From the beginning of the time when man could think and imagine, he has held various perceptions of his universe. Often, as history has shown us, many of those perceptions have been incorrect and sometimes even silly.

As an example, it was once believed that the earth was the center of the universe and that everything revolved around it, including the sun, the moon, the planets and the stars. Early scientists like Copernicus and Galileo, in attempting to refute this perception, were persecuted for their statements to the contrary.

Other scientists, desiring to hold that the earth was indeed the center of the universe, devised complex mathematical equations to explain the odd and sometimes mystical behavior of the planets and the stars. Even though these equations failed terribly at predicting planetary motion, people held to those beliefs because anything else seemed to defy their perceptions-"Of course the earth is the center of the universe, how could anyone believe otherwise?"

Eventually, they were proven wrong (if indeed we can be assured that Earth is not at the center of the universe-we actually do not know where the "center of the universe" is) and science prevailed, but old fears and perceptions had to be overcome first-and perhaps a bit of arrogance. In those days it was a strongly held belief that mankind was the centerpiece of God's creation and should therefore be at its highest place, which in this case would be the center of the universe.

This book is all about perceptions. We do not realize that the many things we take for granted in casual living occur as a result of truly deep and mysterious processes, few of which we even remotely understand. A simple process, such as seeing with our eyes, becomes a tapestry so complex that even the best scientists have difficulty understanding it. And yet, the biological side of this process is perhaps the simpler one. The physical side of it, which is the primary pursuit of study by physicists, is so sublime that it baffles the imagination.

This chapter is mainly concerned with what we perceive about the structure of our universe. I will begin by carefully examining certain aspects of our universe and recalling very clearly what we have stated about them, what is not known and what can be concluded as a result. First I want to talk about light.

The most informative statement that we can make about light is that we do not know what it is. We know, however, a great deal about its nature. We have many perceptions of light. We see it as rays, as particles, as waves and we even visualize it as moving through a continuum-an aether, as it were. We cannot help but think that there must be something in-between a light source and a lit object that "carries" the light. We perceive light as something that moves. It was there, now it is here-it moved! Our observations of earlier physical phenomena tell us that this must be the case. The ball was over here, we rolled it, and now it is over there. It must be true because our perceptions have told us so.

We even have physical laws that govern a ball's motion. But what about light? We cannot see it. We can see an object which is illuminated by light, but if you look between a light source and an object, you will see nothing there. You can place a detector between the light source and an object, thereby showing that there is light between the two, but doing so simply makes the detector the new object, and we come straight back to the original situation. Since this is the case, we have a very serious problem.

Our perceptions tell us that something is there-that light is actually carried through some undetectable medium from one place to another. Could it be that our eyes are playing tricks on us? Or could it be that, as biological creatures, we see only what we need to in order to survive the roughness of nature? When you say, "The ball is over there", you point, I look, and I see the ball. But is it really over there? Your statement has implied that the ball is in a place. If I look in that place I will see the ball.

There is nothing between me and the ball except air, and if you remove the air, then there is nothing but empty space. And light goes through empty space "easier" than it goes through air. How is it even remotely possible that light is emitted from a source, reflects off the ball, travels through nothing (or something), and somehow strikes the biological detectors in the back of our eyes, allowing us to see the ball?

So we perceive space. We pick the ball up, it has weight and we perceive matter and gravity. The ball is cool and we perceive energy. It takes a few moments for the ball to roll and we perceive time. But when we throw the ball up into the air, and it stays there, it is time to challenge our perceptions. And light is a "ball" that will not come down.

Now, in order to deal with this problem, we will need to reach back through some of the history of physics to find out why our perceptions are acting so strangely.

In this chapter I will embark upon an argument that was begun more than 2500 years ago by the great thinkers of ages past and that has continued even through this day. Many mathematicians and physicists are familiar with the name Zeno and with at least one or more of his many paradoxes.

