 Points and Motion 
It
is my purpose in this paper to expose in a very simple fashion the most
basic errors that were made. One basic error involves the idea of
motion itself.^{13}
In
formulating concepts of motion, the geometers used a "point in
motion" to determine or specify, for example, velocity. Now a
"point" is a static concept a priori. To determine (or
even to think and perceive) motion, one must determine that it occupies
two different points (positions or locations) at two different times,
yet consider both points at the same time. Indeed, that is
precisely what the arrow means that is used to represent a vector.
A "point in motion" therefore represents a contradiction of
opposites. That is, it represents the idea that "that which
is motionless has motion.".^{14}
Even with this, there is
a difference in a spatial point and a spatiotemporal (spacetime) point.
To exist at all, a spatial point must be moving in time; in other words,
it is a spatiotemporal line, even if it is a static spatial point.
Vector analysis was
constructed in the abstract  again, a massless point in motion
possessed or constituted a velocity vector, etc. In massless ( and
timeless) space, FIELDS were defined: "scalar" fields
constituted the assignment of a simple motionless number (magnitude) to
each spatial point, while "vector" fields constituted the
assignment of a "simple vector" (magnitude and velocity) to
each spatial point. But the MATHEMATICAL vector system consisted
of massless (point) motional relationships, recognizing zero motion as a
special case of motion.^{15}
Of course
mathematics development was also always intertwined with practical
problems. With the sustained application of mathematics to gross
physical material problems, mechanics slowly arose.
These
developments required decades and even centuries to occur completely.
All along the way, innovations and changes  and additions to the
mathematical formulism were being derived and taught to students as the
"natural" system of reality. A permanent mindset was
being forged.
Indeed,
mathematics was regarded as THE single human expression of fundamental
truth. Not until Godel's work in the twentieth century did it
become evident that MATHEMATICS IS SIMPLY A GAME PLAYED ACCORDING TO
ASSIGNED RULES, AND THERE IS NO ULTIMATE TRUTH IN MATHEMATICS ALONE.^{16} It is a most useful game, of course, since it is the game
fitted to perception. Thus it applies, essentially, to whatever
can be perceived. But to be applied to physical systems, it must
be changed, altered, updated, and fitted as the perceiving/detecting
instruments become ever more subtle.
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