return to homepage Why Non-Euclidean Geometry is a Cheat In my title here I choose the word “cheat” on purpose, since my intention is not to show that non-Euclidean geometry is necessarily false. My intention is to show that non-Euclidean geometry is necessarily less efficient, less transparent, and less exact. Even Poincaré—the grandfather of non-Euclidean math—admitted that, in part. He said,
Why Non-Euclidean Geometry is a Cheat
In my title here I choose the word “cheat” on purpose, since my intention is not to show that non-Euclidean geometry is necessarily false. My intention is to show that non-Euclidean geometry is necessarily less efficient, less transparent, and less exact. Even Poincaré—the grandfather of non-Euclidean math—admitted that, in part. He said,
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 “non-Euclidean” 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 con-job. 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 19th 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 top-level 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 free-floating 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 non-straight 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 free-floating, 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 straight-line 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 so-called 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 four-vector 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 hocus-pocus.
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 willy-nilly. 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| = √x2 + y2
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 thatz = x + yi is not algebraic. It is not analogous in form to |z| = √x2 + y2, 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 la-la 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 two-dimensional 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 x-distance 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 y-component of (x,y) is imaginary, but he doesn’t tell you that the x-component 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 electro-dynamics 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 mis-direction 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 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 de-expands, 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 de-expanded, but it is de-expanded in a different way. By the time it is de-expanded, 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
z1z2 = (x1,y1)(x2,y2) = (x1x2 - y1y2, x1y2 + x2y1)
But according to that equation, y can never be in the first position. Look again at the middle part of that triple equation: (x1,y1)(x2,y2). 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:
(x1x2 - y1y2, x1y2 + x2y1)
We need to ultimately find (0,y) there, but the only way you can get 0 in the first position is if x1x2 = y1y2. And the only way to get “y” in the second position is if x1y2 + x2y1 = y.
If the second point is (0,1), as given here, then x2 is zero, which means that
x1x2 = y1y2 = 0
Since y2 is given as 1, then y1 must be 0.
So the correct equation must be
(0,y) = (x,0)(0,1)
And, since x1y2 + x2y1 = 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.
z = (x,y) = x + yi
Only if y1 = 0,
x2 = 0,
y2 = 1
And x1 = 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 so-called 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
z1z2 = (x1,y1)(x2,y2) = (x1x2 - y1y2, x1y2 + x2y1)
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 i2 = -1. To be rigorous, we should write
i2 = (-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
z1z2 = (x1,y1)(x2,y2) = (x1x2 - y1y2, x1y2 + x2y1)
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 20th 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 non-starter. 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 well-founded, 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 non-historical, but because it was 200 pages. A 200-page 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 over-enthusiasm 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:
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. . .
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 one-dimensional 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 side-effect. 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 post-it 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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