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Why
NonEuclidean Geometry is a Cheat (including
a critique of the “complex number plane”)
by
Miles Mathis
Today's
scientists have substituted mathematics for experiments, and
they wander off through equation after equation and eventually
build a structure which has no relation to reality."—
Nikola Tesla**
First
posted May 26, 2008 Abstract:
I will show that nonEuclidean geometry, although potentially
valid, has been used historically as a cover for bad math. Its
needless complexity, its definitional opacity and incompleteness,
and its inherent lack of rigor have opened it up to broad and one
might say universal misuse. Here I show the primary and
fundamental problem at the core of curved geometry. I then make a
similar case against complex numbers. I attack the definition of
complex number directly, exploding the fundamental derivation of
the math by going step by step through the first pages of a
textbook. Finally, I show why and how curved geometry and complex
numbers are used to purposefully hide the mechanics of the
electrical field.
In
other papers I have shown specific errors in the use of
nonEuclidean geometry by Einstein, Minkowski, and others. I have
also shown many problems in the use of the tensor
calculus as applied to physics. But these papers leave open
the question of the overall status of nonEuclidean geometry. Is
it true? Is it false?
In my title here I
choose the word “cheat” on purpose, since my intention is not
to show that nonEuclidean geometry is necessarily false. My
intention is to show that nonEuclidean geometry is necessarily
less efficient, less transparent, and less exact. Even
Poincaré—the grandfather of nonEuclidean math—admitted
that, in part. He said,
One geometry cannot be more
true than another; it can only be more convenient. Now, Euclidean
geometry is, and will remain, the most convenient: 1st, because
it is the simplest, and it is not only so because of our mental
habits or because of the kind of direct intuition that we have of
Euclidean space; it is the simplest in itself, just as a
polynomial of the first degree is simpler than a polynomial of
the second degree; 2nd, because it sufficiently agrees with the
properties of natural solids, those bodies which we can compare
and measure by means of our senses.
Because
nonEuclidean geometry is less transparent and far bulkier, it is
much easier to fake. It is far easier to hide slippery
manipulations under a blanket of confusing and undefined
operators and spaces. And because nonEuclidean geometry is not
tied into our "direct intuition of space," cheats are
not as easy to spot. Furthermore, nonEuclidean geometry is
utterly dependent upon Euclidean geometry for all its definitions
and for any and all exactness it retains. In mathematical terms,
nonEuclidean geometry is a function
of Euclidean geometry. It is completely dependent upon Euclidean
geometry, since a rectilinear field must lie under every curved
field, whether that field is curved in the hyperbolic, elliptic,
or any other sense. Finally, I will show that although
nonEuclidean geometry could be consistent and logical, it almost
never is. If a mathematician kept scrupulous track of his
curvature during and after every manipulation, and refused to use
slippery operators or functions or variables, nonEuclidean
geometry could be used to get the correct answer. But to keep
track of the curvature like this, a mathematician would have to
“measure” his nonEuclidean manipulations with Euclidean math
all along the way—which obviously undercuts the entire raison
d’etre of the new geometry. If you have
to check nonEuclidean geometry against Euclidean geometry, why
not just use Euclidean geometry to start with?
I used standard terminology in that
first paragraph, just to get us started, but I am going to switch
now to simpler terms. I have already complained of needless
complexity and terminology, and I will do so again below; so to
prove my point I will begin jettisoning baggage now. First of
all, I will get rid of the terms “nonEuclidean” and
“Euclidean,” replacing them with “curved” and “straight.”
That will be my first simplification, although there may be
others.
We have been told that curved
geometry has been used for the last two centuries because it
allows us to solve problems we could not solve before. This is a
false claim. Any problem that can be solved with curved geometry
can be solved with straight geometry, and it can be solved much
more quickly and transparently with straight geometry. If
problems have seemed to be solved with curved geometry that could
not be solved with straight geometry, it is only because those
problems were too subtle for mathematicians of the time. They
could not solve them with rigorous, elegant proofs, and they
needed some room to fudge their way through the proof. Curved
geometry was chosen because it gave them this latitude, this room
to move. As I will show, curved geometry allowed for numbers to
be squashed and stretched, and this allowed for solutions to be
hammered into place. By and large, curved geometry came to the
fore not for honest reasons, but for dishonest reasons. It has
become pandemic not because it is better but because it is easier
to fake. It has flourished for the same reason legalese has
flourished and for the same reason propaganda has flourished and
for the same reason advertising has flourished. It has flourished
because it has proved to be a successful conjob. It is an
impressive and opaque parlor trick that fools almost everyone. It
fills blackboards and makes people rich and famous.
Up until about 1912, Einstein
didn’t trust curved geometry. In fact, many physicists at that
time were still wary of it. It might be said that this proves
nothing, since Einstein wasn’t much of a mathematician, then or
later. If he was afraid of it, it was only because he hadn’t
mastered the manipulations. In fact, I myself have shown,
regarding the equation x’ = x  vt, that Einstein wasn’t even
clear on the foundations of straight geometry, so his fear
of curved geometry proves nothing. But what is interesting about
this case is not that Einstein didn’t understand curved
geometry, but that those who were schooling him on curved
geometry didn’t understand straight geometry. Some of the
biggest names in the history of curved math were active at the
time, including Minkowski, Weyl, Hilbert, and Klein. Not one of
them noticed that x’ = x  vt was false, or that Einstein had
made a hatful of other basic Euclidean errors. Minkowski not only
used this equation, he verified it using curved math. He
verified a series of false equations and a false derivation. This
is the ultimate proof that curved math is dangerous. The masters
of the medium didn’t use it properly, either because they
didn’t want to or because they couldn’t. None of their
students could recognize their mistakes, and no one in math
departments now can see them. If this is not a dangerous
situation, I don’t know what would be.
