Ignorance of One – Part 1


A Note on Real/Imaginary Mathematics

Copyright 1953-2020 © by A. Ray Johnson


I. The Fundamentals

APART from metaphysical speculations, there are no more delusive illusions than may be encountered in the field of mathematics, ancient/modern. In mathematics, the complex of Concrete/Abstract comes to its sharpest and also its most diffuse focus. In mathematics, we have the whole field of Illusion/Reality under both microscopic and telescopic observation. Here we are confronted with unlimited Fact and Fancy, with end-less fancies about facts, with equally end-less facts about fancies. It is no wonder that the most fantastic books in English (the Adventures of Alice in Mirror- and Wonder-land) were the works of a mathematician. Fancy is a prime fact in the structure-nature of man, and mathematics is the most systematically abstract product of man’s fancy or re-creative imagination. Poetry and statistics are twins. Tom-toms and adding machines measure the rhythms of peace and war. At the point where fancies and figures meet, history and mystery make their throbbing appearance. Mystics and masters of the high-er calculus, back to back, search the heavens in opposite directions for the unified field formula. The multiplication table, music, and the molecular structure of Form/Energy: Matter discourse along the same course. It is obvious that a definitive answer to the question, What is the relation of count to reality? would involve a solution of the riddle of the universe. To a fun-da-ment-alist, that ought to be fun.

The fun begins in the recognition of the obvious fact that figures are fancies that have no existence in nature save in the heads of mathematicians (who may here be redefined as figure-heads). There are no such things in earth or heaven as 1,2,3,4,5,6,7; search high and low and you will find that figures exist nowhere but in your mind. In concrete nature such abstract operations as addition, subtraction, multiplication, and division are merely figures of speech. Two white horses times four red-headed girls, four monkeys plus six stars, one hundred cabbages divided by ten kings, forty oysters subtracted from one bed, — such propositions are nonsensical. They are impractical fancies; you can say them, but you can’t really think them: they are as close to Platonic or immaterial ideas as the race will ever come. Two bankers plus two robbers equals four bank-robbers, — this is a formula that can produce nothing more tangible than a ghost or spirit. One teacher of mathematics plus thirty scholars equals a dull half-hour and thirty-one headaches, is a piece of poetry, not a piece of mathematics. Paint the figure 15 in red on the rear end of an elephant, but it won’t change the actual number of elephants in the circus. There may be fifteen elephants in the circus, but they didn’t come into the world with the numbers 1 to 15 stamped on their foreheads.

Search nature: you will never find anything but analogues to mathematical operations and geometrical designs. The closest, visible analogues to geometrical designs are the heavenly spheres and the terrestrial crystals, but there is no tangible evidence that abstract geometry has anything to do with the formation of these concrete forms. The closest analogy to numerical values is music, but it would be difficult to prove that abstract, mathematical formulations enter into the construction of the nodes in the human throat and ear. Atomic structure and the periodic table of the elements would seem to argue that matter embodies mathematical harmonies, but it is obvious that there is nothing abstract about the embodiment. Other analogues to mathematical operations-formulations are readily seen to be nothing but figures of speech. One analogue to addition is growth, and another is the swelling of mountain streams in the course of a heavy rainstorm; you can compute mathematically the growth of a tree or the rise in the volume of a river over the spring months, but abstract addition has nothing to do with the real processes involved in the growth of the tree and the increase in the volume of water in the river.

In nature one analogue to multiplication is generation, but there is nothing at all abstract about the process by which one peach be-comes fifteen annual crops amounting to tens of thousands of peach-es. One male guinea pig times one female guinea pig equals hundreds of guinea pigs, but that, albeit natural sense, is mathematical nonsense. Analogues to division and subtraction are decay, disintegration, erosion, and evaporation; you may also say that the passage of clouds over the sun subtracted a number of degrees of heat from the thermometer: but these are all figures of speech, and are that even if you compose very precise mathematical formulas expounding the processes and the rates of decay, disintegration, erosion, and evaporation. Mathematical formulas dealing with these natural processes are practical fancies: although they leave the actual processes really unformulated, they enable men to deal with the phenomenal, the superficial changes wrought by the processes.

In the regenerative process, cells “add” and “divide”, but both natural operations are biological, not mathematical. As yet there are no formulas reflecting the bio-electro-magnetics involved in syngamy, — the penetration of the egg by the sperm, — and if there ever are any such mathematical formulas they will prove to be fictions about an essentially unformulated reality, — fictions perhaps a little more practical than love stories in “explaining” the sexual process, but not a whit more realistic. Mathematical fictions are useful when elements, compounds, and organic tissue are subjected to artificial analysis-synthesis, but the natural realities thus dealt with remain essentially, or really, unformulated. Mathematical formulations provide men with a wealth of practical information, but with no true understanding of the structure-nature of Matter/Life. And the general failure to realize this fact is of crucial significance. Men generally mistake their fictitious formulas for apprehensions, — even comprehension, — of the realities, and in this are victims of a delusion, — a delusion which, if undiscovered, could destroy the race. Persistent and stubborn ignorance of the fanciful nature of mathematics could be disastrous. The subject is worthy of full attention and thorough inquiry-research. The fundamentals should not be funereal. LET us get a more detailed picture of the interplay of mathematical fancies and natural facts. First, let us consider the mathematical fancies that may be in exact accord with simple, obvious, natural facts. Of this order are the number of eggs in the basket, the number of sheep in the fold, the number of persons in the family, the number of citizens in the town, the number of soldiers in the army. For every unit in these classifications, there may be a real, natural fact; eggs are eggs, sheep are sheep, etc., and each one of these is a natural integer. This kind of mathematics is simple arithmetic, — primitive accounting, — and is very useful in dealing with elementary commercial and communal relationships.

