The Measure of Apollo

According to the measure of Apollo
or after the ecstasy of Dionysos, embracing the paradox
of the orthodox and that which foils it, what is left
when the right path is qualified (less than whole)
The burden is light---photons in the quantum approach
What is left is wegh, the weight and the way, turbulence
of the wake, what it is one sees having awakened, upon
reflection (it is done with mirrors. Ars memoria, for which
in the garden of flowers Jordan Brown was burned in 1600.
Feedback functions of the temporal lobe from the free‑
association areas of the neocortex.

The standard units of measure.
Strings on a lyre. Count them: seven for the intervals of an octave.
Invention of Hermes the culture bringer,
who found a tortoise on a beach, dried with the sinews
stretched over the cheloon  for a sounding box.

What did Sinan have to do with the bar of platinum in Paris?
An old builder, he trusted his eye for correct proportions.
And was it a platinum nucleus that pierced the plate, suggesting
the imaginary quantum of a "monopole?"

Apollo, having gained an instrument with which he could sing,
defeating Marsyas, who could not as he blew his
more popular flute to the ancient Pan dance,
refused to offer Hermes his oracle of the Golden Bough.
Apollo the Destroyer, messenger of Death, Yanantaka,
whose authority is the staff of gold, pole of the ecliptic,
for the orbit of the sun.

Symmetries of self-referential (i) systems in science & art.
Interference patterns as bases for computing higher order complexities--
generation of information space from space-time functions.
Hologram model (Pribram: of the brain). Feedback has a delay.
"Un retard en verre": Marcel Duchamp. Elec­tronic delays (Wally) 
In optics, parallax. Stereopsis. With the idea of temporality,
we may construct a CROSS, symbol of the Cartesian coordinate,
plane (plain) view. It is what it was (++), it is not what it was (-+),
it is what it was not  (+-), it is not what it was not (--).
Plainly there are no other possibilities; we have exhausted
the combinatorical options, filled the space.
For a lateral system to manifest closure, it must fill its space--
not because we desire it to be that way,
but because that is the way it is.

**********

If we are to have the words mean anything in the languages we use, or the symbols to have a conventional utility (upaya): spoken languages, formal languages---all of them are algebras, or systems of the relationship between constants and variables. We can map them all as algebras,
not in the narrow sense of what we learn as school "algebra, although this is one illus­tration of an algebra, operating with alphanumerical tokens. We must understand (stand under, inspect the arithmetical foundation upon which these algebras are all based, as L.E. Dickson shows), every algebra has its arithmetic.

The quadratic is the complete form for algebras because algebra is precisely in the realm (T. khams) of four crossings from the void. The degrees of an algebra are the number of crossings made by an equation. Thus the analogy of "dimensions" may mislead those who seek the Dharma, because dimensions may be multiplied ad infinitum, but with the degree we are intrinsically constrained. (GSB on equations of the fifth degree, for which the algo­rithms up to quadratics will not suffice.) This is quite apart from the "solution" by approximations of QUINTICS, (Abel, elliptical functions). What we see with equations of the fifth degree is the new order of indeterminacy in the variable. One may happen to hit upon the solution to a given equation, but one cannot derive solutions in the same way.

We assume one thing, our language of presentation. Here it is "English." Quite an assumption, but how can we avoid it? This is because the void is within, and to communicate about the void, we must use (in this world in which we imagine ourselves) some sort of "language" having certain properties to enter or to describe spaces without them. If we imagine that we are in a world of distinctions (samsara), how is it that we may use a lan­guage based upon distinctions (nouns and verbs, say) to indicate a world where these or such distinctions do not exist. In fact (a fact is something "made,' L. facere) existence is only in this world, not in the world within.

The negation is the psychological device invented by culture. Invented in the technical sense, because "not" is blatant paradox in the so-called real world. Not blue implies blue, as the devil implies God. So it is we come to use other tokens, distinguishing between, for example, "existence" and "being," in which we may allow "being" to denote those states within and without, but constraining "existence" to the world of tangibility. (Note the geo­metrical and the numerical, trigonometric associations of tangent).

Our culture is replete with forms of the self-referential equation. How many novels are published in which the hero is a novelist? "Chorus Line" is the big hit Broadway show about the production of a Broadway show chorus line, and the San Francisco Comical's front page photo (Jan  1976) pointed out this feedback function. John Lilly's "meta-programming" in Human Biocomputer. Sacred scripture often and in one cultural view MUST do the same: not just anything is written on any page. All the surahs in the Qur'an are numbered, make no mistake about it. There is a reason for the numbering beyond incidental historical sequences of their revelation; it might be illuminating to hear how the contents of the surahs relate to their order (cardinal, ordinal). The first Book of the Bible is, most appropriately, Genesis. Every mark on every page of Laws of Form, we believe, is deliberate, reflects consciousness, reverberates with meaning: EIDETIC/HARMONICS (and FUNDAMENTALS).

We may make arrangements of convenience with our social brothers and sisters. Indeed, we may imagine that we have secured some sort of spiritual contract with the guru, but with the assumption of responsibility for our own actions, it is clear the teacher's interest is to let us go as soon as we are able to proceed toward our on liberation.