Curved math has been gaining power
since the time of Bolyai and Lobachevsky and Gauss, in the
1820’s. It is now used for everything, down to counting apples
and adding up bills. No doubt it will soon replace basic algebra
in junior high. No one, including schoolchildren, wants to be
seen using the old math: it is not sexy enough; it might keep
them off TV. But for two millennia before that, all
mathematicians had avoided it as either intuitively or
demonstrably false. When you and I are reminded that almost all
professional mathematicians now accept it, remember that
historically they are still outnumbered. Counting heads is no way
to determine the truth of any matter, but if contemporary
mathematicians want to start talking about popularity, remember
that at least two millennia of famous mathematicians would think
they are a bunch of equation finessers. Living mathematicians may
believe that dead mathematicians don’t count, but they
themselves will soon be dead, and can therefore, by their own
arguments, be dismissed as easily as anyone else. Besides, no
doubt the dead believe that the living are but a mannequins of
the passing fad, and thereby inconsequential.
At any rate, it was in the 1820’s
that the tide began to turn. This is interesting, since that is
the same time that mathematicians got tired of arguing about the
calculus. Cauchy finalized the modern interpretation of the
calculus at about that time, and very little about the calculus
has changed since then. It might be argued that this is because
Cauchy did such a great job of explaining things, but no one who
studies the question with any rigor will buy that explanation.
The truth is that mathematics turned a corner at that time for
very human reasons: mathematicians were bored. They desperately
wanted something new to do, and curved geometry was that new
thing. Mathematicians had been toying with it for centuries. Most
recently Saccheri had kicked it around the room in a very
suggestive manner, implying that years of fun could be got from
the beast, legitimate or no. Saccheri finally decided that curved
geometry was a flabby balloon, since you could basically prove
anything and its opposite with it, but this did not stop the
mathematicians in the early part of the 19^{th} century.
On the contrary, the fact that curved geometry was so malleable
was just another selling point for them. The least scrupulous
were the first to get involved with the new whore, but soon
everyone was tossing her. Now, 180 years later, she has been with
everyone, and she has the protection of all. They will become
indignant if you call her a whore, or suggest testing her for
disease. Like the teenage son of a sailor, you will be taken to
her for your initiation, and whacked by the older men if you look
too closely at her teeth or her fingernails.
My enemies—which must now be
everyone in every math department—will say that there is no
intuitive argument against any math, and I agree. They will say
that the proofs by demonstration against curved geometry were all
flawed, and I agree. No one in history really got to the heart of
the matter one way or the other, either with straight geometry or
curved geometry. They will say that math is judged by internal
consistency and that curved math has been proved to be internally
consistent. I also agree with this, as far as it goes. It simply
doesn’t go far enough.
We are told that Felix Klein proved
that curved math was consistent only if straight math was, and
that this means that both are on the same footing. In other
words, curved math is just as good as straight. But there are at
least two things to point out here. One is that Arthur Cayley,
one of the last toplevel mathematicians with any integrity (and
a contemporary of Klein), argued convincingly that Klein’s
proof was circular. The other is that, even as stated
here—compressed and interpreted for the modern reader—the
argument does not support the popular conclusion. I actually
agree that curved math is consistent only if straight math is.
That is what I have been saying above, in fact. But logically
that is because curved math is dependent upon straight
math. Curved math relies upon straight math for its very
existence. By this way of looking at it, there is no equality
between the two. One is clearly primary and fundamental; the
other is clearly secondary and derived. Which means that curved
math is not just as good as straight math. If it is performed
scrupulously, it may be just as valid, but that is very
far from saying it is just as good. It is less clear, more
unwieldy, less efficient, and far easier to fake.
Internal consistency is not the
sole requirement of a geometry or algebra, either. Although math
is judged on internal consistency, it is not judged only
on internal consistency. It is judged both on internal
consistency and on the truth of its postulates or axioms. As
Gödel—one of the heroes of modern math—has shown, all math
rests upon assumptions; and if the assumptions are not true, the
math is not true, no matter how consistent it is. What I will
show is that curved math almost always rests upon false
postulates. It is also almost always used inconsistently. The
proofs that show that curved geometry is consistent only show
that, used perfectly, it can be
consistent. But it is never used perfectly. Historically it has
always been used in a very sloppy manner, and the most famous
uses of curved geometry have been both inconsistent and based on
false assumptions. I have already shown this in specific cases,
including Minkowski's
proof of Special Relativity and Einstein's
proof of General Relativity; but in this paper I will show
the more general and fundamental way that curved geometry is used
to fake a proof.
I have claimed that the lady is a
whore, so I should have to prove it. I will not prove it in the
common mathematical way, with 200 pages of fudgy equations. I
will prove it by turning the lights up and removing the veils.
First of all, curved math is based on curves. Hyperbolic math is
based on a curve called the hyperbola and elliptic math is based
on a curve called the ellipse. Mathematically or physically, a
curve has content of a certain sort, and that content is wholly
in its curvature, of course. The information that the curve
carries is defined by its curvature. A curve, by its nature, can
tell us how much it is curving and nothing else. Given a curve,
that is the only question we can ask and it is the only question
it can answer. “How much is it curving?” We can assign the
curvature to various parameters, and ask the question over
various longer or shorter intervals. For instance, the curve can
stand for momentum, and we can ask how much it curves over 1
second or 1 meter or 1 angstrom. But beyond its curvature, the
curve can tell us nothing.