Even so, the numerical fancies have little to do with the real facts, and only the fact that each abstract Fancy can be paired, in a one-to-one relation, with a natural, easily identifiable Fact keeps this basic unreality of arithmetic from being immediately discerned. Consider the superficial realities involved in the simple counting of objects. For example, it is possible that a man could own three hundred sheep without being able to count them. In a primitive society, the reality of his ownership is not at all affected by his inability to count. As a matter of fact, learning to count may confuse him and impair his grasp of the realities. For him to know how many sheep he owns, he must master a relatively abstruse system of abstractions called numbers. To do this, he must bring into correlated play two kinds of fancies about the sheep that graze beneath his eyes on the hill-side. For each sheep he must have both a Concrete image and an Abstract or numerical image. To use these images he must reflect twice and ultimately thrice.

First, he must reflect subjectively each actual sheep; second, he must reflect upon that reflection, — identify it as separated from the general image-idea of many sheep; third, he must mentally affix a numerical-symbolic idea or thought-form to the first two reflections. The first reflection is immediate and factual; every time he sees one of his sheep there is one image in his mind of the sheep. The second reflection is a long step away from the concrete fact in the direction of ideal abstraction; subjectively he divides his Many into two artificial groups, — 1) the one, separated sheep, and 2) the remainder. The third reflection introduces relatively pure abstraction: he assigns a numerical symbol to the counted sheep.

He must be able to project these three steps in abstraction and formulate a series of them, progressively separating the groups of the counted from each other and from the remainder. He must avoid assigning two or more number-fancies to one group of real sheep, and also avoid allotting two or more groups of real sheep to one number-fancy. Though practically useful (especially in exercising and developing the powers of imagination and abstraction), elementary mathematics is really very elusive; at the primitive level, the strong man who can’t count fingers and toes is almost certain to have a more realistic grasp of the sheep situation than the dreamer who can count to a thousand. The technical con-fusion of Concrete reality and Abstract fancy begins on the primitive range, and becomes more and more subtly confusing as men progress in sophistication, civilization, and learned ignorance of their origins. The more you know about fancies the less you know about facts, and the more elaborate your fancies for the superficial control of facts the more likely you are to lose realistic contact with the facts.

The basically fanciful nature of all mathematics becomes more patently evident when we consider artificial, not natural units. There are no grams, centimeters, ounces, pounds, pints, quarts, bushels, horse-power, megatons, or light-years in nature. In the invention of such units of measurement, man took a long stride forward in his characteristic work-play of mixing Abstract-Concrete cocktails. Out in the woods and the fields, there are no inches, feet, miles, acres, or sections. These, and all such fancies, including counties, states, and nations, are pure fancies imposed on natural reality. They possess a secondary reality so long as their inventors score the face of nature with these fictions and abide by the fictions in dealing among themselves with the realities. These fancies enable men to handle space-matter as if the fictions did exist in natural fact. The plastic nature of reality/matter makes it possible for men thus to realize their fancies. (Note this fundamental for future reference.)

Although they love these fancies and often mistake them for true realities, most literates of the modern world, even the most scientifically educated, are able to understand and, under gentle but firm pressure, admit that all the units of artificial mensuration I have just mentioned are pure fancies with no reference whatsoever to any natural realities. This nearly universal but sometimes uncertain and often reluctant sanity disappears, almost totally, at the next level of abstract-concrete, systematic, technical con-fusion. Hardly anyone can realize the essentially fanciful nature of the more complex practical inventions, the ones in which mathematical imagination plays a hidden, though never truly subdominant, role. That houses, automobiles, airplanes, typewriters, skyrockets, cathedrals, telephones, radio, TV, books, H-bombs etc. are, au fond, pure though practical fictions, relatively unreal, is an idea that strikes most minds as having all the earmarks of being a suspicious and probably dangerous foreign-er. They just “don’t get it”, and seldom can be induced to consider it seriously, even for a second.

