What’s the Point?

Preston Harold gives us a mathematical lesson on the concept of a “point.” In this installment we will quote him at length as we prepare to discuss how Jesus understood the concept, which we will explore in the next post. Now, for your pleasure, Preston Harold:

Zero must be seen as the whole, beyond examination, and therefore its measure, unapparent, is expressible only in negative terms, so that zero’s division must correspond to: (-/- = +). But this simple division is equivalent only to taking the diameter of a circle; to define the center point in its being, the product of this division must be divided by itself, so that the whole equation of zero-divided must correspond to: (-/- = +/+ = +). Again, the positive sign (+) is presented as answer, just a s the configuration of the cross appears when one determines the center point of a circle by bisecting its diameter. The center point defined by the cross cannot be seen as a correspondence to negative-one or positive-one; it is neutral in its position and must correspond to one-neutral for one-whole. The point itself cannot be defined, except as it is defined by the cross (+), that is, in the definition of the cross itself.

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Nicomachus said that a point is “the beginning of a line, or an interval, but is not itself line or interval.” The point enfolded in the cross is not the beginning of either line, but is an interval in both lines, and is the one point so arranged and sustained by the opposing horizontal and vertical lines or “forces.” One might say that this point is defined in negative, positive, and specific terms, so that it is the only point in actual being.

Until next time, peace.

The Cross as Mathematical Solution

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When all of Jesus’ statements are applied mathematically, and when one considers the full import of His presenting the cross (+), the positive sign, as His symbol and all-embracing answer it would appear that His mathematical concepts moved beyond those of His day, and that He attempted to describe the reality of the operation of the energy that mathematics attempts to describe…this study poses His words against the concepts of Nicomachus of Gerasa…both He and Nicomachus use “father and son,” and “teacher and pupil,” to symbolize “greater and the lesser,” or the concept of opposites, unequal; but in Jesus’ concept these opposites meet and are reconciled in the unity of one, or sameness.

Here Preston Harold describes the similarities between the teachings of Jesus and Nicomachus, striking as they are. But whereas Nicomachus writes that “when a point is added to a point, it makes no increase, for when a non-dimensional thing is added to another non-dimensional thing, it will thereby not have dimension…,” Jesus took another point of view.

This is to say, when the cross (+) as symbol and all-embracing answer is applied to the problem of “sum of nothing added to nothing,” it would say that Jesus did not overlook the word added – the symbol (+) introduces the problem of organization, or, as Eddington put it, of “and.” The cross as answer indicates that there cannot be a “non-dimensional thing,” –or, one might say, until a point has dimension or until it is defined, a point is not a point: the cross has dimension and it also defines a point, or gives a point dimension. The cross (+) as answer and symbol calls forth the concept of negative (-) and positive (+) numbers.

Harold now goes onto describe how one becomes itself the unit of measure when seen in contrast with zero. It shifts in its capacity from a mere digit to an active function, from a noun to a verb.

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In the view of this study, Jesus realized that the correspondence between one and the whole, zero, and between one and each other ensuing digit, differing only by one, makes of one, measure itself. This is to say, one is not merely a unique digit: it is, rather, a principle, or action involving opposites, “minus-one” and “plus-one,” upon which the whole operates. Thus, Jesus saw that in the definition of one as an operative principle, the definition of the underlying principle upon which one and all operate could be grasped. And as Jesus examined His own mind the “sum of nothing added to nothing,” which must perforce involve the division of “nothing,” or the whole or zero, the positive (+), the cross, arose as the only possible answer to the problem of “naught divided by naught.” This, because the problem itself is posed in terms  that may be seen only as a negative divided by a negative which produces a positive answer. Or, one might say, that if through use of zero-0-an infinite increase in number may be drawn, then unlike the number one, when zero multiplies itself it produces more than itself: it must forever reproduce itself plus.

Until next time, peace.

Jesus’ Mathematical Influences

Preston Harold speculates on other means by which Jesus may have acquired his mathematical knowledge.

But one does not have to look altogether to the unconscious for Jesus’ source of mathematical knowledge. Within His reach was Alexandria, the center of mathematical studies and of Neo-Pythagorianism. Here, Nicomachus of Gerasa, one of the “golden chain” of philosopher-mathematicians, is presumed to have studied, for Gerasa was a city in Palestine, primarily Greek – it is near to the place where Jesus cast demons called “Legion” into the swine – and it is probable that Nicomachus did not receive all of his education there… Nicomachus is thought to have flourished between the middle of the first and second centuries, but it is possible that he was a contemporary of Jesus, and he could have brought Alexandrian mathematics to Palestine, placing his knowledge within easy reach…

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Many of Jesus’ statements regarding one reflect Nicomachus’ thinking, which, in turn, rests upon the mathematical knowledge of his day. Nicomachus had much to say of one, which he saw as unity. Jesus’ mathematics came to rest in His concept of one, which appears to have arisen from His grasp of the operation of signed numbers and the concept of zero.

