LESSON TWO:

A True Wizard begins a study of the mystic arts by first studying the self, the mind, searching for flaws and errors and adjusting them, fine-tuning thought and thus correcting the world that is seen outside.

In this second lesson we will play a mind game, a game of the abstract where the concept of infinity is weighed and measured against the background of reality and human perception. This lesson will show that the concept of infinity is problematic not simply as a property of mathematics in general, but as an inescapable property of the external world, a problem that will bring into doubt the validity of our perceptions and conception of the external world in terms of space and time. We will soon discover that physical material reality as we know it may be wholly subjective, a mere construct of the mind! Tread lightly and just read on.

Mathematics is regarded as the foundation of all the physical sciences and all physical laws have been deduced mathematically using arbitrary measurements of Space, Matter (or energy) and Time, physical units so theoretically small as to be imperceptible to the human senses (or by any instrumental means whatsoever!). Mathematical units used to measure hypothetical quantities on molecular, atomic and subatomic levels are examples of these fantastically small measurements.

Unfortunately, numbers in any form do not signify anything substantial or concrete at all and if we try to associate numbers as definite units of measurement for things and events in the physical material world we are at once confronted with the problem of infinity. Infinity, then, is more than a mathematical problem; it is a physical one, an inescapable contradiction in terms when it is regarded as a property of the external world, and even more so when it is not. Are space, matter and time infinite? Or are they finite? In fact, does the physical material world as we perceive and conceive it make any sense at all?

During my freshman year of college, I attended numerous social gatherings with weekend philosophers and mathematicians alike who would debate about mindless subjects, chasing them down from time to time with beer and elicit substances for hours and hours. During one of these chemically-altered discussions a physics major and devout empiricist, Jim, was hotly arguing with Mike, a math major with in an interest in metaphysics, about which object was rationally larger in size: an apple or a grape.

Jim of course made it quite clear that the answer was obvious to him and to everyone else in the world except Mike.

“Well, look at them, man!” Jim violently snatched the apple and grape that Mike had earlier hunted down and placed before them on a beaten pressed-wood coffee table in order to begin the debate. Jim held them up close together, the apple in his left and the grape in his right, only a few inches from Mike’s eyes, twirling them in his fingertips, separating them with his hands and bringing them back together again.

“See!” he shrieked maniacally with a stiff and humorless grin, eyes held wide and glaring.

“The apple,”—which he nudged a little closer with his left hand--”is bigger,”—and now the right which held the grape—”than the grape!” His head joggled wildly with the last word “grape” as if to emphasize the profound logic of his point and the utter stupidity of this Mike guy who just didn’t seem to get it.

Mike, however, who was a mathematician with a keen interest in and understanding of certain abstract ideas of Euclidean geometry, simply smiled a comfortable smile.

“Well,” he said coolly, “if I could shrink directly proportionate to the distance I traveled towards that grape in a futile attempt to reach its center, I’d be shrinking forever and traveling forever because I would never be able to reach the true center-point of that grape. In fact, we can never pinpoint the exact center of any three-dimensional object in the physical world.” Mike seemed quite pleased with this. “And ‘Why not?’ you may ask,” he said, gently nodding his head to emphasize the significance of his next statement. “Because a point has no dimension! Therefore the grape is just as big as the apple!” Mike jerked his head once as if the point had been taken, got up and went to the fridge for another beer.

Jim and I, and everyone else in the room stared at each other blankly for a long moment with our mouths open, totally perplexed!

Mike, of course, may have been chemically motivated towards this final and ultimate conclusion, and this line of reasoning may be a bit abstract under ordinary psychological conditions, but his argument seemed valid to me nonetheless (an opinion I wisely kept to myself!). It seemed to me that Mike was really expressing in a roundabout way the problem of infinity in a microcosmic sense, the idea of infinitely small space which, if true, would make even a grain of sand seem as big as the entire universe, only, of course, on a different scale. Infinity in a macrocosmic sense, as infinitely large space, is just as problematic and we find that human thought is intrinsically plagued by this irritating problem of infinity.