This point is worthy of some illustration. Men much prefer fanciful fabrications to natural realities, manufactures to living creatures. Tell the proud owner of a Cadillac that his shiny car is a mechanical myth that would disappear from the earth if men did not continue to believe in it and serve it, — then watch with cool and scientific but compassionate eye all the mystical and metaphysical antics the devotee will perform, all the absurd fairy tales he will tell, in the defense and adoration of his sacred cow. (Automobiles in the United States surely are as sacred as cows in India.) The idea that his house is not at least as real as any of the trees that went into its construction would strike a professor of literature as a piece of poetry, unworthy of even such masters of line typical perversity and obscurity as T.S. Eliot, H. W. Auden and e. e. cummings. In truth, all the products of men’s arts and sciences are the practical Idols of the mystique of technology. Their secondary reality is unquestionable. Nevertheless, they are masks that disguise pure fancies, the offspring of the re-creative imagination functioning in relative ignorance of prime reality, the outer children of the inner marriage in confusion of Concrete and Abstract thought.

This note is designed to dissipate the confusion and reveal the real facts in their right relationship, — insofar as fiction about fact can do that. This is a serious/humorous undertaking, a comedy of intellect-ion which could easily become a farce or a tragedy if I fail to realize, at any point in the narration, that I am exorcising fancies with fancies, fictions with fiction. I may not be able to clarify your mind, but at least I should be able to avoid confusing myself.

THE preliminary picture is far from complete. We have yet to survey the somewhat special field of money and the 1-2-3 levels of the Higher Mathematics, but I pause to refocus the attention and more clearly define the scope of observation-inquiry.

I have defined the factor in natural reality that makes it possible for men to realize in local practice their fancies as the Plasticity of Form/Energy: Matter. Protyle is fictile. Hyle is impressionable, and may be molded; it receives and to a ponder-able degree sustains-retains artificial forms. Basically, the relationship between the two, between Art and Nature, is not difficult to observe and understand. Men’s artificial forms and formulas are no more naturally real than the sand-castles built by children at play on the seashore. They are a little more practicable, a little more durable; that is all. The children’s hands, their pails and their shovels are essentially identical with their elders’ mathematical formulas and technological designs. The work of men and the play of children are, at bottom, the same thing, but there is a difference between the men and the children. Children never more than fancy that their sandcastles are real castles, but men are completely deluded by their idols, and only in fits of despair or aberration ever surmise that all their arts and sciences and religions are nothing more real than daydreams.

Man’s infatuation with his fancies is an important fact. The realist who doesn’t realize this is not very realistic. Only facts can really dis-illusion, and so to You who are now reading these words, I say: Before you disagree with me, make sure that it is with Me and not with your own mistaken idea of Me that you are dis-agreeing.

I shall not always do so, but at this point, I am going to call my next shot. I am going to deal with the subject of Money in a somewhat fanciful and disorderly fashion.
Men’s deluded worship of their idols, their absurd adoration of the works of their hands directed by their re-creative imagination, is most easily analyzed and illustrated in a consideration of that masterpiece of mythical-mystic mathematics, Money. In Money, man has come as close as he ever will to manufacturing something that is nothing. Plato shot his immaterial Idea into the Attic air one balmy afternoon two thousand and four hundred years ago; in that hour High Finance was born in the mind of an anonymous Greek, who happened to overhear Plato’s immaterial prattle, and has been going strong ever since. Compared with the nonsensical metaphysics of the mint and the bank, the systems of hyper-spacial hocus-pocus invented by Buddha, Saint Augustine, Saint Thomas, Swedenborg, Calvin, Missus Eddy, and Einstein are infantile. When it comes to converting fancies into facts, Freud and Marx and the Pope are tyros compared to Santa Claus, the Secretary of the Treasury, and the girl at the cash register. If there is any matter in creation in which “there is no life, truth, intelligence or substance”, it is Money. It is far more holy than holy water, and when it comes to stimulating mass action and class consciousness, it reduces the Communist Manifesto to muttering and imbecilic impotence; yet it is a Thing that is next to No-thing, a some-thing like the square of the square root of minus 1. It is mathematics made tangible, “in”finity made finite, pure poetry made ponder-able, — the most popular fanciful fact in the whole Pantheon of technological mystification and practical metaphysics. No one in his right mind would exchange the real figure of a fine horse for the figure 300, whether that number was stamped on a piece of paper or metal or not.

What is the real value of any coin, of any unit in any monetary system? No one has ever been able, no one will ever be able to give any save a purely fanciful and basically nonsensical answer to that question. The only realities involved in the exchange of 1 penny for 1 lollipop are a childish lust for sugar and an adult lust for counterfeit. Money is magic, myth. Its very nature is that it is fiction and has no real value, unless you equate it to the complex of greed-fear-ignorance-superstition-credulity that is its source and substantiation. Its hypo-stasis is irrational faith, — credit. It is the every-day religion in every land. Currency is the bread-and-wine which ceremonious belief transforms into body-and-blood. It is frozen sweat, manufactured muscle, the corporeal ghost of an incorporeal idea. It is a vicious circle of ciphers kept in motion by a ring of enchanted integers. At the highest level it is the hocus-pocus performed by the high priests of financial theology, as insubstantial as a pen-scratch; at the lowest level it is the sacred wafers, the Host made visibly veritable to reassure the unenlightened and half-doubting layman, — the government-banker’s Word made tangible coin so that incredulous Thomases may handle it and be convinced.