To measure the curve, we have to
apply a measuring stick of some sort to it. Originally, the
hyperbola and the ellipse were both measured with straight lines.
The common shape of the hyperbola and ellipse are both relative
to straight lines. If you measure these curves with straight
lines, they look like they do in textbooks. The hyperbola is
“hyperbolic” relative to a straight line. The ellipse is
“elliptical” relative to a straight line. If you measure the
hyperbola with curves, it isn’t really a hyperbola anymore,
since it isn’t even hyperbolic. Depending on the curve you
measure it with, it can be almost any shape. If you measure a
hyperbola with the same hyperbola, it is a straight line, for
instance.
This is because all curvature is
relative. A curve in a curved field is not necessarily a curve.
The word “curve” only has meaning relative to a straight
line. The only way to know how much a curve is curving is to put
it next to a straight line. This is why all curved geometry is
absolutely dependent upon straight geometry. Without a straight
line, all curvature is freefloating and undefined.
Think of it this way. Say you are
given a ruler that is a curve. You are given a bent yardstick.
But you aren’t told what the curvature is, and you aren’t
allowed to try to discover it. Instead, you are told to just
measure everything relative to that bent yardstick. Can you know
how much other things curve? No. If you don’t know the
curvature of your measuring stick, the curvature of everything
else is equally mysterious. The knowledge attained by measuring
cannot exceed the knowledge of your measuring stick. The only way
to know the curvature of the things you are measuring is to
measure with a straight stick. If you don’t measure with a
straight stick, then “curvature” has no meaning. All
curvature curves relative to a straight line, and all
nonstraight geometry is knowable or known only relative to
straight geometry.
This is the reason that all
background independent curved math is a cheat. When you are given
a background independent curved math, like the math of General
Relativity, you are being given a curve that is not dependent
upon any straight line. The curved math is background independent
because it does not have a rectilinear or Euclidean field
underneath it, defining it. It is freefloating, which means that
the curvature is trying to define itself. But this is logically
impossible. A curve cannot define itself with its own curve
equations. A curve can only be defined by a straight line. If you
have no background, or if you have "background
independence", which is the same thing, what you really have
is a license to cheat. You have a curve that is not only
metaphysically ungrounded, you have a curve that is mechanically
ungrounded. You have math causing motions in the field, rather
than mechanics causing motion in the field. This is the first and
greatest mistake of General Relativity.
But it goes far beyond that. Let us
say you are given a rubber ruler. You measure it next to a
straight ruler and discover it is 10 centimeters long. Since your
ruler is rubber, you can measure curved things with it, and you
feel very superior. No matter how much it curves, it is still 10
cm long, so you can’t really go wrong. You can even measure
around corners. Modern mathematicians have tried to convince us
that this is basically what is going on with curved geometry. We
have all been issued rubber rulers and life is good. But that is
not what is happening with curved geometry.
To see why, we have to go to the
triangle. In curved geometry, a triangle may have less than 180
degrees. You may ask, How much less? The answer is: It varies,
depending on how hyper your hyperbola is. But it basically means
you can choose from an almost infinite number of curves from one
corner to the other, as long as they curve in rather than out. If
you like, you can define your field so that your triangle has
less than one degree.
The thing to notice is this:
curving one of those sides of a triangle is not like using a
rubber ruler. If you are using straight geometry, you must draw a
straight line from corner to corner, defined by the least
distance from corner to corner. But even more important than that
least distance rule, is the rule that there is only one line. It
cannot vary. There is no choice in picking a line from corner
to corner. No fudging is allowed. We must choose the shortest
distance, we must call it a straight line, and if our triangle is
a unit triangle, that distance must be one.
In this way, the number one is
determined by the straight line.
The distance “1” is defined as
the distance from corner to corner, and that distance is straight
and may not vary. But in a hyperbolic triangle none of this is
true. An infinite number of curves may be drawn from corner to
corner, and precisely none of them can be measured with your
rubber ruler. Let us say our triangle is 1 meter to a side and
that our rubber ruler is also 1 meter long. Then we make that
triangle hyperbolic. Our ruler will be too short to measure any
of the possible sides of that triangle. To measure the hyperbolic
triangle would require that our ruler not only bend, but also
stretch. Unless we move the corners closer, all curves
from corner to corner will be longer than 1 meter. The length of
the unit one will be infinitely variable. In fact, it will always
be greater than one, but it will never be equal to one. This
means that the VALUE of numbers is determined by straight
geometry. Integers are straightline values, and if the field
is no longer measured with straight lines, the numbers lose their
absolute value. In curved geometry, the number 1 is no longer 1
unit in size.
This is one of the places that
mathematicians cheat with curved geometry. With hyperbolic
geometry, the number 1 itself is or can be stretchy. It is not
just that the length is bendable; the length is actually
variable. It can be pushed and pulled, but because it is
hidden within the number, no one notices. A lovely bit of magic.
This is true in socalled pure
math, but applying math to physics doubles my argument. In
physics, numbers apply to parameters. At the most basic level,
they apply to differentials, and differentials are lengths. Even
the second is operationally a length. For instance, look at
Minkowski’s fourvector field. All his basic variables or
functions in that field are lengths. As I have shown, x, y, and z
are differentials, and differentials are lengths. The time
variable is also an interval, which is operationally a length; so
when it is transformed by i to make the field symmetrical,
it must take its character with it. Time is operationally a
length both before and after Minkowski makes it imaginary. My
point with all this is that lengths, like numbers, should not be
stretchy. Once we are given a certain object to measure, the
length is no longer a variable, it is an unknown. A variable only
varies in a general equation, but once we apply that equation to
a certain object or event, the variable no longer varies. It
stands for an unknown number, and unknown numbers are just as
stable and invariable as known numbers. But in many curved
manipulations, you will find numbers, both known and unknown,
varying. This is a sure sign that you are in the presence of
hocuspocus.