In mathematics, infinity is a matter-of-fact property o numbers in two directions—infinitely large quantities and infinitely small quantities (generally designated by negative numbers or fractions)—and poses no problems in purely mathematical terms. However, when we try to use numbers to calculate space in both a macrocosmic sense (like the size of the universe) or in a microcosmic sense (like the size of molecules, atoms and subatomic particles), we find that we still have to deal with this inescapable concept of infinity. How big is the actual size of the universe? How small is smallest particle of matter or the smallest unit of space? With respect to time we have similar problems and thus find ourselves asking these same questions about the nature of time: How long ago was the beginning of time and how far away is the end of the future? How big is the biggest unit of time and how small is the smallest? In attempting to answer these kinds of questions we soon realize that it is impossible for us to logically keep infinity out of the equation because leaving it out seems even more incomprehensible: How could there possibly be a limit to space in the universe at all? How could there possibly have been a beginning to time? How can there possibly be an end to it? The more we think about such questions the more we find ourselves faced with problems in logic that cannot be ignored and, suddenly, the physical material world as we perceive and conceive it becomes less and less rational.

The problem of infinity, therefore, should be the beginning of our investigation into the validity of our perceptions and conception of the external world and, although the importance of mathematical infinity will seem difficult to understand at first, the point will become much clearer when we establish that infinity is more than a mathematical problem: it is a physical one that asserts itself when we step outside the scale of ordinary human perception.

The concept of infinity is essentially beyond our comprehension. We, of course, understand what infinity means: a state or condition that goes on forever without end, and sometimes without a beginning. We understand the definition but are unable to capture the idea with our minds as a concrete or substantial image. Infinity, then, eludes our scope of comprehension. In mathematics, numbers may be continually added to, subtracted from, or multiplied and divided by any number to produce ever increasing or decreasing quantities that may continue to grow or shrink to infinity. Numbers, however, are really only symbols of thought having no physical equivalents. Assigning numbers to things, then, is completely arbitrary and any thing may be assigned any number: ‘This will be number one, and this will be number two, and this will be number three, and so on and so on.’ In other words, there is no actual thing that can be considered, in fact, the one and only number one, or the one and only number two, or number three, and so on and so on. In fact, numbers are used only to designate, divide and relate individual things and to order and limit them within a finite group or operation, and on a certain scale. They represent nothing real or concrete in a physical material sense and only in their relationship to one another as arbitrary numbers do they have value: ‘This is number one because this is number two, and this is number two because this is number three, and so on and so on.’ Moreover, numbers may be divided into smaller parts called fractions: 1/2, 1/4, 1/8, and so on and so on. From a purely mathematical standpoint, each of these fractional parts may be divided infinitely into smaller and smaller parts, making every number infinitely divisible.

As we can see, infinity is an ever-present property of numbers on many different levels, thus reaffirming the purely practical significance of numbers only within a given set of laws and on a certain scale. Understanding the relationship between scale and the concept of infinity, therefore, will be the key to understanding the problem of infinity itself.

In my junior year of college I tried explain my thoughts on this subject to my Philosophy of Reality professor in a strange little paper I entitled “As Above, So Below!” The paper itself was a weak attempt to show that mathematical infinity might represent, in a general way, a possible connection between the microcosm and the macrocosm referred to in certain mystical philosophies that suggest that the whole universe may be contained within a grain of sand. The paper was lucky to get the C+ that it did and my professor criticized my thoughts for their abstruseness, or rather, for his inability to decipher the following passage:

Since each whole number is only one in an infinite number of whole numbers possible in the mathematical universe, each whole number represents nothing substantial or concrete. And, by themselves, whole numbers are really insignificant in the infinite scheme of things. In the face of infinity, all individual whole numbers may be regarded as the same in size and scope whether it is the number one, two or one million and two! In the case of fractions, the initial beginning of this division of whole numbers is theoretically supposed to be the whole number being divided. However, since each whole number may be divided infinitely into smaller and smaller fractional parts, all whole numbers in this instance are equal in size and scope because infinite division has no end and, therefore, the apparent value or size of the number being divided becomes just as insignificant. Taken individually, each whole number when divided infinitely in this way becomes equal in size and scope and equal to the sum of all whole numbers taken together as an infinite whole simply by virtue of this potential to be infinitely divided into smaller and smaller parts forever, infinitely! Moreover, each fractional part of a whole number is equal in size and scope, by virtue of this potential to be divided infinitely, to the divided whole number that is itself only one in an infinite number of whole numbers possible in the infinite mathematical universe!

I have to admit, I can barely understand the passage myself!

Put simply: The problem of mathematical infinity reveals its true problematic nature when we discover that position and size appear as insignificant properties in the face of infinity.

Let’s examine in simpler terms the condition that arises when we relate the idea of scale to the concept of infinity.