It is a system of signs designed to perform the continuous miracle of transubstantiation labor into goods and goods back into labor. It is the imaginative animal’s inability to understand and control his imagination. You cannot eat it, drink it, wear it, or build anything of it, but unless you are a mere animal or an angel you can’t eat, drink, wear or build without it. It is materialized mirage. The mint is the factory that turns out the great fairy tale, Money, the most charming of all fables. It is one with the serpent in the garden of Eden and Balaam’s ass, and greater than any such: though inorganic, it talks. Money is Time, the fourth dimension. Money is counterfeit. It reduces every real fact to a fancy. It is the most obvious proof that men are creatures who believe far more in Fictions than in Reality.

In finance you can observe and study the trinity of human ignorance, — Mathematics the ghost, Machinery the father, Money the son, — in impractical, religious-scientific action. Money is price-less and in-valuable. It is the delusive potential of everything, and you really don’t know any-thing or something until you know the nothingness of money. Men will remain highly civilized savages, very learned idolaters, deluded devotees of technological animism, so long as they are unable to project a communion not dependent upon money, so long as they fail to transcend faith in Fiction and come to a realization of Reality. It is obvious: those who really love lies cannot really know the truth. But man is the imaginative animal. Imagination is a part of the whole of reality, and all illusions, whether objective or subjective, are aspects of reality-imagination. Myths and other lies are re-creative “truths”, which the plasticity of Imagination/-Matter: Reality will substantiate so long as men believe in and serve them. The understanding and control of illusion-imagination is the negative complement of the understanding and control of fact-reality. And one must admit that such a project is not the province of mere poets, priests, professors of physics and politicians.

THE picture of abstract-concrete con-fusion is still far from complete. We have yet to consider the intellectual money current in the constellations of the Higher Mathematics, — in the heavens of the whole shebang of Im-Practical Symbolisms. This upper world is the realm ruled by the metaphysics of science; here we encounter the more rarefied of the systematized fancies employed by men in their Knowledgeable/Ignore-ant endeavors to understand and control themselves and their environment. In this field I define three levels of reference, as follows: 1) the Mathematics of Form, 2) the Mathematics of Energy, 3) the Mathematics of Mathematics.

Geometry is the ancient-modern core of the mathematics of form. The simple elements of geometry are point, line, plane, and solid. All four are meta-physical, seemingly hyper spacial, ideals above or beyond the gross world of real, three-dimensional things. The first three are merely imaginary, — pure abstractions, or, as the more erudite gobblygook has it, conceptions. (Note that term, conception, for subsequent reference.) A point has position (+) but no (-) dimension; it is the first positive-negative in scientific mythology. In the line the negative becomes a trifle more positive; it has length (+) but no (-) breadth or thickness. In the plane the negative becomes still more positive; it has length and breadth (+) but no (-) thickness. In the solid, ideal abstraction-projection becomes perfectly positive; Imagination here apprehends three-dimensional reality. All four are imaginary, and unless Imagination is operative at every point in natural Reality, there is no place in reality, save the head of man, where these elements of geometry may be found. Even in man’s head, they do not exist as defined; you cannot really imagine a non-dimensional point, a two-dimensional plane, a one-dimensional line, a solid composed of these. If you draw them on a three-dimensional blackboard with a piece of three-dimensional chalk, they will be basically three-dimensional objects. You cannot really isolate any one of the dimensions from the other two, or separate all three from anything. Subjectively, these abstractions exist in your concrete, three-dimensional body. By no magic of imagination can you really transcend three-dimensional reality.

At this point, Imagination and Reality, — “Mind” and “Matter”, — squarely confront each other. At any other point in this note, from the first sentence on, I could have engineered such a confrontation, but for two reasons did not choose to do so. First, in the interest of precise and comprehensive communication, I desired to pose this crucial meeting of the prime poles of the field under observation-integration against an elementary, general survey of the Order of the Whole, — against a picture in perspective of the major factors and their relationships. Second, I desired the inevitable confrontation to happen in relation to terms of such a nature as to facilitate exclusion from the focus of your attention of the greatest possible number of distracting associations. For example, had I precipitated the confrontation in relation to money, I would have compelled myself to formulate my exposition in terms most heavily charged with distracting associations. I not only did not choose to make money the verbal lens of focus, but in the interest of a strong contrast, I chose to treat money in a way both frivolous and disorderly. I trust my non-sense about the non-sense of money, –in which I said nothing about the price of labor as correlated with the price of eggs, and also, which is even more to the point, refrained from introducing the factor of symbolism, — will serve to define more clearly the very rigorous order of discourse I am about to indulge and divulge. (If all that I have said thus far does not make sense, it isn’t likely that anything I shall say from this point on will make sense either. So — ) … Now — Mind is the most general term for the subjective half, and Matter is the most general term for the objective half of Reality conceived as an Organic Whole. When Mind squarely confronts Matter, what is the first thing that happens? Answer: Matter is reflected in Mind. The best way to study the results of the confrontation is to conceive of Mind as a living mirror. And the very best way to illustrate this conception is to study a non-living mirror.

