plain definition, a
point is reduced to a self-contradiction: a structure-less structure. Predictably, this leads to
meaningless and irrational conclusions:

“ The empty set is said to have dimension -1. What is here to draw? So we start with a
point whose dimension is 0.”[8]

[Dimension negative 1? The mathematicians are truly very creative when it comes to
concocting meaningless nonsense. In Physics, anything that has dimension zero is
called empty space! Dimension -1 is what a mathematician has between his ears!]

These are non-trivial objections, indeed, fatal to the religion of Mathematical Physics. A dot is a
physical object. A location is a concept. An object differs from a concept in that it has shape. What
shape does 'a point whose dimension is 0' have?

Of course, we can see how useful this object-concept duality is to the prosecution’s case. We can
explain anything with it. The object-concept duality converts physical interpretations into
unfalsifiable propositions. It is not surprising that the mathematicians refuse to define the word
point. If the mathematician presents the point as a dot, he can’t use it as a location. If he defines the
point as a location, he can’t use it to construct figures in Geometry. Therein lies the contradiction.

2.0   Christian Mathematics: Of avatars and emoticons

But there is a more fundamental problem with Euclid's point and it is the amusing habit the
mathematicians have of depicting the concept location with a physical marker such as a dot. This
puts Mathematical Physics on the same footing as Christianity. In Christianity, for instance, it is
routine to depict concepts with objects:

the concept ‘love’ with the object ‘heart’  [9]
the concept ‘piety’ with the object ‘woman’ in a desolate scene  [10]
the concepts ‘salvation,’ ‘suffering,’ or ‘resurrection’ with the object ‘man’ nailed
to a cross  [11]
the concept ‘Good’ with the object ‘angel,’ a winged man usually sporting a gold
ring or a plate on his head  [12]
the concept ‘death’ with the object ‘Grim Reaper’, a skeleton holding a scythe.  [13]

In science, we refer to these surrealistic transformations and transubstantiations as emoticons,
avatars, or incarnations: a concept-turned-object. They make us smile. Some people take them
seriously, but fortunately most of these folks are locked up in padded rooms for their and our
safety.

So what did Euclid do?

Euclid replaced the concept location with the object dot and passed this nonsense down to his
followers. Today, the mathematicians continue with this mythological tradition. The modern
scholars believe that if they can successfully divide a dot in half, they have also split its location
in half (Fig. 2). In Science, we refer to these amusing techniques as ‘Christian Mathematics'.
Mathematical Physics is a discipline that exclusively studies reification.
 Fig. 2   Christian mathematical objects

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