Let’s say there are an infinite number of apple pies in the infinite universe. Imagine that these pies are lined up in a single, infinitely long row and that we could continue counting these pies forever if we thought to do so. Let’s also imagine that they are all numbered and you begin counting them in a race of counting, starting with apple pie number one. I, on the other hand, attempt to cheat in this race and begin counting with apple pie number two, or twenty, or one hundred twenty, the winner being the first to count all the pies first. Unfortunately, I could never win this race by cheating. In fact, neither one of us could ever win this race because we would always be of equal distance from the finish line, a line we could never cross no matter how many pies we counted. Starting with apple pie number two, twenty or even one million twenty will not bring us any nearer to infinity. In the face of infinity, all numerical positions are the same and only in their relationship to one another on a certain scale and within a finite group or operation (say, between numbers one through ten) can the significance of their positions be determined and distinguished.

Similarly, an apple pie that is one foot in diameter and an apple pie that is two feet in diameter are apparently different in size. However, we could divide these pies into slices—halves, quarters, eighths, sixteenths, and so on and so on—and, from a purely mathematical standpoint, we could continue dividing these slices into smaller and smaller slices forever. Each slice would be smaller than the last while the seemingly different sizes of the whole pies with which we began this division become less and less significant. In the face of infinity, both pies seem equal in size and scope and only in their relationship to one another as whole pies on a certain scale, can the significance of their different sizes be determined and distinguished.

We see here how numbers are intrinsically bound to our concept of space in that numbers are invariably associated with positions in space or quantities of space and, therefore, space itself. Space, as we know, is three-dimensional in that it extends left-and-right (width), up-and-down (height) and forward-and-backward (breadth). However, when we take the concept of infinity related to scale and apply it to three-dimensional space we find that the infinitely small world of space seems just as big as the infinitely large world of space. In other words, everything, no matter how small, will always seem as immense (or as small) as any other thing when thought of in these terms. All things in this sense are of equal significance (or insignificance) in the infinite scheme of things. Similarly, with respect to time, we may say that an eon of time is no more significant than a mere millisecond when standing side by side against the infinite vastness of eternity.

Analyzing the problem of infinity even further, we find that the concept of infinity poses more practical problems in logic with respect to measurement, especially when we step outside the practical scale of ordinary human perception. Once we do so, we soon realize that human thought becomes uncomfortably strained by the abstract, almost painfully!

For example, if I measure out an inch on a piece of paper with a pen I have not actually measured out a real inch because the ink lines themselves may be one one-hundredths of an inch thick. The actual lines that would measure out the boundaries of a real inch would have to be infinitely small, if such lines even existed. An inch, then, is an ideal that can never be fully realized in the physical material world, and the pen marks really only measure out the closest approximation to this ideal on the practical and visible scale of ordinary human perception. This approximation of course is sufficient and useful on this scale, but the ideal of the inch, the actual measurement, is quite beyond us.

Similarly, if I say I have timed and measured a minute with a stopwatch, I have not actually measured out a real minute because the clock is always ticking and the actual dividing lines separating the seconds between one minute and the next are impossible to capture. An real minute, then, is impossible to measure precisely because there are milliseconds and nanoseconds and possibly infinitely smaller units of time to be considered. What I have actually measured out is the closest approximation to a minute possible for me on the practical and perceptible scale of our ordinary experience of time.

We may argue here, then, that modern physics is a science of approximations, not necessarily an exact science. Physics bases all of its findings on what it believes are precise measurements of invisible or imperceptible phenomena (e.g. subatomic, atomic and molecular bonding) deduced mathematically through experimentation and observation of visible or perceptible results (e.g. silver electroplating). The visible or perceptible results of these experiments, however, still conform to a given set of laws determined by imprecise measurements of space and time on this perceptible human scale and, thus, may not provide us with a correct picture of these imperceptible worlds. This situation is of course inescapable because numbers are by nature infinite and, in attempting to apply mathematics to the physical material world, we are at once faced with the possibility of infinitely small (and infinitely large) worlds of space and time that may transcend and resist any truly precise, mathematical means of measurement.

The important idea to be considered from all that has been said is that the problem of infinity with respect to scale in both a microcosmic and macrocosmic sense implies that all things, no matter how small or great, may be equivalent somehow in the infinite scheme of things, an important doctrine of many ancient mystical beliefs.

Let me describe a strange vision I once experienced long ago:

It was Sunday morning, around 10 a.m. I was in a seedy hotel room after a long night of driving. It was spring and the world outside me seemed bright, profound and meaningful. My own life, on the other hand, seemed tragic, hopeless and meaningless, teetering on the brink of nonexistence, dark and sad. I was slumped in an armchair propped against the foot of a rusty metal frame of a single bed that cradled a damp and musty mattress, bare and stained by years of use. The room itself was small and confining. It smelled of dankness, sweat and disease. I was staring blankly outside a small, open window, a window that appeared to me to be a doorway to another place, the real world, a world whose beauty and mystery would forever stand outside me and painfully beyond in unrequited love, a love never to be realized. And I was quite sure that these thoughts and feelings were lingering traces of separation I felt the night before.