II  Reflections and Symbols

IMAGINE a large, clear mirror on the wall of a beautiful drawing-room. Step before this mirror, and examine it systematically. As you do so, realize that your reflective body-mind is face to face with its prime analogue in the relatively impersonal, objective field of overall reality. The mirror is full of images of the things before it, and your body-mind is full of images of the things around it. The mirror and your mind are similar and also dissimilar. (Note this Positive/Negative relationship, for it is a prime factor in the analytic Dis/Integration of Real/Imaginary form-energy.) Let us consider the similarities first.

The mirror is a three-dimensional plate of glass, to the back of which a coat of silver paint has been applied. It is the coat of opaque silver at the back of the glass that transforms the otherwise transparent glass into a relatively perfect reflector of light and the objects before it illuminated by light. The uniform coat of silver together with the uniform transparency of the glass makes the reflective medium, a brilliant blank. The silver is not (-) transparent, but the glass is (+); the two opposites complement each other in structural union to compose a surface-depth capable of reflecting the many different forms of the objects in the drawing-room. The brilliant blankness, the product of a uniform Positive in on-to-one opposition to a uniform Negative, is the mirror’s three-dimensional zero (0), the potential of all reflections. The first, simple facts about the Mind are that it is exactly like the mirror and does exactly what the mirror does. The mind is the reflective function of a three-dimensional body. The uniform darkness of the body’s interior and the uniform sensitivity of its exterior combine to create a reflective medium, basically neutral or “blank”, capable, like the mirror, of receiving and reproducing in-differently all manner of different impressions. The mirror faithfully reflects form-color-motion, and so does the mind; both have a relatively opaque back, and neither one can “see” objects that are not in front of it.

For more similarities, let us now examine the reflections in the non-living mirror. In the mirror is an image of yourself and images of all the chairs, tables, rugs, and other objects in the room back of you. As you stand facing it, the mirror is like a pair of eyes in the back of your head. The images you see in the mirror are so real that, apart from your experience with mirrors, you might mistake them for real objects. Indeed, in their own system of reference they are real objects, — objects tangible to your sense of sight. They are actual models of yourself, the chairs, etc., models created by light reflected in-from the brilliant blankness of a three-dimensional, silver-backed plate of glass. They appear in true but reverse perspective. They possess length and breadth, and a very real illusion of the third dimension. But they are dead images, and have no life apart from the life of the things they represent. Only to the sense of sight are they tangible. You can touch your nose with the tip of your finger, and the image of yourself in the glass will reflect that touch, but you cannot touch the imaginary nose in the real mirror with the tip of a real, not a reflected finger. If you know its powers and its limitations, the mirror is a faithful but non-living witness.

The images in the Mind are very like those in the mirror. Within their system of reference, they are very real; primitives sometimes find it difficult to distinguish between dreams and objective realities, and civilized men find it all but impossible to realize that their artifacts are projected dreams. Mental images or thought-forms are brain-cell patterns created in the electromagnetic field of your three-dimensional body by stimuli originating in the world outside the body-mirror-mind. If you are not one of the unfortunate ones who do not have subjective visual projection, you can close your eyes and see images of all the things around you; just as you cannot touch the images in the wall mirror with your actual hand, so you cannot touch the images in your mind with your actual hand, but, just as a reflected hand will touch a reflected thing in the mirror, so you can touch the images in your mind with the hands of imagination. Provided you know their system of reference and their limitations, the real illusions of mental reflection are faithful models of three-dimensional realities.

These are the first, simple Facts about Fancy, — the first Realities of Imagination, — the first Matters of Mind. The more complex facts about the re-creative imagination come to light in the study of the dis-similarities that distinguish the living from the non-living mirror. The dissimilarities are radical and far-reaching. The reflection in the mental mirror is not only a subjective impression, but an objective projection; you see things by means of a two-in-one reflection, — a reflection that is at once subjective and objective. In this respect, the mind is less like a mirror and more like a motion-picture camera, — a camera that instantaneously takes, develops and projects moving pictures. The universal sensitivity of the mirror is limited to light, but the universal sensitivity of the body-mind is a complex of light-warmth-touch. Moreover, the body-mind-mirror has special organs of sense; it can reflect not only form-color-motion, but sound, savor, and odor. It does not merely mechanically reflect; it (un)consciously re-presents. Unlike the wall mirror, it can retain and recall images. The Vital Factor in the human body-mind transforms reflections into recreations, by a process of conscious awareness. This process is a vital complex of Three orders of progressively integrated reflections: 1) you first reflect the object, producing a subjective image; 2) you reflect on the image, transforming it into a subjective object, from which you abstract properties, qualities, parts; 3) you re-reflect on the image and the original object, a process that establishes Identification and evolves Nomination. These three-in-one orders of conscious reflection amount to re-creation, — the ability to project into the objective field not only images-models of the things in that field, but images and models of things that have their origin, not in the objective field of Nature, but in the subjective field of Art. The first cycle of the re-creative reflective process ends when the Re-creator projects into the objective field an image of the image in his mind of the objective thing. The archetype of this end-product is the Picture. The vital mirror in man 1) sees, 2) reflects, 3) recreates, and 4) projects, — an image, a picture. And the picture, once projected, becomes a symbol of both the objective thing and the subjective image thereof.