[For
more on how curved math is used to cheat in physics, you may go
to my newer paper on
General Relativity.]
Now let us look at complex numbers.
Curved geometry is often used in conjunction with complex
numbers. Well, complex numbers can also be stretchy. A complex
number is in the form x + yi, where i is the
imaginary number √1. Now, like the number 1, this number
should be firm. It should not vary. The √1 should always be
the √1, and it should not change size or shift value
willynilly. But in modern manipulations, i is not always
used as a firm value. No, it is sometimes used more like an
infinitesimal. It can change size depending on the needs of the
mathematician. In other words, it is a fudge factor, hidden by a
letter that confuses almost everyone. Many people seem to think
that i is a variable, since it is dressed as a variable
and sits next to variables. But it is not a variable. It should
not vary. Treating i as a variable is like treating the
number 5 as a variable. I hope it is clear that the number 5
should NOT be a variable in any possible math, since in any
problem the number five should have a firm size.
Complex numbers have an even more
important role than supplying this room to move. Complex numbers
were invented to hide something. What are they hiding? Let us
see.
Wikipedia, the ultimate and nearly
perfect mouthpiece of institutional propaganda, defines the
absolute value of the complex number in this way:
Algebraically,
if z = x + yi
Then z = √x^{2}
+ y^{2}
Surely
someone besides me has noticed a problem there. If i is a
constant, there is no way to make that true. That equality can
work if and only if i is a variable. But i is not a
variable.
Let x = 1 and
y = 2
i =
.618
Let x = 2 and
y = 3
i = .535
Let x = 3 and
y = 4
i = .5
But i is
a number. A number cannot vary in a set of equations. Letting i
vary like this is like letting 5 vary. If someone told you
that in a given problem, the number 5 was sometimes worth 5.618,
sometimes 5.535 and sometimes 5.5, you would look at them very
strangely. I don’t think you would trust them as a
mathematician.
I will be
told that you cannot solve for i in these equations, as I
did. But Wikipedia says outright that the equations are
algebraic. That is what the word “algebraically” means, does
it not? If these equations are algebraic, then I should be able
to solve for i. If I can’t solve for i then these
equations are not algebraic. But, of course, we should have known
something was fishy even without the variance of i. The
fact that i equals anything is a major axiomatic problem,
since it can’t equal anything but √1, and √1 is nothing.
The √1 is like a unicorn or a fairy. We should put a picture
of a griffon in the equation instead of a cursive character. Or
how about a clover as our lucky charm here? My “special
characters” list has a clover which we can insert:
z = x + y♣
Which brings
us to the question, “How can you multiply a variable by a lucky
charm?” A modern mathematician will say it is alright as long
as you define your charm, but that begs the question, “How can
you define something that does not exist?” Defining something
that does not exist as “something that is imaginary” and then
claiming that is a tight definition is a bit strange, is it not?
Also, I seem to remember that √1 used to be undefined,
in a strict mathematical sense. It was a discontinuity or
singularity on a line or curve, and mathematically undefined. How
can the same value be undefined in one mathematical situation and
defined in another?
No, the
reason I am not supposed to solve for i here is that if I
do, I discover that all this “math” is bollocks. What they
should say instead of “you can’t solve for i” is
“you aren’t allowed to solve for i; please
don’t solve for i; we forbid you to solve for i;
look at my watch swinging, you are getting sleepy, you don’t
want to solve for i; oh dear, all our work!”
Wiki also
tells us that complex numbers were discovered by Cardano. Cardano
was a famous gambler and thief who was arrested for publishing
the horoscope of Jesus in 1554. He cropped the ears of his son,
and his son was later executed for poisoning his wife. I would
say that there is no irony in the fact that modern mathematicians
are intellectually and morally descended from such people: it is
purely in keeping with the odds.
The
real reason you cannot solve for i
here is that z = x + yi
is not algebraic. It is not
analogous in form to z =
√x^{2}
+ y^{2},
so the whole “if/then” claim above is false and misleading.
The second equation is algebraic, but the first equation is a
vector addition. I will be told that vector addition is part of
vector algebra, so it must be “algebraic.” But I don’t like
that use of the word algebra. In algebra, the mathematical signs
like “+” should be directly applicable, without any
expansion. In algebra, you should be able to solve for unknowns.
As I have just shown, you can’t do that here. That plus sign
implies a sum, of
a certain sort, but does not stand for straight addition.
Contemporary math is sloppy not only in its manipulations, but in
its terms. And this sloppiness is not an oversight. Math now
conflates any number of things in any number of situations, and
it does it to purposely confuse you. With complex numbers, you
could just be told you are doing vector math, but instead you are
taught that you are dealing with imaginary numbers. You are being
misdirected for a reason, as I will now prove.
Let’s look
at the “definition” of the complex number, to see how tight
it really is. In Churchill’s textbook from 1960, the complex
number z is defined as the ordered pair of real numbers (x,y).
The real number x is then defined as the real component of z, and
it is expanded this way
x = (x,0)
We haven’t
even gotten to the imaginary part of z and we are already in
lala land. If x is a real number, what does the 0 stand for in
this ordered pair? What does the ordered pair stand for? How can
you write a real number as an ordered pair? The variable x was
already part of an ordered pair, so it can’t be an
ordered pair itself. The ordered pair (x,y) was a point on some
graph, with x representing some number that itself is
representing an interval from zero. You can represent a
point on a twodimensional graph or metric space as an ordered
pair, but you can’t represent a single interval
or length as an ordered pair. And the reason is clear: the
second half of the ordered pair doesn’t represent
anything.