Zero, that non-number number that is both nothing and everything.  At the end of his introduction to his book, “Zero: The Biography of a Dangerous Idea,” Charles Seife writes, “The clashes over zero were the battles that shook the foundations of philosophy, of science, of mathematics, and of religion. Underneath every revolution lay a zero – and an infinity… Yet through all its history, despite the rejection and the exile, zero has always defeated those who opposed it. Humanity could never force zero to fit its philosophies. Instead, zero shaped humanity’s view of the universe – and of God.”

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Preston Harold writes:

About the time Euclid was stating his axioms (300 B.C.) an unknown scribe jabbed into a wet clay tablet a point to make the space that zero would come to occupy about a thousand years later when Hindus brought to the court of the Caliph of Baghdad the digit 0, still used today. To the mathematician, zero – 0 – is indeed a perfect pearl for the possibilities opened through this symbol are limitless. Did the digit 0 take shape in Jesus’ mind – or was it another gift of the Magi? In speaking of the “eye” of the needle, Jesus called to mind this configuration: 0, and related it to “naught,” for the “eye” of the needle is the  “nothing” of it that makes it operable; and in this enigmatic statement, He brought God, the absent or “minus” one into correspondence with man, the present or “positive” one, and brought both one’s into correspondence with this “hole,” or whole of “nothing” that takes on a “circular” shape, through which God, “minus” one, draws man, “positive” one, into infinity. Through this correspondence, any one-thing is vested with zero’s enigmatic, unmeasurable properties. But Jesus appears to have realized that although one and zero are corresponding unities, they are not the same in action and reaction.

It is this difference between one and zero that we will look at in our next installment. Until then, peace.

Jesus the Mathematician?

We now begin to look at Jesus’ ministry in quite a unique, “unorthodox” way; as that of a mathematician. Rest assured, when one considers how often Jesus used the number “one” as a part of his teachings, one must wonder at his mathematical knowledge.

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Jesus has never been considered a mathematician. He made but few statements dealing with number. Yet, He made many statements about one, the number that is the basis of arithmetic through which all branches of mathematics become possible. If He described one’s inner structure and the principle upon which one, as measure, operates, He was a mathematical genius. How could this come from the man of Nazareth?

How indeed? At this point I am reminded of the 12 year-old Jesus amazing those present in the Jerusalem temple with his questions, answers, and understanding. From where does his wisdom come?

The realization could have arisen from His unconscious, as has been the case with other great mathematicians. Jung felt that a fruitful field for further investigation was the study of man’s basic “mathematical axiomata – which Pauli calls ‘primary mathematical intuitions,’ and among which he especially mentions the ideas of an infinite series of numbers in arithmetic, or of a continuum in geometry, etc.” Dr. von Franz writes that “William James once pointed out ‘the idea of an unconscious could itself be compared to the ‘field’ concept in physics.’” She says:

“In other words, our conscious representations are sometimes ordered (or arranged in a pattern) before they have become conscious to us. The 18th century German mathematician Karl Friedrich Gauss gives an example of an experience of such an unconscious order of ideas: he says that he found a certain rule in the theory of numbers, “not by painstaking research, but by the Grace of God, so to speak. The riddle solved itself as lightning strikes, and I myself could not tell or show the connection between what I knew before, what I last used to experiment with, and what produced the final success.”

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Keeping in mind that for Harold, the Father dwells in the unconscious, and that Jesus “can only do what he sees his Father doing, because whatever the Father does the Son does also,” and “My Father is working until this hour, and I am also working,” and one can clearly see that Jesus is receiving his mathematical revelations from his unconscious, or Father. But is this the only explanation for his mathematical genius? We will look at other possibilities in our next installment. Until then, peace.

Striving Towards the ONE

To see how the three and four might be transcended we begin by looking at Erwin Schrodinger’s observations of cell division.

In his work, What is Life?, Schrodinger is not concerned with investigating the ancient dilemma of three and four, but he describes cell divisions, and in his descriptions one sees that the “triadic” family structure, mother-father-child, is involved with a “tetradic” pattern – as Schrodinger discusses the hereditary “code-script” that rests in the chromosomes, he says:

…this whole four-dimensional pattern is known to be determined by the structure of that one cell, the fertilized egg.