In all this, we discover four orders of vital reflection or re-creative reality: 1) the reality of the objective Thing; 2)the reality of the subjective Image; 3) the reality of the subjective-objective Re-Construction; 4) the reality of the projected image conceived of as a Symbol. All four are interdependent, but each is prime in its own field. In none is there anything less or more than three-dimensional. These four-in-one realities are present in any account of anything. You can’t count anything without using these four realities at every point in the process of counting. This is the Re-creative Order of the Whole of Reality, and it is impossible to abstract any of the four-in-one realities and treat it as an isolated reality. As you stand before the mirror on the drawing-room wall, all four of these realities are present in one: 1) there is the objective whole which includes yourself and the mirror; 2) there is the reflection in the mirror; 3) there is your reflection of there flection in the mirror; 4) there is your re-creation of the Order of the Whole, which includes yourself and the mirror. You are the ONE that integrates the four-in-one recreation of the three-dimensional, objective-subjective universe.

APART from the vital factor which empowers the mind to perform these are creative operations, the whole “mystery” of the relation of the Mind, or subjective field, to the objective field, or Body, is now cleared up. Nowhere in either field can be found anything that is not embodied in some portion of three-dimensional Form/Energy:-Matter. Organic body and organic mind are complementary “halves”of organic matter. Immateriality is unimaginable, and all metaphysics, religious or scientific, is absolutely anoetic.

In arithmetic, mathematics, and geometry, as in mysticism and theology, misunderstanding of reflections, abstractions, and symbols induces belief in ghosts, spirits, pure ideas, and incorporeal principles. There are no body-less entities, — no non-dimensional points, no immaterial powers. All such are products of mirror-land, — reflections, imaginings. All the abstract-symbolic thinking in the universe takes place in concrete heads, and heads that imagine that such no-things have any save a symbolic reference to overall reality are very concrete indeed. You can’t shake hands with your image in the mirror; for this reason, you will never really see a spirit or a ghost, — an event apart from a material eventer.

The prime gimmick in religious-scientific metaphysics is the symbol. The fact that you can write-speak such terms as immaterial, non-dimensional, and infinite might mislead you into opining that there really is something immaterial, non-dimensional and infinite, — If you use these terms, — these verbally manufactured negations, in a positive way, you talk non-sense, not sense. You can forgive the monkey for mistaking his reflection in the mirror for another monkey, but you shouldn’t make the same mistake yourself. (In this, I am patently talking to myself, and the you addressed is my alter-ego; as for you, you are free to do as you please, of course.)

However, the fact that re-creative imagination is a prime function of the human body-mind, which is itself an integral part of the Order of the Whole Universe, is a certain indication that imagination-conception and abstract-symbolic design are involved in the order of Prime Reality. Before the end of this discourse,I shall come to grips with this crucial fact.

There is no Form apart from Energy, no Energy apart from Form. Light, electricity, magnetism, and imagination never manifest themselves save in the three-dimensional manifold. Inseparable form and energy spell matter, — ponderable, three-dimensional some-thing. Reality is some-thing, not no-thing, and matter is the most familiar, general, impersonal term for that some-thing. Matter reflects itself, both in the organic and the inorganic systems of reference. When you look at the moon and its reflection in the still lake, four moons are present, one of which is the real, objective moon, one of which is the real, objective reflection of the moon, the other two of which are subjective reflections, one of the moon itself, the other of its image in the still waters.

That is all that I need to say at this point about the mathematics of form.

COME we now to consideration of the Mathematics of Energy, — to an analysis of the abstract-symbolic models men have invented and use for the exploitation and control of the Power in all real Form. Zhe best introduction to this subdivision of real-imaginary mathematics is a brief history of the evolution and use of abstractions and symbols. The first symbol was a picture. It may have been simple or elaborate. It may have been drawn with a finger in moist sand or with a charred stick on the surface of a flat rock. It represented some object in nature, — a man or another animal. It was an ideal model, an outline reconstruction, and it did not exist in the objective field until the mind-and-hand of man put it there. Its making was accompanied by human sounds. All of man’s arts and sciences were involved in the making of that first picture, and from it, by a process of progressive simplification and stylization, have come all symbols. Obviously, it was not the real object which it merely re-presented, and therefore, even there in the beginning, was nothing more or less than a symbol. And it is significant to note how intimately related the ultimate symbols are to the first pictures and to all form, abstract or concrete.

T was involved in the first picture of a tree, and so was Y; M was somewhere in the first picture of mountains; W was there in the first picture of waves; H was part of the first plan for a house; O was the first picture of the sun; S is still a pretty good picture of a serpent; and I, the straight line, the universal symbol for one or first, was in them all. All that man, — first or last, — thinks, says or does is immersed in reflection-abstraction-symbol.

