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It has been known for millennia that the Earth rests upon the back of a giant turtle. Only in recent centuries has this knowledge been added to. In 1794, in one of the high valleys of the Himalayas, one of the wise was asked, "Master, what does the turtle rest upon?" The Master answered: "It is turtles all the way down, my son." But now that scientists have finally succeeded in mapping the universe, a turtle controversy has arisen. It turns out that level 7,484,912 is occupied not by a turtle, but by a man dressed as a turtle. It is not known how this will affect our other equations.
You probably aren't used to having a book on science and math open with a joke. But a sense of humor is crucial to existing in a world where even our greatest accomplishments contain large elements of the absurd. Some contemporary thinkers are of the opinion that we are very near to a complete understanding of the universe. I am far from agreeing with them. We have made some wonderful discoveries and are due a small dose of pride, I suppose. But the things we don't know so overwhelm the things we do that any talk of a full understanding is just bombast. Worse, it is hubris. It may even be a scientific sacrilege, with real curses attached to it. When we become too secure in our knowledge, we stop questioning. Failure to question is the ultimate scientific failure. Answers quit coming precisely when they aren't sought, and they aren't sought precisely when they are (erroneously) thought to be in hand. We are like the dog who discovers how to use the little flap-door and now considers himself master of the house. He lies in front of the fire and congratulates himself for his cleverness. He would be better outside chasing rabbits.
this book I propose solutions for several of the greatest errors currently existing in physics and mathematics. I do not propose to solve all the greatest errors, of course, or even to know what they are. I only present the ones that have become known to me in my years of research. Many may find my list surprising or even shocking, since I do not seem to choose problems that are commonly acknowledged to exist. Rather I choose problems that are believed to have been solved. This, I realize, can have the appearance of caprice or insolence, but I have simply gone where my nose leads me. I suspect that the whole history of science has moved in much the same way, so I will not apologize for seeing problems where I see them.
Lest I be dismissed as a crank before my first equation hits the page (and this sort of dismissal has become pandemic in the field), I rush to add that I am not a so-called classicist, bent on refuting Relativity and Quantum Mechanics simply because they disturb my sense of balance or my love of Newton.* I attack Newton as well, long and—I like to think—shockingly. Beyond that, I am convinced of time dilation and length contraction and the necessity of transforms. I simply do not believe that Einstein provided the correct transforms. Likewise, I believe in the accuracy and usefulness of many of the equations of QED. But QED is still in large part a heuristic math posing as a theory. Even Feynman admitted this before he died, to the chagrin of most in the field. QED is not “the final solution” until it is fleshed out with a coherent theory. I believe, contra current wisdom, that QED will be provided with a coherent theory, one that makes sense even in the macro-world.
I am not a classicist, nor am I in any of the other dissenting groups that are opposed to the standard interpretation of Einstein. That is to say, I am not proposing supra-luminal theories or any other theories that go beyond the math and theory of Einstein. I am not proposing any new particles, forces, fields, or maths. All the major chapters and findings in this book deal with straightforward mathematical analysis of famous historical papers and theories. For the most part, this analysis is high-school level algebra applied to these papers. In critiquing the calculus, some rather subtle number theory is used, but no higher math at all. This means that this book is unlike anything you have read or heard of before. It is not allied to the status quo, but it is also not allied to any of the dissenting groups. It is completely outside the 20th century argument, since it cannot be said to be ultimately pro-Einstein or contra-Einstein, pro-Newton or contra-Newton. It is pro-Einstein in that his theory (and Lorentz's and Poincare's, etc.) is shown to be correct in many important ways. However, it is contra-Einstein in that my algebraic corrections falsify some fundamental assumptions and equations. How you would classify my correction is therefore more a matter of your own allegiances than mine, since I have none.
This book differs from all the other critiques I have seen of current theory in that my arguments are not mainly philosophical or even theoretical. They are mathematical. I rerun the original equations in the original papers and show where the specific mathematical errors are. In this I believe I may be the first. Especially as regards Relativity, there has been a massive amount of criticism and absolutely no mathematical proof to back it up. A few mathematical variants have been put forward, some with a certain amount of validity; but no one has shown where Einstein’s math is wrong in itself. Herbert Dingle, perhaps the most famous critic of Einstein in the 20th century, said in the 60’s that he was astute enough not to search for mathematical errors in the theory. Whether his astuteness was based upon the recognition of his own mathematical limitations or upon some other factor is less clear. I suppose current wisdom is that because they are assumed to have been combed by everyone from Bohr to Feynman, the equations must now be unassailable. But nothing in this world is unassailable, as Einstein’s refutation of Newton was supposed to have proved. Newton survived two hundred years of geniuses before Einstein appeared. If Einstein had been cowed by genius, I would now have nothing to critique. But Einstein did not see problem-solving as an attack upon genius or upon the status-quo, or as the solution to his career aspirations; he saw it simply as problem solving, let the cards fall where they may.