Say you have
a Cartesian graph and the ordered pair (x,y): in that case x is
the horizontal interval from the origin. If this Cartesian graph
is representing a physical situation, x would be a distance
from zero. Therefore, x is just a naked cardinal number. How can
you write a naked cardinal number as an ordered pair? An ordered
pair of whats? If we solve and find a value for x, then that must
stand as the xdistance from the origin. If we write that as
(x,0), what does the 0 mean? It means nothing. It is meaningless.
Not only undefined, but meaningless. We could just as well write
x = (x,0,0,0,0,0...) or (x,♣♣♣♣♣...). For instance, if
x = 1, then this definition says we can write that as (1,0), but
the zero has no physical or mathematical meaning. It has no
potential assignment. Where are you going to put it on the
Cartesian graph or any meaningful metric space? The ordered pair
(x,y) exists in some metric space. Where does the ordered pair
(x,0) exist? In a subspace? Shouldn’t that space have to be
defined, or at least recognized? As it is, the definition of
complex number simply conflates the two x’s in x = (x,0),
treating them as if they exist in the same way in the same metric
space. But they can’t do that.
Look at it in
another way: give a value for x and then look at the equation
again. Say x = 1.
1 = (1,0)
Does that
mean anything? No. That equation is meaningless. The number 1
cannot be equal to an ordered pair. The number 1 is 1 and that is
all there is to it. In the definition of complex number, these
equations are being finessed.
The truth is,
this expansion is just preparing you for the next step of magic.
It is massaging your brain to accept this expansion into ordered
pairs. Churchill doesn’t care if this makes sense, he only
wants you to accept the next step, which is that y = (0,y). Once
you accept that, you can forget that x = (x,0). In fact, they
would prefer you did forget it, since they want to keep
your attention on y = (0,y). Churchill tells you that “it is
convenient to assign the ordered pair (0,1) to i,” but
obviously that is the second order of ordered pairs, not the
first order. Churchill has created two orders of ordered pairs,
you see: the first order being z = (x,y), where x and y are real
numbers, and the second order being an expansion of that, where
nothing is real. Churchill admits that the ycomponent of
(x,y) is imaginary, but he doesn’t tell you that the
xcomponent is just as imaginary. Once x is expanded into (x,0),
neither x nor 0 are real. This expansion is imaginary, therefore
both sides of the imaginary ordered pair are imaginary. You can’t
really write a single number as an ordered pair, since you can’t
assign those numbers to real intervals in space or on the number
line. Therefore the ordered pair is imaginary. (x,0) is just as
imaginary as (0,y).
Now, the
reason this sort of math works in electrical
engineering and in quantum electrodynamics
is that this expansion into ordered pairs gives you four
dimensions from only two initial variables. You get a sort of
matrix with four degrees of freedom. As a heuristic device, it is
clear why this would be useful. But it is also useful as a device
of misdirection and obfuscation, since two of these dimensions
or degrees of freedom can exist in the dark, undefined and
unnoticed. This serves to hide the mechanics, and the very
existence of the mechanics. Because these dimensions are
imaginary, no one asks mechanical questions about them. The math
hides real interactions under confusing variables or functions,
and no one ever looks at them. This is just one more reason that
the foundational electrical field or the charge field has been
defined as "virtual." It is smashingly easy to turn the
switch from "imaginary" to "virtual", since
they are basically the same thing. Virtual particles are
imaginary particles, ones that do the mechanical fudging where
the imaginary variables did the mathematical fudging.
But
why would physicists want to hide the mechanics under confusing
math? Because they know how to do the math, but they don't know
much about the mechanics. They have fit the math to the data
after the fact, but can't explain the data mechanically. To be
specific, they have followed Lagrange
in this creation of extra degrees of freedom out of nothing,
and it is because, like him, they recognized they needed them.
But since neither Lagrange nor anyone since could assign those
degrees of freedom to anything physical, they have hidden the
physics. They have buried the physics under the math, so that
students of both math and physics would not ask them the hard
questions.
To be even more specific, no one has seen that
both the data and the math require a second field in physics
beyond the gravitational field. Or, if they have seen it, they
have not been able to say what that field is. So it was best to
hide the fact. But the field is just the charge field, as I have
shown in dozens of papers. The old equations already contained
it, including Newton's
gravitational equations, the Lagrangian,
Laplace's field
equations, and so on. What is more, the electromagnetic
equations have also been unified all along, containing gravity.
Coulomb's equation,
like Newton's, is already unified. That is why both celestial
mechanics and electromagnetics have required these extra degrees
of freedom: the math of both has to represent the unified field,
whether the mathematicians and physicists can assign the
variables or not.
But let's
continue with the "definition" of z.
z = (x,y) =
(x,0) + (0,y)
(0,y) =
(y,0)(0,1)
z = (x,0) +
(y,0)(0,1)
(x,0) = x
(0,1) = i
z = x + yi
Good god! Who
could be satisfied by such nonsense? First of all, notice that
(x,0) has been simplified back down to x. The derivation expands
then deexpands, and it does so just to finesse your brain, as I
said: x is taken from one dimension to two dimensions and then
back to one. Why? Simply so they could do the same with y. The
variable y is expanded and deexpanded, but it is deexpanded in
a different way. By the time it is deexpanded, it comes back
married to i.