Over the centuries, men have developed many different systems of symbolism. Without Something that stood for Something-else, they would have no way to relate their reflections to reality or to communicate with each other at any level above the basic semantics of the brute. To rise above the jungle vocabulary of grunts, groans, shouts, slaps, blows, nudges, grimaces, and gestures, men had to invent articulate symbols, a higher order of elementary signs. This they did. The picture was the first symbol, and the Word, for all practical purposes, was its coeval. The non-verbal picture was accompanied by the verbal equivalent. All systems of symbolism are either Verbal or Non-verbal. For every word there is a non-verbal reference, and for every non-verbal sign there is a verbal reference. There are many, different systems of non-verbal symbolism,but for every sign therein there is a word. It is therefore accurate to say that all of man’s symbolism are Non/Verbal (another example of the Positive/Negative complex at the base of all reflect-ion.)

In general, the non-verbal systems of symbolic reference consist of two, closely inter-related systems, viz., the Spoken and the Written (or printed). Since all symbols are either words or signs for which there are words, it follows that the one, general sign for all human symbolism is Word. The Word is man’s prime tool. The word is a name. All words are names, — the names of objects, groups of objects, actions, motions, qualities, quantities, places, relationships. The distinction between Words and Numbers is nominal: 1, 2, 3, 4, etc. are the non-verbal or mathematical signs for words that name the imaginary-abstract operations indicated. This is equally true of all other mathematical signs. Mathematical formulations are merely short-hand versions of verbal exposition-narration. It is well to remember this, whenever you are tempted to mistake a blackboard or a book full of mathematical formulas for anything other than signs for material-mental operations, — for abstractions divorced from concrete demonstrations.

From the first, numbers were more particularly associated with and referred to power-energy. The number of sheep in the fold,the number of horses in the barn, the number of lamps or candles in the house, the number of children in the family, the number of shekels in the bag, the number of men in the army, — these are simple computations of obvious power. Similar or identical examples of the elementary mathematics of Energy may be found on virtually any page of modern newspapers, magazines, and books. Such formulations compute superficial, natural and relatively unorganized energy. For thousands of years the race’s interests and needs were satisfied with nothing but Obvious Energy and the Simple Arithmetic required for its symbolic calculation and direction. Windmills, water-wheels, and sailing ships, though crude power machines, called for no great understanding or computation of the energies exploited. Neither, in the long days of their retarded development, did levers, screws, pulleys, etc. The eolipile, an ancient toy prototype of the steam engine, failed to interest primitive mathematicians in speculations that might have led to the invention of internal-power machines hundreds of years before their time.

Later, the invention of gunpowder and the telescope kindled no minds on record at the time to imagine ways to tap the energies hidden in all forms-modes of matter. It was not until a little more than a hundred years ago that man began to investigate the inner energies of matter, and invent ways to harness and use them. All the mathematical formulations for the generation and application of steam-power, oil-power, electrical and atomic power have been imagined during the past century and a half.

The exploration and exploitation of non-organic energies, of course, was accompanied by extensive and intense inquiry into the structure and nature of organic energies, but the work in the second field, though productive of much important information, has led to no formulation-s for the generation and release of vital energy. During the past hundred and fifty years, men have learnt more about physiological-biological fact than the race learnt during the whole of recorded time down to the XIX Century. Much as that is, it is not concerned with the Vital Factor, the central and prime reality. The Vital Factor is utterly ignored; it is as if the one thing that makes the difference between a pile of sand and a professor of semantics did not exist. (Maybe the difference isn’t as great as I think it is.) The difference is generally unobserved, and so today, in the hour when the manufactured release of atomic energy threatens to destroy the race (perhaps the planet), nothing, — absolutely nothing, — is known about the power of re-generation that was not known, in practical essence, to the ancient Chinese.

The human race ignores the one thing it ought to be most interested in. For that, it has one, shame-faced symbol, — sex.

ALTHOUGH seemingly better informed about the relatively non-vital energies, men are, in reality, just as ignorant of the realities involved here as they are of the realities involved in the field of vital energy. The authorities in the fields of physics and chemistry freely confess that they do not know what really are electricity, magnetism, gravity, — electrons, protons, photons, neutrons, atoms, and molecules. They intimate, and often explicitly state, that they are not interested in the What, only in the How. And there is no denying that they have considerable know-how now,– at least enough to blow themselves to Nirvana, if not enough to keep from sacrificing thousands of lives annually to the Moloch of Machinery. Their imagined formulas for the release of real energy are practically very efficient, to say the least. Elements and compounds respond as if atoms and molecules were present there-in according to the formulated patterns; the plasticity of matter sustains these imaginings as surely and neatly as water takes the exact form of any receptacle from a thimble to a tank. There are no ohms, volts, amperes, etc. in natural reality, but the Reality does not reject these Fictions. It is now evident that Matter, like Mind, has depth reflection and something very analogous to vital response. If you poise the model of a molecule before the mirror of Matter at the right angle, Matter will reflect that model-symbol. Just as you can project a picture on paper or a motion-picture screen, so you can project into the matrix of matter a dynamo-graph or mathematical formula of energy, and get the predesigned results. Within certain limits, you can impose on Nature-Matter, rape it, and get such bastard children from the enforced mathematical-mechanical nuptials as TV, the airplane, and the H-bomb. But it is not-able that all the imaginary knowledge thus far formulated swiftly goes beyond the limits of Nature-Matter’s tolerance, and ultimately proves itself to be destructive.

