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-- It Started With Geometry and Grew --
At
the very beginning of what we call the "scientific period,"
mathematics was both king and queen, and Euclidean geometry was its
handmaiden. So we ask, "What precisely is
geometry?" Here we are not interested in a
"textbook" answer, but in an answer indicating what geometry
really does.3
In other
words, with what does geometry concern itself, and what is the
fundamental nature of those things with which it concerns itself?
Briefly,
geometry -- at its foundation -- is totally spatial. It is fitted to, and
expressed in terms of, the TOTAL ABSENCE OF MASS. Thus the geometer
deals in abstract, massless entities called "points,"
"lines," "planes" etc. When the geometer speaks of
"motion," he speaks of a time-smeared, length-smeared point.
Geometry at heart is massless, and a "geometer's vector" is
a highly specific type of "system." In fact, it represents the
"time-smearing" and "length-smearing" of a point.
A priori, the fundamental concept of the geometrical vector has taken a
"spatial" entity and introduced a hidden involvement with
"time."
Modern
mathematics and physics have followed an intertwined development for
several hundred years. And both sprang as offshoots of the original
work of the geometers. Let us briefly sketch the overall path of
interest taken by these two developing disciplines.
With the advent of
Descartes's fundamental work, algebra was combined with geometry to
yield analytic geometry, a new and powerful mathematical tool. With
the invention of calculus by Leibniz and Newton, both mathematics and
physics received a giant impetus. Differential geometry and vector
mathematics arose in full splendor and, in physics, mechanics leaped to
the forefront with Newton's profound work.
But the
mechanics made a most fundamental error when they simply applied the
geometer's vector to a mass, to produce -- so they thought -- a mass
vector. That which rigorously applies only to the absence of mass
cannot be so lightly applied to the presence of mass without the risk of
serious limitations in the resulting theory. The precise
difference between a geometer's massless vector and a mechanic's
mass-vector is one of the issues to be developed in this thesis.
As rapid
development continued in mechanics and mathematics, certain physicists
were involved in intense experimental work on charged matter, becoming
the first electricians. Both the preceding mathematical ideas and
constructs as well as the preceding (partially erroneous) mechanics
constructs and ideas were applied by the electricians, struggling with
their pith balls, cat fur, and glass rods to understand, quantify, and
model electrical forces and the phenomena of charged matter. In
other words, the electricians strove to formulate the physics and
dynamics of charged matter and its interactions by simply "adding
to" the work of the geometers and mechanics. Here again, a
fundamental logical error was made. That (geometry) which a priori
applies only to the absence of mass, and that (mechanics) which a priori
applies only to the absence of charge, cannot be lightly applied to the
presence of charged mass (both mass and charge)4
without risking the incorporation of grave limitations in the
resulting theory.
After the
profound work of Maxwell, the idea of FIELDS OF FORCE became more
prominent, until the field concept ruled the day5.
The electricians
continued, pushing the idea of fields into space and vacuum itself,
along the way inventing the idea of "charge effects" existing
even in the massless vacuum, with concomitant fields. Meanwhile, they
had thoroughly confused chargeless point-smeared, chargeless
mass-smeared, length-smeared and time-smeared vectors.
After a set
of fundamental experiments designed to detect motion of the material
ether yielded essentially null results6, Michelson and
Morley were regarded as having completely disposed of the ether -- even
though the experiments only disposed of material ethers, and not
Lorentz-invariant non-material ethers7.
Maxwell's equations and the
field concept were elevated to profound importance.8
Then, after Einstein's fundamental relativity work shortly after the turn of the
century, the ether concept faded away and the field concept reigned supreme.
Indeed, in their enthusiasm the interpreters of relativity went so far
as to affirm that one can have a wave without any medium; that is. that
something can be moving (waving) without anything there to move!9
And
with great glee they pronounced the final end to the idea of
"ether" as a medium, even though Einstein himself never did
any such thing.10
With the advent of Einstein's General Theory of
Relativity, even matter came to be regarded as just a special
"kink" or curvature in spacetime or "vacuum nothing."
Quantum
mechanics arose and even certainty and determination fell. Chaos,
probability, and randomness now assumed the ruling position. Probability
waves (and probability fields) arose,11 as did quantum fields of various
kinds. The intermingling of these concepts with the concepts of
electrodynamics pushed the idea of the field even farther into esoteric
realms.
The point
is, each of these developing disciplines incorporated and built on the
foregoing disciplines. From the beginning of geometry, there was no
rigorous definition of a vector, and there is none today.12
From the
beginning of mechanics, in their foundations the theorists made grave
logical errors by incorporating the geometer's vector; errors so great
that today mechanics and electromagnetics are severely flawed, as is
everything that came after them and built upon their illogical
foundations.
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