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He points out that physical laws “rest on atomic statistics” and their “precision is based on the large number of atoms intervening,” whereas the living organism is under the control of “incredibly small groups of atoms, much too small to display exact statistical laws,” but they play a dominant role, control observable large-scale features, determine important characteristics of its functioning, and “in all this very sharp and very strict biological laws are displayed.” These laws insure that each one is always an event in himself, unto himself.

Here we can see that the three and the four both exist within the one and work together towards becoming one.

Old folk-wisdom has also acknowledged the predicament of the three and the four…

The strangest thing about the dilemma of three and four is that some time ago those on the side of three, embracing Euclidian geometry which says that the whole is equal to the sum of its parts, were contradicted in the nursery:

Humpty Dumpty sat on the wall,

Humpty Dumpty had a great fall.

All the King’s horses and all the King’s men

Couldn’t put Humpty together again.

This bit of wisdom is restated by Kluckhohn, the anthropologist, “A whole is different from the sum of its parts,” and it is restated by Koestler, “A whole is defined by the pattern of relations between its parts, not by the sum of its parts…”

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It is also worth noting that the rhyme has also been interpreted as representing the second law of thermodynamics in that after his fall and shattering, the inability of the most powerful men in the world to reassemble Humpty is representative of the high unlikeliness to return him to his previous state of lower entropy as the entropy of an isolated system never decreases.

And in the nursery, those on the side of four, insisting upon the qualitative aspects of the whole, were reminded that time changes the aspects:

Hickory, dickory, dock.

The mouse ran up the clock.

The clock struck one, the mouse ran down.

Hickory, dickory, dock.

In these rhymes both the devotees of three and the devotees of four were given the clue: the whole must be described in terms of one, whole, and one must be seen as a working principle through which as time changes the arrangement of material forces within and without, one remains, itself a unity and the measure of unity.

This is to say, the answer to the dilemma of three and four rests in the answer to the mystery of one. This answer must be given in a mathematical statement that describes the composition and inner operation of one, itself. Jesus stated the Equation of One which must be seen against the briefly sketched background of the dilemma of three and four in order to appreciate the magnitude of His thought, for He transcends the dilemma, giving as the measure of one or wholeness, the number five.

We will begin to look at “Jesus the mathematician” in our next post. Until then, peace.

A Musical “Number”

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Merriam-Webster’s Dictionary defines music as “the science or art of ordering tones or sounds in succession, in combination, and in temporal relationships to produce a composition having unity and continuity.” Notice how “science” is used as part of this definition, along with the usual category music falls under, “art.” Any musician knows that music’s aural qualities can be described in terms of numbers, for instance the way rhythm or time is notated – ¾, 4/4, 6/8, etc. Most western music is based on a 12 tone chromatic scale, with numbered intervals denoting the spaces between the tones. There are major thirds, minor thirds, flatted fifths, dominant sevenths, etc. Numbers play a major role in how musicians communicate with one another and notate their compositions. Of course music cannot be reduced to numbers as they leave out the emotional and immaterial impacts of the SOUND, which can lead us to a higher plane. As a musician myself I know the importance of understanding theory, but also understand the fact that theory must be secondary to the actual aural experience and feelings invoked by the tones themselves.

Preston Harold explains the dilemma of the three and four and how it manifests in music.

The dilemma of the three and four is not confined to the world of science. One comes upon the play between three and four in the world of art. Here one sees that musicians, like ancient alchemists, strive to bring together four “elements” to make possible the quinta essentia, the lapis, that brings forth a new element or a whole new statement. Although musicians do not discuss this in terms of the dilemma of three and four, one can hear the play between the two numbers in Beethoven’s Symphony #5 in C Minor.

This symphony begins with a four-note rhythm, described by the composer as “destiny knocking on the door.”  Deems Taylor calls it an “ominous” signal. This signal announces an agitated main theme which changes to suppressed tragedy, to resignation, to a lyrical movement during which the dread reminder of the four-note rhythm is played. It continues as a sinister reminder throughout the work. The third movement is a dream of terror wherein the four-note beat “sounds noble,” but the nobility fades and it becomes a caricature – the restated theme becomes a “macabre joke.” Reduced to its original skeleton, it becomes suppressed tension, increasing in rhythmical intensity, releasing tightly harnessed fury which becomes a bridge to the last movement. Here triplet note rhythms are introduced, artistic unity is achieved, the symphony ends on a one-two-three beat and a magnificent chord in a triumph of grandeur. In that chord is expressed wholeness: one sound based upon the number man’s hand is based upon – five. Abt Vogler says of music:

“I know not if, save in this, such gift is allowed to man

That out of three sounds he frame, not a fourth sound but a star.”