Across the street was a birch tree, strong and healthy with thick green leaves that fluttered with the force of small winds that chanted in soft whispers, impelling me to look. I stared at the tree intently, waiting for a sign that appeared as quickly as I knew it would. The tree itself became something surreal as the sputtering sounds of cars in motion down the street faded away. The tree grew closer somehow, bigger and the branches and leaves increased in size and significance. I felt them touch me somehow, caressing inside me while remaining as far away from me as I believed them to be. Suddenly, I saw a symbol appear halfway up and in the center of the trunk, a symbol whose meaning I vaguely knew. The symbol of Pi, a mathematical symbol, I thought. At the fringes of my vision new, dark shadows of symbols appeared abruptly and as my eyes moved to capture and focus on one of them another symbol of Pi was revealed, only this time halfway across and in the center of a branch, and on each and every branch of the tree. And, once again, there was the appearance of the same symbol in the center of each and every leaf!

I was overcome by a sense of profound realization that I understood but could not grasp. I began to wonder if within each leaf there existed the essence and history of a branch and the essence and history of a tree; and within each branch the essence and history of a leaf and the essence and history of a tree; and within the tree the essence and history of a leaf and the essence and history of a branch and the essence and history of all trees and all branches and all leaves since the beginning and until the end of time! And I began to wonder if each part of the tree--the branches and the leaves--were equal to it and to everything else somehow in the eternal mystery of things.

There is an ancient belief that if a man can learn everything there is to know about a tree then he will know everything there is to know about everything else! Each thing, no matter how small, is an infinite and eternal universe of knowledge, a microcosmic image of the macrocosmic whole.

Consider also this script idea for short animated film I once considered producing, which is based on this problem of infinity and my strange vision:

A rather droll man is sitting comfortably in an armchair positioned in front of a window. Through the window there is a view of an elm tree across the street. The man begins staring dully at a small dried up wad of gum that sits eternally motionless on the paint-cracked windowsill. The man’s eyes begin to sag and he eventually falls asleep, at which point begins a strange little dream. In the dream, he shrinks very small and the wad becomes massive. We see him soaring across the slick, cragged surface of the wad, up and over large hills that seem to roll for miles and miles. He plunges into the mass, penetrating its rocky surface as huge boulders of hardened gum break away, whipping by him. Darkness falls as what look like molecules appear, thick elastic globules veering around him, repelled almost magnetically to avoid a collision. Strange blue lights appear in the distance. As he approaches the hue of lights change to red, and then to a blinding white as the site of great celestial bodies appear, encircling a nucleus of radiant energy in perfect orbits: they are electrons revolving around a nucleus of protons and neutrons! The man shrinks even smaller as he penetrates the center, suddenly seeing galaxies and solar systems. Small planets emerge, planets that look strangely familiar. One planet in particular, a blue and vibrant ball of light, stands out. He descends into it, penetrating its atmosphere and plowing through its massive clouds into a clear, bright sky. He descends deeper into the visible skies of a particular land mass. Deeper he drops into a particular valley and a particular city and a particular neighborhood where a particular elm tree sits on a particular side of a street across from a particular window revealing a view of a particularly droll man staring at a particular dried up wad of gum! The young man is jolted from his sleep as his consciousness plops abruptly back into his head! He begins laughing uncontrollably.

Setting aside all mystical inferences that could be derived from this odd little film idea, the most interesting conclusion here is that it illustrates perfectly our utter inability to comprehend the concept of infinity, and any attempts to understand it in an ordinary way always leads our thoughts back, in vicious little circles, to where we started, back to the inescapable and enigmatic concept of infinity.

In Lesson Three: Space and Matter we will analyze this problem of infinity in greater detail as it applies to our perceptions and conception of the physical material world; specifically, how it completely shatters our ordinary concept of three-dimensional space in both a macrocosmic and microcosmic sense while ultimately unveiling the true metaphysical nature of our widely accepted concept of matter. We will demonstrate in a reasonable and logical way that space, as we ordinarily perceive and conceive it, may be nothing more than a mental construction, and the external world, the real world, may actually extend far beyond three-dimensional space. Ultimately, we will show that the essential nature of all things in the physical material world may actually be metaphysical, and possibly purely mental in substance, form and function!