There is a thing however, disturbing to me, that the true beginnings of Physical Science (Physics) lie here, in the thoughts and observations of the ancients, which are severely neglected as young people begin their formal study of Physics during high school and college. For, even though their observations may seem philosophical, they are the groundwork for modern thinking, and an essential platform from which to embark upon greater ideas and concepts.

Instead, modern classroom studies begin, even now from a historical standpoint, in the 1600's, and build from there, except when the study of Geometry is undertaken, in which case, the early Greeks are referenced. Unfortunately, even here, the basic premise' for such mathematical structures are ignored, but more destructively, are assumed, thereby failing to allow students, learners and thinkers to decide for themselves which to choose. Note that I am introducing a faulty assumption here; that, given what we know, a logical and correct choice can be made.

Essentially, we are told which mathematical or physical interpretations of the universe are "correct," and are told to blindly accept such representations or suffer the consequences of reduced grades in our classrooms. This is partly the fault of our teachers, for not standing up to the politicians, and partly the fault of our politicians, for telling our teachers what to teach, thereby interfering with the growth, progress and diversity of science.

Specifically, Phusiologia, or Physics, was referred to by Aristotle as The Study of Nature. The thinkers of ancient Grecian times were called the Phusikoi (the plural form-the singular form is Phusikos), or Student(s) of Nature. These were the first physicists, and they can be named: Thales, Anaximander, Anaximenes, Pythagoras, Alcmaeon, Xenophanes, Heraclitus, Parmenides, Melissus, Zeno, Empedocles, Hippasus, Philolaus, Ion of Chios, Hippo, Archelaus, Leucippus, Democritus and Diogenes.

These are referred to by modern philosophers as the Pre-Socratics, meaning of course, that they pre-dated Socrates, one of the more "known" philosophers. Certainly the list here is incomplete, and while I may be accused of "name-dropping," the point of this argument is to expound the magnitude and life-span of the study of The Physics, or Phusiologia.

A more important statement can be made by what they studied and said about the world and the universe. For one might be amazed to find those thinkers bequeathed of remarkable insight, even for their lack of scientific instrumentation or experimental data. Eventually there was a splitting, of sorts, between the ancient thinkers and, while some turned to Philosophy, or, the "study of wisdom," others turned to Natural Philosophy, or, the "study of nature."

This separation remains intact, even today, and occasionally physicists and philosophers alike, have a tendency to "cross the line," to the other side, neither being "perfectly" qualified to do so. The two groups as a whole however, have a general inclination to think in similar ways, their primary division being their respective scholarships.

One of the greatest arguments of the Pre-Socratics and later, the Aristotelians, was the question of universal singularism versus universal pluralism-that is-whether the universe exists as a single "object" or a multitude of "objects." This author, of course, has examined the arguments, but does not wish to discourse at length upon them at this point (the argument will be taken-up shortly). Rather, I wish only to point out that they existed then and they continue to exist today, even so, without resolution.

The only unfortunate thing about this is that modern scientists, mathematicians and physicists seem to have already made the decision concerning the issue, as evidenced by our Mathematica (the calculus). And while one might believe that this is not such a large issue, it is crucial to the definition of the underlying structure of the universe.

For those not familiar with it, the Greek philosopher, named Zeno, once suggested that a runner racing to a finish line should never be able to cross it since he would, at first, have to run half the distance to get there. Having done that, he would again have to run half of the remaining distance.

This process continues on and on as the runner closes on the finish line. But since he must continue splitting the distance in half, there would be an infinite number of splits, and therefore the runner would never be able to cross the line.

This is the version commonly passed on to modern students-that is-the ones who bother to ask about it, and most learners and modern thinkers shrug it off as a curiosity which can be explained away or, more precariously, assume that it has been explained away.

In fact, Zeno's discourse on the subject of singularism versus plurality was far more comprehensive than this and his paradoxes are fairly difficult to solve.