*I am also not any sort of conspiracy theorist. I do not believe that Einstein plagiarized anyone, not even his own wife. I have no special regard for German philosophy or special disregard for Jewish scientists. I am not here to bury Einstein or to praise him. I am here to mathematically evaluate his equations. I find it a shame that the field has already been so muddied by politics and other petty misunderstandings that an objective critique has become a near-impossibility.
My critique of Relativity was begun to solve a problem—that of the Pioneer Anomaly. I therefore approached the problem as both mathematician and physicist. I saw the final equations of Einstein as applied mathematics. Not esoteric theory, but physical equations. They therefore must be made to make sense not only as abstractions but as predictors of motion. In this they were failing. The physical community had finally been forced to admit this in 1999, when, after almost 30 years of fiddling, they had still been unable to solve the Pioneer Anomaly. So the Jet Propulsion Lab allowed Newsweek to report on the anomaly. Unfortunately, from the point of view of theoretical physics, this only brought the final cranks out of the closet. Physicists were inundated with new theories but none of them were seen to be at all promising. A good percentage were apparently written on the back of paper napkins, if the horror stories we hear are to be believed. So the walls went back up, and this time they were forbiddingly high and reinforced. The physical community wanted to waste no more time with paper napkins.
In some ways this was understandable. In other ways it was tragic. It has become a common feature of modern life in almost all fields—publishing, art, science, airport security, etc. The presumptions and unmannerly behavior and outright sociopathy of some have restricted the communications and movements of all. We all of us have had so many bad experiences that we begin to doubt the possibility of a good one. And there are other factors, ones which the physical community must take responsibility for. Closed doors and closed minds are not found only in town councils and corporate meetings.
For this reason and many others, Relativity is now the strangest sub-field in all of physics. In the universities, it barely exists. As a living field, it does not exist at all. What I mean by that is there is no sub-department of Relativity at most universities. It is not taught as a sub-field that you can enter and hope to make a contribution to, like all other sub-fields in physics. Relativity is taught as dogma—as a finished field. You learn it only to use in other fields. At the university and research level, Relativity is only a defensive field. Most of the work now done in the field is in keeping away pests. Look at Physical Review Letters or ArXiv, and their positions regarding Relativity. No research papers are published. None. None are even considered. In the past two decades, the editors of most journals have fortified all means of approach, in order to fend off invaders. These invaders, rather than give up, have instead multiplied. The internet has allowed for the mutual support of a vast sub-culture of doubters, nay-sayers and theorists. As would be expected of any large group, most are deluded. But the sheer size and persistence of this group has forced the status quo to extreme measures, including blacklisting. The major journals have blacklisted not only pesky outsiders, but also marginal characters from within the field. As part of this blacklisting, the field of physics has quite simply shut the sub-field of Relativity.
This all goes to say that it is a very different world intellectually than the world Einstein entered when he began publishing with Annalen der Physik in 1901. The field of physics had not yet closed itself off from “amateurs.” It was remembered then that Newton was an amateur—a self-taught mathematician and physicist—as were many of the greatest scientists and mathematicians of history. Einstein was a bit of an amateur himself, as the stories of his patent office imaginings confirm. The “university professional” was still a thing of the future. Forty years later amateurs still existed, though in fewer numbers. Karl Popper was resented maybe, but he was respected by most. Einstein himself understood the necessity of philosophy in the intellectual sciences, and he tied his theory early on to various epistemologies and metaphysics. He found it just as important to learn to speak of Kant and Hume as to learn the equations of Riemann. He was the last to do so.
The next two generations of physicists would lose all respect for the past. First Relativity and then Quantum Mechanics were seen to supercede all the theories of the past, and history became a clean slate. Richard Feynman could speak of philosophers with open disdain, and even Einstein was given only lip service. Einstein’s “regression” into philosophy and his quarrel with the Copenhagen interpretation of QED made him a dinosaur in his own lifetime. TIME magazine may have voted him the most important person of the 20th century, but physicists considered him a befuddled old classicist by the 1940’s.