Perhaps Sibelius was trying to say this in another way in his mighty Seventh Symphony – here, symphonic expression is not given in four movements following each after the other, posed the one against the other: one movement encompasses the whole expression that transcends the dilemma of three and four.

The question becomes – how does life transcend its play between triad and tetrad to produce one, wholeness?

This we will begin to explore in our next post. Until then, peace.

Four Dimensions

Picking up where we last left off, our dilemma of the three and four continues into the realm of mathematics. Preston Harold explains:

In the mathematician’s view, the physicist deals with four continua, or – more precisely – with four dimensions. In geometry, four dimensions would mean that one has four independent directions. For example, this could be seen by drawing four lines through a given point, all perpendicular to the others. In this narrow sense the universe has only three dimensions. The mathematicians extended the concept of dimensions to any situation where events can be described by independent coordinates, and where certain simple laws hold. In this broader sense Einstein found it convenient to use four independent coordinates, with time playing the role of a fourth dimension. In pure mathematics, as well as in its applications to physics, it is often convenient to use many more dimensions, even infinitely many.

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But the idea of wholeness, or of continua itself, as one dimension greater than any number of dimensions has not broken through the tetragrammaton – through the confines of four. For example, Einstein speaks of the “bold” interpretation of the modern quantum theory associated with de Broglie, Schrodinger, Dirac, and Born – he says their interpretation “is logically unobjectionable and has important successes to its credit. Unfortunately, however, it compels one to use a continuum the number of whose dimensions is not that ascribed to space by physics hitherto (four) but rises indefinitely with the number of the particles constituting the system under consideration.”

For how long does the number not ascribed to space rise? Is it infinite? Jesus will have something to teach us here but before we get to him, we will detour into the world of music.

Until next time, peace.

Triad or Tetrad?

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Difference emphases on either the three or the four can be found within varying cultures, the beginnings of which are lost in the sands of time.

Although scientists move away from the quantitative view toward the qualitative view and acknowledge the validity of both positions, the dilemma of three and four is by no means resolved – its beginning is lost in antiquity and its end is not yet in sight. As to its beginning, Jung says that number helps more than anything else to bring order into “the chaos of appearances…primitive patterns of order are mostly triads or tetrads,” and he points to I Ching, Book of Changes:

“…the experimental basis of classical Chinese philosophy…one of the oldest known methods for grasping a situation as a whole and thus placing the details against a cosmic background – the interplay of Yin and Yang… there is also a Western method of very ancient origin which is based on the same general principle as the I Ching, the only difference being that in the West this principle is not triadic but, significantly enough, tetradic…”

He refers also to the alchemists’ tackling of the problem of three and four, seeing the dilemma stated in the story that serves as a setting for the Timeaus and extending all the way to the “Cabiri scene in Faust, Part II…recognized by a sixteenth-century alchemist, Gerhard Dorn, as the decision between the Christian Trinity and the serpens quadricornutus, the four-horned serpent who is the Devil.”

Of course western religion and culture has been based on the tension between the three and the four, both being primary factors in the Holy Scriptures. The four is stated outright: YHVH, even translated into English as a four-letter word, LORD. The three is implied in the three visitors to Abrahm, the Christian Trinity, etc. Returning to alchemy’s approach of the problem, Preston Harold says:

Wolfgang Pauli discusses the controversy between Johannes Kepler, discoverer of the three famous laws of planetary motion, and Robert Fludd, in his day a famous alchemist and Rosicrucian. Pauli says that Kepler’s ideas “represent a remarkable intermediary stage between the earlier, magical-symbolical and the modern, quantitive-mathematical descriptions of nature,” indicating a way of thinking that produced the natural science which today is called classical. Kepler, a devotee of Euclid’s geometry, insisted upon strict mathematical methods of proof. His premise was that “Mathematical reasoning is ‘inborn in the human soul’…” His is a trinity-concept, his symbol “contains no hint of the number four or quaternity.” Fludd, however, was a mystic with great aversion to all quantitative mensuration: “It is significant for the psychological contrast between Kepler and Fludd that for Fludd the number four has a special symbolical character, which, as we have seen, is not true of Kepler.” Fludd drew his inspiration from Moses, and he brilliantly defends his stand on the nature of the soul. Kepler, however, appears to best him in all scientific argument until one realizes that Kepler considered the quantitative relations of the parts to be essential while Fludd considered the qualitative indivisibility of the whole. Pauli says, “modern quantum physics again stresses the factor of the disturbance of phenomena through measurement,” as Fludd (and Goethe) insisted upon. He concludes that the only acceptable point of view appears to be one that recognizes both the quantitative and the qualitative, “the physical and the psychical” as compatible, embracing them simultaneously.