Here, Zeno argues against the infinite continuity of matter-particularly that if the states of matter and things in existence are more than one, then they must have continuity in order to be so. However, with keen intellect, he argues that if such infinite continuities exist, then there must be an infinite range in the proportions of the things which exist. This is no small argument to deal with. For it implies that if such is indeed the case, even a small amount of motion could not take place. Before the motion of any object could take place, one part of it would have to actually overlay another part, indeed, two parts of the same object occupying the same space at the same point in time.

The works (or words) of Zeno were preserved by Aristotle, not so much for their value in a philosophical light, but for ideas to condemn. The discussion between Zeno and Socrates is characteristic of Zeno's concept of singularism. Later philosophers, or phusikoi, denounced the works and words of Zeno, regarding his ideas as little more than insignificant musings.

Such ideas eventually led to the rather incomprehensible Runner Paradox. Here, Aristotle puts Zeno's Arrow Paradox to rest.

Aristotle continues his argument, but this first response is crucial to the remainder of it. In his final comment he has revealed a false assumption on Zeno's part; that time is made up of indivisible instants. This is, of course, perfectly correct. Zeno has made a false assumption. Zeno may only suggest that the nature of time can be this way; he can neither prove nor disprove his statement, given the information which he has about nature.

However, Aristotle has also made an error; he has made a statement that he can neither prove nor disprove, and by remarkable coincidence, it is the exact opposite of Zeno's statement. He has stated that time is not made up of

indivisible instants. He has made a statement about time itself, for which he can provide no evidence.

In this way, Zeno's paradoxes were set aside for many ages, the decision having been made, the word of Aristotle having been spoken with no one remaining to stand against him. Certainly the issue was taken up later, particularly since his arguments are the very basis for the concept of the non-existence of space, and presented themselves as a challenge to the concept of the space-time continuum, that being the very thing Zeno had argued against.

Even Einstein, as we saw in an earlier chapter, relented on this issue, calling it "pre-scientific thought," thereby giving in to the ideas of Descartes, Minkowski and even Newton. Einstein's arguments, though well-constructed, also failed to either prove or disprove the existence of space and the space-time continuum.

It is not a surprise, however, that he was forced to choose between one or the other. Certainly the political pressure on him had to be enormous, since, by that time, almost all of mathematical physics was based on the works of Sir Isaac Newton and Gottfried Wilhelm Leibniz, which relied upon the "proofs" of Aristotle against the arguments of Zeno.

Our difficulties were protracted several hundred years ago with the introduction by Newton and Leibniz of the calculus.

While Newton is predominantly credited with the invention of the calculus, his desire was to formulate solutions to the problems of gravity and mechanical systems, including the calculations of planetary orbits-for which they worked remarkably well (except for Mercury, the precession of which was discovered many years later, and then only many years later, explained by Einstein).

Leibniz, on the other hand, was more of a pure mathematician than anything else, but was fascinated by the ideas of the infinitesimals and solutions of infinite limit problems.

At any rate, both men came up with fairly similar systems and both systems were eventually named The Calculus. No student today can get through an advanced physics course without a very good knowledge of the basics of the calculus, and no mathematician can call himself such unless he is thoroughly familiar with it. It is the most used tool of science, today.

It is also no surprise that, nowadays, the calculus is even being taught to business students and used as a tool of economics and finance. The calculus is used in mathematics, physics, chemistry, economics, biology, engineering, finance and many other fields of endeavor. Suffice it to say that few educated (college-degreed) individuals have not, at least, been exposed to its rudiments.

In this sense and many others, the use of the calculus permeates our society and civilization. Such a blow it would be to find something wrong with it. But then, such a hardy and useful tool it has been that no one would dare to challenge it. We have used it to solve many, many problems covering a wide range of specializations. The calculus to date, has an unblemished reputation in this respect.

The calculus has been so thoroughly successful that most folks (scientists and mathematicians, too) blindly accept it and the basis from which it was constructed. The calculus goes unquestioned, unchallenged and unmaligned, so dear it is to the hearts of many.

So I shall question it, challenge it and malign it. Now, I will prove it is wrong. At first glance, the calculus seems like a beautiful garden with several varieties of plants, trees and bushes, each with different flowers, roots, leaf shapes and so on. It is true that the applications of the calculus come in many varied forms and shapes.