Also notice
that while your brain was in shock, our “mathematician” here,
Churchill, suddenly expanded his initial ordered pair into a sum:
(x,y) becomes x + y. Since when is it legal to add terms in an
ordered pair, without any explanation? On a Cartesian graph,
(x,y) is not equal to x + y. Algebraically, (x,y) is not
equal to x + y. Normally, an ordered pair is not a sum. It
is a point in some space. In fact, this complex number derivation
is done this way in order to create a metric space, and both Wiki
and Churchill assign z to a point in that space. But an ordered
pair as the representation of a point in a metric space is not a
sum. Therefore this derivation is not valid. It is magic. As you
will see, only Δ(x,y) is equal to x + y, and only if x + y is
understood as a vector addition.
Here is
another problem. Churchill says that x and y are real numbers;
then he says "a pair of type (0,y) is a pure imaginary
number." How can y be real and (0,y) be imaginary? Why is
(x,0) real and (0,y) imaginary?
And another
problem: where did he get (0,y) = (y,0)(0,1)? That is just
equation finessing. He claims to have gotten it from here
z_{1}z_{2}
= (x_{1},y_{1})(x_{2},y_{2}) =
(x_{1}x_{2}  y_{1}y_{2}, x_{1}y_{2}
+ x_{2}y_{1})
But according
to that equation, y can never be in the first position. Look
again at the middle part of that triple equation: (x_{1},y_{1})(x_{2},y_{2}).
Do you see a “y” in the first position there? No. We need
some explanation of (y,0)(0,1), but historically we don’t get
it. Then look at the last part of that triple equation:
(x_{1}x_{2}
 y_{1}y_{2}, x_{1}y_{2} + x_{2}y_{1})
We need to
ultimately find (0,y) there, but the only way you can get 0 in
the first position is if x_{1}x_{2} = y_{1}y_{2}.
And the only way to get “y” in the second position is if x_{1}y_{2}
+ x_{2}y_{1} = y.
If the second
point is (0,1), as given here, then x_{2} is zero, which
means that
x_{1}x_{2}
= y_{1}y_{2} = 0
Since y_{2}
is given as 1, then y_{1} must be 0.
So the
correct equation must be
(0,y) =
(x,0)(0,1)
And, since
x_{1}y_{2} + x_{2}y_{1} = y,
then x = y
You will say
that Churchill and Wiki are vindicated, since if x = y, then
(0,y) =
(x,0)(0,1) = (y,0)(0,1)
And the
derivation is saved. But if you think the derivation is saved,
you are delusional. Yes, the derivation is saved as long as you
accept that the entire complex number plane is dependent on x =
y. I have just proved that complex numbers are not based on i,
they are based on x = y. The √1 is just a decoy, a pretty
diversion to keep us from noticing that x = y. And x doesn’t
just happen to equal y; it must equal y.
Beyond this,
z = (x,y) = x
+ yi
Only if y_{1}
= 0,
x_{2}
= 0,
y_{2}
= 1
And x_{1}
= y = x = 1
The entire
complex number system is defined on that invisible equality,
hidden by a terrible mess of number juggling. There is nothing
imaginary about these solutions, and they have absolutely nothing
to do with √1. Complex numbers are a smokescreen and nothing
else.
You will say,
why do they work and what are they hiding? They work because the
addition and multiplication operations work. But if you look at
those operations, they are no different than real number
operations. There is nothing complex or esoteric or imaginary
about them. You can use those operations without ever learning
about i or complex numbers at all. The only difference
between those operations and the operations using real numbers is
that in the multiplication of points on a real metric space, you
don’t get negative numbers. If you multiply real ordered
pairs—which in physics stand for lengths—you end up with
positive numbers and lengths. But in these socalled complex
number operations, you get negative numbers and different angles.
Your lengths or numbers turn out to be the same absolute value,
but may end up in negative parts of the graph. This works great
for electrical solutions, but it actually has nothing to do with
imaginary numbers or i. Mechanically it has to do with
vectors. In other words, the important, crucial, and defining
part of the complex number definition and derivation is the minus
sign in the multiplication operation, not the use of i.
To say it
again, that minus sign in the equation
z_{1}z_{2}
= (x_{1},y_{1})(x_{2},y_{2}) =
(x_{1}x_{2}  y_{1}y_{2}, x_{1}y_{2}
+ x_{2}y_{1})
is what
determines the entire process. That minus sign means that the
point (0,1) times the point (0,1) puts you at the point (1,0).
This is a vector outcome and nothing else, but since this math
has let (0,1) equal i, it means that i^{2}
= 1. To be rigorous, we should write
i^{2}
= (1,0)
But of course
this means that i is not imaginary at all. It exists at
the point (0,1) by definition. It is not really the square root
of negative one, it is the square root of the point (1,0). There
is a big difference, for two reasons. One, that point is found by
the equation or operation
z_{1}z_{2}
= (x_{1},y_{1})(x_{2},y_{2}) =
(x_{1}x_{2}  y_{1}y_{2}, x_{1}y_{2}
+ x_{2}y_{1})
Which is a
vector operation, not an algebraic or normal exponential
operation. The length of the vector is not negative one,
it is positive one. The negative is not telling us we have a
negative length from zero, since a vector cannot have a negative
length. The negative is telling us a direction, not a size.
Square roots are normally used on numbers, not on ordered pairs.
An ordered pair is not a naked number, as I pointed out above. An
ordered pair is a point in a metric space, and a point in a
metric space is two lengths or intervals from zero or the origin.
The other
reason is that the √1 is not (0,1). You can’t define
something as imaginary and then assign a real ordered pair to it.