Each new discovery, each new masterpiece of fancy formulation, each new invention implementing the fancy is always heralded as a boon or a potential blessing,but without exception, each has proved to be destructive of truly human qualities, of true natural harmony, an unmitigated menace to all life on earth. You can imagine that the speed of light is 186,300 miles per second, and matter will reflect your formula in a way that will deceive you into believing that the image you have held up to the mirror of universal form-energy formulates the reality, but if you persist in foisting this Fiction upon the field of Reality, the real facts will explode the fiction and expose your ignorance. This is the truth, though you may not live to discover it. The obvious reality is that the universe is not a mathematical machine; it may be treated as such for a brief time, with seeming impunity, but in the end the result, invariably and inevitably, is a highly destructive explosion of unformulated Energy.

IT is unnecessary, here, to analyze the never-never-land of mystical geometry and mathematical metaphysics, countless bibles of which you will find on the shelves of the most holy temples of what is called (most solemnly) the sciences of physics, chemistry, astrophysics, and social psychology. The vain imaginings in these volumes are as unconsciously absurd as the speculations of Medieval mystics, alchemists and Schoolmen, but they are far more practical and this-worldly. The miracles of modern machinery, manufactured from the mundane, symbolic visions in their pages, prove that matter, though not informed by conscious awareness, is in its way as volatile and plastic as mind; these machines prove that if you can imagine it, you can find a way to materialize it, whether it be heaven or hell or an in/sane mixture of both. One glance at the mystical mathematics, the maze of symbols to be encountered on the pages of this order of scientific anagogic, — one glance at these precisely diagrammed prayers of devoted technologists, thrown off in the throes of intense mental effort to realize an ideal-statistical-mechanical construction, — just one such glance overwhelms one with the surmise that one is examining the graffiti of lucid lunatics, the rational ravings of dangerously intelligent madmen. This surmise is absolutely confirmed when one turns from the books to examine the objective results of these cortex-curdling, hair-splitting and hair-raising, almost hyper-spatial imaginings, — turns from these typographical terrors, surveys the field in which these fancies have found objectification, and beholds the hosts of modern,mythical-mechanical monsters, — beholds the droves of countless chimerical constructions, the armies of fire-breathing, automatic dragons, — the myriads of engines that now affright the air, the earth and the sea, and devour the minds and the bodies of men, the world around. I repeat, you can materialize a fancy if you fancy the matter analytically and systematically enough, but you can’t realize reality so long as you permit reason to be the slave of either religious or scientific symbols and illusions. And not the least of the monstrous illusions modern men have materialized are the machines of social order, the totalitarian-democratic church-states of communism, fascism and republicanism, all of them the products of psycho-mechanics, the application of statistics to human values, the reduction of men to the status of numbered things in a politico-ethnic-economic equation. The social psychologist is the ultimate engineer of human power, the technological thaumaturge who now seeks to impose his vain Imaginings upon the Reality of man.

It is not necessary, here, to do more than mention these matters, for the essence of the whole matter is to be found in a consideration of the Mathematics of Mathematics, — the mathematics that utilizes and formulates the highest mental energies generally known to man, — terrestrial-celestial mathematics, so to speak.

At this top-level, in the very heavens of the abstract, mathematics encounters itself, mirror face to mirror face, and reflects upon itself in-finitely. At this level, the basic fact, that mathematics is a product of the imagination, ends and loses itself in an end-less maze of count-less mirrors. I define in this field three levels of reference, as follows: 1) the mathematics of space time; 2) the mathematics of number-s; 3) the mathematics of infinity. I consider them in that order.

The least “pure” of the three is the first, for it is chiefly concerned with the nature-structure of the universe as conceived-observed by men on Earth. Except in the common sense of the word, space is imaginary, an abstraction as unreal as the non-dimensional point. As a matter of fact, generalized space is merely an extension of the non-dimensional point, — the brilliant blankness of the mirror conceived as not reflecting anything. Vacuum and 0 do not exist except as convenient symbols of general potentiality, distribution, relative quality, quantity, and composition. Three-dimensional objects are not in space; they are real space: the reality is an end-less but not in-finite manifold of three-dimensional objects. Some of these three-dimensional portions of the universal, three-dimensional manifold are very elastic, even diaphanous, so to speak, but all of them, including the finest gases, are essentially three-dimensional. The universal, three-dimensional manifold itself is a very elastic continuum, capable of in-definite but not in-finite expansion and contraction, about which I shall have some-thing to say in the closing pages of this discourse. Common space, like common time, is such a familiar reflection-conception, — one that is so firmly fixed in the mixture of every-day, symbolic reference, — such a real illusion, — that all but all men would find it impossible to dis-illusion themselves about it and come to a realization of the reality. As commonly conceived, space, time, and money have a lot in common.

Continue to Part 2