My mathematical critique of Special Relativity therefore arrives at a rather inauspicious time. It could not be less welcome. This is ironic considering the mixed respect that Einstein has in the field of physics. He is believed to have been mistaken about almost everything important, in the grand scheme of things; and yet the equations of Relativity are sacrosanct. They are sacrosanct not because they are understood and admired—they are sacrosanct because they are the foundation of so much current research. Relativity theory is a miniscule part of modern physics. Very few people know anything about it. The few that do are working on billion-dollar projects—to discover the graviton or launch the next satellite. The last thing they want is some theoretical controversy to get in the way of funding. Even these scientists know very little about the theory. Most are glorified engineers. Theoretical physicists do not work in Relativity, since there is believed to be nothing left to do. The big names are in QED, especially in string theory and other esoteric modeling. They are also not interested in Relativity. It is no longer sexy. It is a settled question. It is not up for discussion.
So you can see that the field, despite seeming to be at a very creative time historically—due to the theoretical freedom that the top physicists would seem to have—is actually quite rigid and dogmatic. There are certain things you do and certain things you do not do. Superstring theory is prestigious. Looking at basic algebra is not. Looking into the distant future is progressive. Looking at old dusty papers is not. Tying esoteric theory to time travel and science fiction and Star Trek and the Dalai Lama is au courant and cool. Tinkering with ancient history is not. Stephen Hawking can claim that physics will be over in ten years, since ten years is still in the future (and apparently always will be, by some paradox), and not break any unstated laws. But a scientist who claims that Einstein or Newton or Feynman may have made a verifiable mathematical error is seen as monomaniacal and anti-social.
Despite all that, I am confident that my math will speak for itself with those who have eyes to read. It is to be hoped that I have left very little room for argument in my equations. Metaphysics may allow for endless bickering, but algebra was invented to finalize the argument. Even the tensor calculus may allow for some movement: there are places to hide amongst the matrices. With algebra there is no shelter as large as a shrub to huddle beneath.
Concerning my critique of the calculus itself, my argument there is likewise unobstructed. A chart that lists differentials is not open to much interpretation or misinterpretation. I do not open myself up to deconstuction. Even if you don’t like my comments regarding the historical method, or my explanation of graphing, it is hard to deny that I have solved the calculus “without the calculus”. This, by itself, is news on a grand scale.
I began this book when I stumbled across the first great error many years ago, in reading Einstein’s Relativity. Although it soon became apparent that the error was both elementary and profound, I thought at the time that it was an isolated error. But my naivete evaporated as I subsequently reread other important theoretical papers, and my awe of the past evaporated with it. What I came to realize, with rising disbelief (as well as some excitement), is that my faith—the faith of all scientists—in the basic theory and math of physics has been unfounded. It became apparent that the theory and math of many famous and influential papers, both classical and modern, had never been checked closely—or not closely enough for my taste at any rate. Buried in these papers were algebraic and geometric errors of the most basic kind. Suffocating beneath dense, often impenetrable theories and unnecessarily difficult equations of so-called higher math were errors that a high school student could understand, were he or she presented with them in a straightforward manner.
My goal became to do just that. To strip physics of its mystifying math, its unnecessary proliferation of variables and abstract concepts, its stilted language and dry jargon, and to speak in clear everyday sentences and simple equations. Einstein is famous for stating that a theorist should be able to explain his theory to an eighth grader, but he did not practice what he preached. Like his precursors, he could not explain his theory even to his peers. Relativity has remained uncorrected for a century not because it is flawless but because, as written, it has been impervious to understanding. Nor was this imperviousness an accident. Some might argue that Einstein simply fell a little short in places—no theory is born in complete and perfect form. But this belief cannot hold: Einstein imported the tensor calculus into Special Relativity himself, though it was completely unnecessary and ill-advised. He did this mainly as a public relations move, to impress the mathematical elite, to dress his theory up for the trip to Princeton. But this move has been disastrous, since it buried the math of the 1905 paper, making any correction almost impossible, especially by those who had taken the time to learn the new math. Those most likely to be able to correct the initial mistakes—the brightest minds in the field—had been diverted. They have been diverted ever since. No one who had spent five years learning General Relativity and its math would want to waste any time looking at basic algebra. It would be like Mozart stooping to think about scales. The math of Minkowski was another unfortunate addition to the mess, as I show in my paper on him. The false symmetry he gave to the time variable, and then the loss of that variable altogether, further cloaked the theory and algebra of Relativity. Very early in its history Relativity had already become the most esoteric of esoterica, and, despite its inherent mathematical simplicity, it was sold to the world as if this were its strong point. Bohr said that by the ‘20’s only six people understood it. I now know that he overstated the case by six. Anyone who had understood its theory would have corrected its math, since the mathematical errors are so simple.