This attitude eases the argument, but it does not resolve the dilemma of three and four, as may be seen in a mathematician’s explanation of continua.

We will explore this mathematical explanation in our next post. Until then, peace.

The Dilemma and the Pearl

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Chapter 8, “The Dilemma and the Pearl,” begins with Preston Harold asking us what type of outlook we bring to the world around us – are we a “three” or a “four?”

Wolfgang Pauli says that two types of minds have battled through history: first, the thinking type that considers the quantitative relations of the parts to be essential – and secondly, the intuitive type that senses the qualitative indivisibility of the whole.

The first type mind is posed on the side of three. This type took its stand with Euclid, resting upon his well-known axiom: the whole is equal to the sum of the parts. This axiom, along with the rest of Euclidian geometry, dominated Western thought until the late 19th century. One might say that Euclidian geometry still dominates, for the revolution in mathematics that tumbled it from sacred pre-eminence has not yet seeped down to the layman’s level, and many students will learn first, by rote, Euclid’s axioms, imbedding in the subconscious mind these fallible statements which have been presented as unquestionable truth…

One might say that the three represents Rene Guenon’s “reign of quantity,” the historical manifestation of the descent from form (quality) toward matter (quantity), and the “nothing but-ness” of stark materialism. Tradition calls this period the Kali Yuga, the age of the demon, Kali, or the iron age.

Today, the second type of mind, posed on the side of four, insisting upon the qualitative indivisibility of the whole, regains much of the standing lost in recent centuries. As regards the sum of the parts in relation to whole being, scientists, dealing with one whole atom and the sum of its parts, have found that in the formation of a nucleus from protons and neutrons some of the mass of the particles apparently is converted to energy. The chemist sees that the combined action of several elements taken together is greater than the sum of them taken separately. Mathematicians working with transfinite number theory confront the concept that the whole can equal one of its parts. In short, one is forced to alter his concept that a discrete whole within the universe can be divided and its parts regathered to equal the sum of the whole…

Anyone wishing to look further into the “qualitative indivisibility of the whole” would do well to search out the works of the Goethean scientist Henri Bortoft. You can thank me later ;-).

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Of course both the three and the four have their place in our world but how do we go about regaining the balance between the two? This is what we will continue to explore in Chapter 8. Until next time, peace.

Trinity or Tetragrammaton?

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At this point we are finishing up chapter 7 and getting ready to move on to chapter 8. In order to make the transition a smooth one, Preston Harold takes us from the mortal sin via blasphemy into pronouncing the name of God. I will leave the rest of this post to Harold as he transitions us so well.

That the mortal sin is nameless, that it is a derelict from the forgotten past which can be any sin a man has repressed or has been unable to forgive himself, rests upon Jesus’ contradictory words. He says any blasphemy will be forgiven – and He, Himself, blasphemed if “pronouncing the forbidden name of God” is to say, “Ani hu,” and if to say, “Ani hu,” is blasphemy.

The forbidden name of God is indeed a mystery. What is this name? The definition of blasphemy (from Webster’s Collegiate Dictionary) reads: “In Jewish law, cursing or reviling God or the king…pronouncing the forbidden name of God. See Tetragrammaton.” Tetragrammaton? “The four letters (variously written, without vowel points…) forming a Hebrew tribal name of the Supreme Being…too sacred to pronounce.” What is this mystery having to do with four “unpronounceable” letters, IHVH, or JHVH, etc.? Does the blasphemy rest in rendering the form of God in a four-dimensional concept such as consciousness can know? Could one state the forbidden name in numbers, for example:

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Could this be “blasphemy” because although the equation might bespeak a four-dimensional concept, it does not coincide with the “odd-even” division of a light wave group and thus it cannot truly and fully satisfy life’s situation? Is there anything in the realm of physics and mathematics that might explain this mystery?

The answer is yes. But to solve the riddle one must enter into an argument that engages the scientific world. The gist of the argument can be very simply told: it hangs upon what Jung calls, “the dilemma of three and four,” and one may grasp its general outline in Jung’s work and that of the physicist W. Pauli…The “dilemma of three and four” deals with a very old dispute, but it is one that is examined to this day. Trinity or Tetragrammaton? Triad or Tetrad?

Until next time, peace.