But a closer examination will reveal that it is all one plant (not even offspring) which has all grown from a single seed, has only one root system and only one body. What if our beloved garden is really only a very cunning weed in disguise? What if we have been awed and then fooled by the pageantry and the fanfare? What if there is something wrong with the seed?

This, of course, hails the great weakness of the calculus-that it is borne of a single seed. And if that seed is wrong, then the whole of the calculus-all its roots, branches, leaves and flowers-is wrong.

The first thing everybody wants to say about the problem with the calculus is that almost nobody understands it (even those who have studied it extensively). To be certain, the "average" person can remember having difficulties getting through algebra in high-school, much less try to tackle the more abstract mathematical constructs bound up in this form of computation.

This however, is not the problem I am thinking of, and we need not really go into the more complex aspects of this subject (do I hear a sigh of relief?), because the problem lies in its most basic precept which just happens to be the concept of the limit.

The very heart of the calculus can be found in the philosophy of one of its founding fathers-Gottfried Wilhelm Leibniz. Leibniz believed in the infinitesimally small and, in this sense, believed in the continuity of existence of space, time and matter. Sir Isaac Newton, no doubt, had similar ideas as he developed his laws of motion and wrote the Principia (the basis of the calculus- including the laws of motion).

With these ideas in mind, both men applied the concepts of limits and the infinite continuity of existence to the abstract construction of mathematical systems of scale. In this, there was certainly no problem so long as it remained nothing more than an abstract study in the mathematica. Unfortunately, its application to the physical world (and the universe) renewed the problem. The seemingly innocent assumption originally made by Aristotle allowed us to believe that the infinitely microscopic could be directly applied to the finitely macroscopic world, thus solving many of the most perplexing problems of physics.

For example, suppose that we have a basket of (whole) apples and a bucket of water (consisting of whole [H₂O] molecules), and we want to know two things: first, we want to know how many apples are in the basket, and second, we want to know how much water is in the bucket. We are not allowed to count the apples and we are not allowed to weigh the bucket, but we do know the dimensions of each and (assuming that the apples are all identical in size) the amount of space taken up by each apple.

There must be some rules though. Let us suppose, in this example, that no part of any water molecule and no part of any apple may extend above the rim of either bucket. If one of either does, then it cannot remain in its respective bucket and must be removed. Further, we have (somehow) managed to arrange all of the water molecules and all of the apples in their respective buckets for maximum efficiency of space.

Now, instead of being smart and simply finding the volume of each by use of simple formulas (which, incidentally, will yield incorrect answers in both cases), we wish to make these computations using the calculus-which we can do. We can do this by a simple method of the calculus called integration, where we simply integrate a function of a known area (as the incline of the respective basket and bucket-smaller at the bottom, larger at the top, etc.) around a circle at the bottom and then into a volume as we integrate bottom to top.

So, at the end of this computation, we get the total volumes of both the basket and the bucket. Now, for the basket of apples, we simply divide the total volume of the basket by the average volume taken up by an apple and get the total number of whole apples in the basket. Let us suppose that we have done this and gotten the number 15.9472674553298 apples. Our calculation has been very precise, and as far as we are concerned (without having noticed an obvious error in our calculation) this number is correct.

Of course, in this calculation of the amount of water in the bucket, there is far more leeway since atoms are much smaller than apples. Suppose then, that we finish our calculations and find we have 23.9474538730098765 cups of water in the bucket. Again we have decided that since we have used an "unquestioned" mathematical procedure for finding this value, we are convinced that our number is flawless.

Obviously, in the calculation of the number of apples in the basket, the calculus has failed us, since, if we are counting the number of apples in the basket (there are no bits or pieces of apples allowed), we must come up with a whole number of apples in order to get the answer correct-we cannot have 15.9472674553298 apples¹, we may only have 15 apples in the basket (we cannot have 16, since that would exceed the capacity of the basket).