As incredible as it may seem that errors have remained uncorrected in Relativity for a century, that time period is actually quite small compared to other errors I will relate here. The errors of Newton have persisted untouched since he made them, traveling unnoticed beneath the noses of the greatest mathematicians in history. And the errors embedded in the calculus are older still. We have to go back to ancient Greece to find the theoretical underpinning of Newton’s and Leibniz’s calculi. This theoretical underpinning was often improved upon in the 2000 years between Archimedes and Cauchy—which makes it all the more amazing that it is false. Mathematicians spent two millennia refining an error. The calculus is true, but its theory is false. It does not work the way anyone has ever thought it does, or for the reason anyone has ever thought it does. It has nothing to do with infinitesimals or limits. But I am giving away the ending of a great story.
It was fortunate that I discovered early on the soft underbelly of modern math, for it allowed me the rare privilege of transcending it. I saw almost from the beginning that esoteric maths such as the tensor calculus had become obstructions to true understanding. If the tensor calculus could build its greatest structure on the false math of Relativity, then it must be an overrated tool. An architect who knows his job does not build a palace on a sand pit, and the mathematician is a fool who spends his college years diddling with a math better done on computers, when he doesn’t understand algebra or geometry.
As a tonic to this chaos, I have tried at each point in my proofs to use the simplest math possible. This runs counter to current dogma, which tells us to impress each other with the most difficult math imaginable at all times. Simple math is considered neither sexy nor imposing. It also cannot be used as ballast, as misdirection, or as obfuscation. It is therefore not of much use to the modern theorist. Careers are advanced by advanced math; nothing is propelled by simple algebra, it is thought. Despite this, I have found that algebra is the first and most useful tool for unraveling the mathematical mystifications of the past. In the beginning, Special Relativity was proved by Einstein with algebra. The 1905 paper has only one line of calculus in the proofs of the transforms, which line is redundant padding. These transforms are exactly the same ones proved today with the tensor calculus. But the obvious tool to critique algebra is better algebra.
In correcting the foundation of the calculus I did not need calculus or any math evolved from it. I only needed basic number theory, which basic theory is now so elementary as to be forgotten. The modern mathematical method for solving any problem is to come at it from above, with more and more abstract math. My method is to come at it from below, questioning the fundamental postulates and often simple math that have been lost to view over time. As an example, the problem of gravity is being attacked now with superstring theory, which preens itself on its mathematical complexity and its theoretical density. But I believe that gravity will be solved by unlocking simple algebraic relations among classical variables. There is very much in the existing theories of Einstein, Lagrange, Hamilton, Newton, Kepler, Galileo, and even Euclid that has not been resolved. Leaving these mysteries in the trash in order to concentrate on new mathematical paradoxes is a grave error in judgment.
Descartes (who also missed seeing the fundamental error of the calculus, by the way) said in his Meditations that he had reached a point of absolute doubt. He felt he could rely on nothing around him. He must start over from the beginning, taking as true only what he could prove himself. Most philosophers now believe that Descartes was only using a convenient method of argumentation, one that did not seem so unique, or so egotistical, in the 17th century. But I believe he was in earnest. I find his doubt highly plausible, even beyond its usefulness in critiquing the unsupported beliefs around him. As more and more of the pillars of my certitude fell, I too reached a point of near-infinite doubt. I found that I could no longer look at any theory or equation, no matter how self-evident it seemed, without checking the math from top to bottom. No more would I take any proof on faith, assuming, as an example, that a short series of equations by Richard Feynman must be correct, simply because I knew that he was famous for being a great mathematical physicist. I have since found absurdly simple mistakes everywhere I looked. In fact, it has been rare that I have checked anyone’s math and found it correct. I have gone through textbooks, finding algebraic errors on nearly every page. The calculus is almost universally misused, even beyond its cardinal error in claiming to find instantaneous values. The newer maths, many of them offshoots of calculus, are likewise flawed in many fundamental ways, from set theory to topology to Cantor’s theory of infinities.
I know that most will be shocked at my presumption, and the rest will question my credentials. But I can only answer that physics has never, in the whole history of science, had anything to do with credentials or false humility. It has to do only with truth. If my equations are faulty, then I am abashed. If my theory is incomplete, I am vulnerable. But no one should have to apologize for having the courage to question, or to present his findings. The overly socialized and pressurized milieu we live in, where intelligent and earnest people are dismissed for the flimsiest of reasons, or for no reason, and where most people are cowed into permanent silence, has more to answer for to history, or to the gods of physics and math, than I ever will for my boldness.
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