Libraries can be burned, monks immolated, thangkas traded for cash and purveyed to art lovers. What is it that remains of Chinese Buddhism? With the completion of the idea of a fully evolved system, the "anatomy" as completed (A. Wright's citation of the Buddhist idea of history: 3 stages to completion, then the dissolution of the church, Buddhism in Chinese History) The diamond is achieved. One travels light into the next degree of "space." The Vajrayana is translated; the Dharma comes to the West. An algebra fills its domains. We are using language self-referentially, therefore we are moving in an algebra. Counting out from the void we fill our space with four degrees (quadratic equations). Then what happens when we apply the same algorithms--no more coming out to meet our basic ground of operation, but as it were, going through ourselves and beyond, but now moving behind ourselves, although without reversing the directions. We can't reverse our directions because all the space has been filled, so we do not solve equations of the fifth degree anymore, but instead generate equations of higher degree. Instead of "solutions" we find we are indicating spaces. There has been a flip/flop.

The Wisdom-that-goes-beyond (Prajnaparamita), was culturally and historically underground in the Orient. The same fiery sword of Manjusri, Gnosis, science, happened to be in the marked state culturally in the West: worlds of duality, distinctions, analysis. When we have three dimensional space, we have all the space we need; the next order of complexity is of time (but not yet with duration). If we go back into Eternity, the first realm encountered is of algebra, in which the being and doing of "existence" are the variables in relation to the constants--actually it is the relationships which are mapped by algebras, the system itself, as a whole, regarded as "constant" in respect to the variables of experience which rest on its formal ground.

Existence gives way to TRUTH, truth to INDICATION, indi­cation to FORM, form to VOID. Algebras deal only with re­lative truth, arithmetic with the truth. But what is true in the deeper sense is that which is the case, Dharma. And it is the case that the foundations of truth may be imagined, visualized. We know precisely how deep we can go beneath the truth, but we pay for each step we take. What we lose or sacrifice is the capacity to make distinc­tions of a certain order.

Ardthmetic is always an icon in quadrature, the mandala. Truth is one of the quadrants. For every true there is a false, a not-true, just as in the illustrative example of the natural numbers, for every minus there is a plus. We have to recognize certain cultural biases overvaluing the so-called truth; in the arithmetic, in the mathematical sense, they are complementary, in a general sense "equal." But true and false are not all there is on the level of arithmetic: let us fill out the space with real and ima­ginary. Plus one, minus one, zero (a real number neither plus nor minus, an essential identity element in the group), and i, represented as the square root of minus one. Now the arithmetic is not defined by these numbers; rather the numerical values are only illustrations of deeper arithmetical formal relations. That is why we must use numerical values possessing the property of UNITY. There are four such, and they may all be represented as powers of i;

1⁰= +1
1¹= +i
1²= -1
1³= -i
1⁴= +1
1⁵= +i  and so forth

We can see here the cyclical nature of the powers of i, and in fact this is a mathematically defined group, illus­trative of unity, represented by real and imaginary values, of plus and minus sign.

1⁰= +1: Anything raised to the ZEROth power is unity, to which we may assign a numerical value of plus one (+1).  1¹= +i: Anything raised to the first power is itself. If we decide to assign a numerical value to I, then we may indicate it as the square root of minus one. Since in the set of so-called "real" numbers there is a rule that any minus number multiplied by itself (the squaring function) yields a product of plus sign, we are constrained to "imagine" some form of unity which when multiplied by itself yields, not the plus form of numerical unity, but the minus form of numerical unity. Thus we ascribe to this number the value of √-1, conventionally indicated by the token "i." It should be remembered that the terms "real" and "imaginary" are technical, introduced in order to make distinctions without which computation would become confused. There is nothing any more real about "real" numbers than about the so-called imaginary ones; nor is there anything less imaginary about them. No physicist can show us a number in reality—numerals, yes, because they are tokens of number, but not the number itself.

Spencer Brown points out that some numbers exist and some do not. The imaginary form of unity (i) certainly exists. Much of the higher mathematics of the last century would be quite impossible without it. It works, it is reliable, it gives consistent results in equations, it has many practical applications. But the "largest prime number" does not exist.

That this is so was already brilliantly proven by Euclid, and has since been corroborated to the complete satisfac­tion of number theorists. It is in a different sense that we may "imagine" some so-called largest prime number in order to write down the words LARGEST PRIME NUMBER. Still, no such number exists in the same way as the square root of minus one, or one, or one hundred, or two thousand three hundred and eleven most certainly do.

1²= -1: That is to say, quadratic working with the numerical values, a square root times a square root is of course the value of the square, as the root functions drop out: (√-1)(√-1)=-1.

1³= -i: To raise i to the third power we must multiply i squared times i: (-1)(√-1)=(√-1), depending upon which notation we choose to employ, the numerical or formal.

1⁴= +1= +1: Since i³ = -i we have (-√-1) (√-1), or (-i)(i) which again yields plus one. More fully, we may represent this as (√-1) (√-1) (√-1) (√-1) = 1. We have seen that 1² equals minus one; and 1⁴ is (1²) (1²), or (-1) (-1) =+1. Another way to represent this is (1³) (i), and since 1³ is the same as (-1) (i) or -1, we have       (-1) (√-1) (√-1) = +1.

And so the value of 1⁰ is the same as the value of 1⁴, that of 1² is the same as 1⁵, that of 1³ is is the same as 1⁶, that of 1³ is the same as 1⁷, 1¹¹  and so forth. 

Let us us call attention to the double oscillation: between the real and imaginary values, and between the plus and minus signs. There is a 90-degree phase shift between oscillations too.

Kurt von Meier
January 1976

Below is a page of Kurt's notes about algebras and calculus.