This is doubly true when your derivation turns out to require
that x = y. If x = y, then if y is imaginary, x must be also. You
cannot put y in the x position in an ordered pair and then claim
that x remains real. If x = y, then either both are real or both
are imaginary. I have shown that both are real. There is no such
thing as an imaginary number or a complex number. A complex
number “plane” is only a blanket covering a simple system of
vector math.
But why the
cover? Why go to all this trouble when you could just write the
equations in a straightforward way? As I said, it was to hide the
charge field. If you start doing electrical and QED computations
with transparent vector math, you beg all the important
questions. A transparent vector math makes physicists and
engineers want to ask about the mechanics of the field. We can’t
allow them to do that, because if they do they will try to assign
mass or energy to the field. If they do that, then the charge
field is no longer imaginary or virtual. If that happens, then we
have to explain why the proton does not lose energy by radiating
this field. If we do that, we have to rebuild QED and QCD from
the ground up. So it is best to keep all the math of the
electrical field under heavy blankets.
As I have
shown with celestial
mechanics and the mechanics
of tides, the mathematical fudges aren’t even well
concealed. Wikipedia is bold enough to put this garbage right out
in the open. Mathematics and physics departments have the
effrontery to teach it and defend it. Do students really not see
this stuff, or do they just ignore it? I can’t say for sure,
but I suspect it is due more to a lack of integrity than a lack
of competence or intelligence. As in any number of other fields,
at some point in the 20^{th}
century the phonies reached a quorum. At that point, anyone with
any integrity was driven out as a nuisance, as everyone’s bad
conscience. The only ones left are those who can’t see or won’t
see the corruption: those who benefit from it in one way or
another. And it is these people who rewrite history. It is they
who decide what historical figures are honored and which ones
slandered. That is why we have to hear endless paeans to Hilbert
and Minkowski and Klein and Gödel and so on and on, even those
these people are not worthy of any real respect. They are little
more than a gang of insiders and equation finessers, the
ancestors of the current batch of mediocrities.
I will be
accused of a lack of humility, but one is not required to be
humble or gracious in the presence of such people. One is
required to be humble in the presence of those one respects and
admires. I have not found any reason to admire these “masters”,
as they have been presented to me, and therefore any talk of
humility is a nonstarter. On the contrary, I have only
discovered mountains of reasons to disrespect these false gods,
and to resist them with every fiber of my being. Every fact I
learn about the Feynmans and Hilberts and Gausses of history
confirms once again my initial estimates that they and all like
them were shallow revolutionaries, selling a false product, and
worshipped by the ignorant and corrupt.
The latest
confirmation of this is the glorification and knighthood of
Andrew Wiles, and the witch hunt against Marilyn vos Savant*.
Wiles is the one who claimed to solve Fermat's Last Theorem in
the 1990's. Vos Savant has or had the world's highest IQ, and she
criticized the form of Wiles' solution (a curved solution). Two
things stand out in any overview, and that is that almost no one
else questioned Wiles and that almost no one supported vos
Savant. In a time of openness, you would expect some disagreement
on large and important issues like this. Never before in history
has such an important claim been met so monolithically. If you do
a web search now, almost fifteen years later, you find only one
poor soul against Wiles, a Philippine named Edgar Escultura. It
turns out that Escultura never actually presented a proof against
Wiles or any other evidence, he only suggested in a paper on real
numbers that Wiles might be mistaken. This has been taken as
reason enough to dismiss Escultura as a loon, a crank, and a
perfect fool.
Vos Savant
has fared even worse. She admitted, in print, some very tame
doubts soon after Wiles’ proof was universally accepted. She
said that because no curved solutions to squaring the circle had
been admitted, she found it strange that a curved solution to
Fermat had been accepted. Beyond that, Fermat could not have
intended such a solution, since he predated the possibility of
such a solution. If his solution implied in the margin was indeed
lost, it couldn’t have been a curved solution. Both these
comments were true and wellfounded, and yet vos Savant was
treated to outrage not seen since Savonarola. The substantive
arguments against her were almost nil. The only one I have heard
of is that "geometry is a tool in Fermat but a
setting in Squaring the Circle." Semantic gibberish,
in other words. No, the dominant “argument” was that vos
Savant was not a “professional” mathematician, and therefore
had no standing in deciding the status of Wiles’ proof.
Political gibberish, in other words. Vos Savant was badgered so
unmercifully, she finally backed down, adding a paragraph to her
book to the effect that all tools were valid in mathematics.
But she was
right the first time. She just didn’t have the fortitude to
stand up against the popular majority, the popular unanimity. In
fact, she never stated the case strongly enough the first time.
Wiles’ proof should have been laughed out of the park, not
mainly because it was curved or because it was nonhistorical,
but because it was 200 pages. A 200page answer to such a simple
question MUST be a disaster, even if it is correct. As with
Goldbach, what we want is an elegant, transparent explanation.
Not an extended torture of the medium. I don’t see any reason
that Fermat cannot be proved with straight geometry and simple
math. In fact, I predict it will be, perhaps in my lifetime.
The matter of
Wiles and vos Savant should be a big red flag to anyone with his
or her eyes open. It is the clearest evidence possible that the
field of math is completely controlled, completely closed, and
completely corrupt. Disregarding the status of Wiles’ proof,
which is beyond the scope of this paper, the way the proof was
received is cause enough for alarm. When intelligent comments
from a proven genius are shouted down with such heat and volume
and, frankly, hysteria, one must ask what is really going on.
Surely vos Savant touched a nerve, and the mathematical community
doth protest too much. Someone who was wrong would not require
such absurd levels of condemnation. Only someone who was right
could deserve such a united front, such a universal outcry. Only
a recognized threat could call forth such a yapping and a
frothing.