The bucket of water however, is a different story since it may be possible to have a whole number of water molecules in a cup of water. Unfortunately the calculus has yet failed us again, since there is something we have not realized-and this is that both of these, in reality, are the same problem! How can this be?

It is because we see water as being so motile that we do not see the difficulty. In actuality, if we follow the rules correctly, then we will only be able to get an exact whole number of water molecules in our bucket.

So although, if there were some exceedingly small probability that, using the calculus we might accidentally compute an exact whole number of apples, it seems even more likely that we would be able to compute an exact whole number of water molecules since they are so much smaller and there are so many more of them. This is not the case however, since the probability of either computation coming out as a whole number is the same; 1:10¥ (actually, I could have made this number "1:¥ " and would have been equally correct, but using infinity at a power helps to emphasize the depth of the improbability. In a moment, I will show that this representation is not far from the actuality of the argument).

The reason for this is the use of infinitesimals in the calculus. Let me see if I can explain what I mean by this. Suppose that we used, instead of the calculus, a system which gave us solutions in terms of tenths of apples? If it had worked properly in our previous example, then our answer would have been 15.9 apples in the bucket-which still would have been wrong. But then we could try several different-sized buckets with the same rules, and for each bucket we tried, there would be a 1:10 probability of our answer coming out correct.

Suppose however, that we do not have the advantage of a computing system that comes out in only one decimal place? Suppose that our computing system comes out in six decimal places (to the right of the decimal point)? Then our solution would come out to 15.947267-which is again wrong, but the probability of getting a correct value for any particular-sized bucket we compute with this system becomes 1:1,000,000. And for each decimal place we add to our computing system, the probability of "hitting" the correct value right on the nose decreases by a factor of 10.

Because of infinitesimals in the computing system we call The Calculus, there are an infinite number of decimal places to use, and so it is impossible to compute an exactly correct answer for this problem. The same is as true for the apple problem as it is for the water molecule problem.

Do not get me wrong in this-the calculus is a wonderful tool for calculations if you only want to get close to the right answer. What I have at this point is no surprise to any mathematician or physicist, but every one of them after reading this, is thinking a little bit harder about it.

There is of course, a method to all this "microscopic madness." And the method is this; We recall from previous chapters that both Einstein and Lorentz used variations of the calculus to solve for their respective versions of the Lorentz Contraction (Equation 2.1).

So how does this affect the relativity theory? First of all, we do not really know what "happens" to light when it is in "transit' between two places in space or why there would or could be a temporal dilation for moving matter. Einstein's equations and logic simply pointed out that things happened this way. On a macroscopic scale his equations may be somewhat valid, but because of the problem of infinitesimally small measurements (which are assumed in order for the value of Δx to go to zero), the differential equation (Eq. 3.8) cannot be perfectly valid.

The most direct and obvious reason for this is the quantization of energy-particularly light energy. Further, we realize that in order for energy to be transmitted between particles of matter there must be at least one full wavelength of spatial separation between particles. If the value of Δx becomes infinitesimally small, this cannot occur.

This is not to say that Einstein's equations do not correctly predict dilation and other effects; it merely indicates that the problem was not properly characterized. More importantly, however, it indicates that the use of these equations in other applications can give other misleading results-thereby confusing things even further. The greatest example of this is the failure of the results of The Relativity Theory and Quantum Mechanics to agree with one-another.

Quantum Mechanics is also not immune to the problems associated with the calculus and limits. While The Relativity Theory may have only violated a minor rule for the microscopic limit of the problem-resulting in its improper characterization, Quantum Mechanics emerged out of certain discoveries (which I will discuss shortly), but most of its mathematica originated out of the Schrödinger Wave Equation, invented by Erwin Schrödinger, which is based upon an extension of the calculus, known as The Hamiltonian Equation.

These two equations will not be discussed in this book, however, since the essence of Quantum Mechanics has less to do with the calculus and more to do with quantization, which will be discussed extensively in the next chapter.

 Go to Chapter 9

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