The
overenthusiasm in support of Wiles is also odd. What living
field in the history of science or thought has ever been
unanimous? Only a dead field is unanimous. The mathematical
community was hungry for a hero. It needed one, no matter the
reason. In the deafening PR barrage that is modern life and
business, mathematics needed to turn up the volume. The Nobel
Prizes had become a media blip, and more was required. Without a
magazine cover, a parade, and a knighthood occasionally, no field
can remain viable. This applies to math just as much as art or
physics or ballroom dancing. If there had been no Michael
Flatley, it would have been necessary to invent him. In the same
way, it was necessary to invent Andrew Wiles, from a pile of
empty papers if need be.
Perhaps the
strangest turn in this whole comedy or tragedy is yet to be told,
or at least yet to be commented on. Escultura printed a letter
from Wiles that was so strange everyone at first thought it must
be fake. It now appears that this letter was genuine. I copy it
here for your edification and amusement:
Dear Sir,
Your work is
incredible, I read all of it just yesterday and let me tell you I
respect you. I am going to review all my ‘proof’ which I am
sure is wrong (thanks to you!).
Would you like
to collaborate with me in this work? I have noticed some
imperfections in your perfect proof (that sounds like you), and
I’d like to create a perfect proof with you, great professor.
Also I’d like
to have the address of the guy who let you get a PhD 30 years
ago. I’d like to discuss few things with him. . .
Very
respectfully,
A. Wiles
The status
quo has explained this letter, in passing, as “dripping with
sarcasm” and “better than Escultura deserved.” But that is
to miss all the delicious psychological undercurrent available
here, for those who are not only onedimensional calculators. It
does not take a Nietzsche or a Freud to recognize that we are in
the presence of honey.
First of all,
Wiles had no need to reply in any way. He was already the
beneficiary of a positive prejudice greater than any in the
history of mathematics. He was, figuratively speaking, already
under a pile of warm bodies, being given a cultural blowjob by
half the planet. Why would he need to crawl out of bed, remove
his thumb from his mouth, and answer this one poor sinner on the
other side of the planet? Surely he could have ignored this one
tiny “yop”, barely distinguishable and unlikely to cause even
a ripple? But no, here we see him acting very childish indeed,
and not even childish in a consistent or edited way. Even for a
personal note or an email, the language here is not what one
would expect from an intelligent person. The content veers
unpredictably from simple and mostly unsuccessful sarcasm to
direct insult. Wiles either has the verbal IQ of a teenager, or
he is under the influence of an intoxicant (or both).
Beyond Wiles’
temporary state of mind here, is the question as to the permanent
state of inferiority and unease that would trigger such a
reaction. I would compare it to Feynman’s unnecessary attacks
on the mostly quiet and powerless philosophy departments. One in
a position of power and authority who is so easily threatened is
always threatened first and most by his own knowledge of the
truth. Had Feynman believed his own PR, if he had been proud and
sure of his own achievements, there would have been no reason to
snip at obsolescent naysayers. Likewise, if Wiles had been
convinced by his own proof, he could have ignored Escultura with
no effort at all.
Likewise, if
the math departments were really convinced by their own
propaganda and press releases, they would have no reason to fear
an “interloper” like vos Savant. If she were truly
inconsequential, she could be ignored as such, with no
consequences. The fact that she was not ignored is only proof
that she was of some consequence, and was seen to be so.
I have
already proved in many
other papers that straight math is faster and more
transparent than curved math. I have proved this in the most
direct way, by solving
real historical problems much more quickly and transparently
with straight math. I have solved many of the biggest and most
famous “curved math” problems of recent history without
curved math, and done it in a fraction of the time. My proofs are
therefore not abstract, they are real. I have not claimed
that
something could be done; I have actually done
it, in black and white.
And now you
have seen precisely why I have chosen to use simple math in all
my papers. Some will have thought it is because I wanted to
appeal to a wider audience, and, as I have said, that is
certainly a welcome sideeffect. Some will have thought it is
because I can’t do “higher math,” or because, like
Einstein, I “don’t trust it.” But it is not that I don’t
trust curved math, it is that I don’t trust curved
mathematicians. I rarely use curved math because I rarely find it
useful. I don’t see any reason to fill a blackboard solving a
problem when I can do it on a postit note. I don’t like to
waste time, and I really admire elegance. I don’t just salute
elegance and then march off behind inelegance; I actually publish
elegant solutions. I want my solutions to be as short and
transparent as possible, with all the variables labeled clearly
and all the manipulations explained mechanically. Using this
method is what has allowed me to clear up many messes. And using
this method is what has allowed me to so easily penetrate the
veil of modern maths and spaces and theoretical claims.
As in any
other field, it is much preferable to have perfect control of a
small bag of simple tools than to have imperfect control of a
large bag of complex tools. Any tool that is mishandled is a
danger. Which is just to say that real mathematics will always be
more powerful than any heuristics. Feynman was wrong: an answer
is not preferable to an understanding. We don’t just require a
solution, we require an explanation. Anyone can force a number
out of a gigantic computing device, but it takes a real
mathematician to provide a clear and concise derivation.
*Strange
that Vos Savant was said to have an IQ of 230 before she
disagreed with Wiles, but now, according to all the internet
"information" sites, she
has an IQ of just 186. Apparently it is possible to push an
IQ more than 40 points for political reasons. Not surprising, but
it must be noted. Wikipedia leads this campaign of disinformation
and slander. Although it admits she did score 230, it does
everything it can to cast doubt on this score, short of saying
she faked it. **Modern Mechanics and Inventions, July, 1934.
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