Qabbalistic Oddments 00
§00 — Words consist of letters, which can be simply tallied. This is the crudest method of numbering language.
It is known, by esoteric tradition, as primitive method. Primitivization (₱) reduces words to values determined by their letter count.
Both one and two are 3. Tally is 5. Number is 6. Primitive is 9.
Despite – or because of – its extreme crudity, the value of primitive method is easily demonstrated. Its incisive contribution to the analysis of the English Numonyms is especially remarkable.
In the ancient Sumerian, Roman, and Chinese counting systems, among many others, formation of numerical signs up to the initial triad are indistinguishable from tally markings.
When counting in the Roman style, at its purest, I + I + I = III. This produces an impression almost of logical tautology. The sum is merely the whole of what has been written. Between summation and simple comprehension there is as yet no difference. One, and then another, and then another, is – at the beginning – the entire procedure. A number is explicitly drawn, shown, or directly inscribed. The numerical figure is at this level a tally sign. It is tempting to note here the residual indices of a prehistorical numerical order (or PNO), one devoid of any capacity for summarization. What it lacks, relative to the ANO (or Ancient Numerical Order), are numerical denominations beyond unity.
The value of primitivization is first locked-in by odd and even.
Primitive consistency is also found in decimalize (and pentazygon).
The only primitively consistent number is four. Upon superficial estimation, this is a result so unimpressive it attests almost to evasion. Sheer chance could have been expected to out-perform it. Nevertheless, ‘four’ satisfies the conditions for a primitive exemplar. This abstract function will be generalized, through application to other problems.
§01 — Primitive discrepancy can be simply tabulated. The PD of any natural numonym is derived by subtracting its letter count from the number it names. Graphed PD exhibits an erratic upward slope, converging asymptotically upon the named number. For example, the PD of one trillion is 999,999,999,989. Zero, one, two, and three have negative PD. Four is neutral (PD = 0). For all naturals greater than four, PD is positive.
Primitive discrepancy crudely captures semiotic surplus value, or notational efficiency. A tally has zero SSV. The number of units indicated exactly corresponds of the number of indicative units. One trillion is tallied by a trillion strokes. PD compares these trillion virtual strokes to the ten letters the word requires. Positive PD is thus a measure of economy. It counts strokes saved. As noted in other terms, this economy scales quasi-linearly with the number considered.
§02 — Primitive method is the key to a riddle, which common arithmetical expectations set.
There was a gate whose lock was a puzzle. Follow the pattern to enter said the inscription upon it. The gate spoke.
“Twelve,” it said to the first visitor, who answered “six” and was permitted to enter.
“Six,” it said to the second visitor, who answered “three” and was permitted to enter.
“Four,” it said to the third visitor, who answered “two” and died in front of the gate.
“It was a fatal error for the third visitor not to echo the gate,” said the first visitor, understanding.
Ruin resulted from mistaking primitive method for arithmetical division, which it initially – and then once again – simulated. The lesson might be thought harsh, were it not only a story.
§03 — There are five twins. Not much is more basic. What might be logically questionable is pragmatically presumed.
Hands mirror each other. Their digits are twinned. In Kantian terms they are incongruous counterparts. Their difference attests to the a-logical quality of space.
Anatomy not only instantiates decimal, but also factorizes it, exhausting its integral decomposition. Two hands times five fingers is the basic plan, modeling the pentazygon.
No logical necessity underpins this. The numerical compliance of the body is exorbitant, which is to say empirical, accidental, or coincidental. It decimalizes radically, without reason.
Bringing the hands together, finger-tip to finger-tip, is a common, unreflective gesture, while also at times being a gestural sign of reflection. Doing this orders the digits into consistent pairs. Each checks its complement. Digital, primordially, means all of this. Even after associative collapse down to binary modularity, half this sense survives.
§04 — The first great achievement of primitive method in regard to the English Numonyms was August Barrow’s Octaves (1753).
First, arrange the elementary numonyms in the Pentazygon. Thus:
Zero + Nine
One + Eight
Two + Seven
Three + Six
Four + Five
The remarkable result is self-evident. In it, Barrow saw the Pentazygon, Anglossic, and Primitive method simultaneously vindicated.
§05 — Any number is even if, when it is added to an even number the sum is even. Zero is then found to be even. This demonstration can be infinitely extended, but only the immediately successive phase concerns us here. Any number is trinomic if, when it is added to a number divisible by three, the sum remains divisible by three. Zero is thus trinomic.
It follows uncontroversially that any complement to a trinomic number is itself trinomic if their sum is trinomic. The trinomic syzygies can in this way be defined.
§06 — Primitive method also unlocks the Iron Law of Six. Barrow undertook this task in his Shelves (1758).
The abstract motor of the Zhouyi (or Yijing) is diplo-triadic. It consists of two trigrams, or twin triangles, locked together in alternation. The structure emerges from the dynamic assembly of three twins. Understood as a path, it is Möbian – double-sided but continuous.
The Star of David is composed of twin-triangles. Its model of the Hex is therefore approximate, simplified, and static. This suffices, nevertheless, to betray its descent.
A purely numeric encryption of the Hex is found in the modern informatic ‘byte’ (eight bits of information is 2^8 but eight is 2^3).
The cycle 1, 2, 4, 8, 7, 5, emerges from decimation of the binary powers. Accelerating growth is succeeded by accelerating collapse (and inversely). Three doubling periods are marked in each direction.
The same cycle emerges when the rows of Pascal’s Triangle are summed and decimated in sequence.
Under primitivization, subtraction of the trinomic tetrad (0, 3, 6, and 9) regularizes the decimal numonyms. The series one, two, four, five, seven, eight incrementally ascends through three terraces (Barrow’s shelves). In this regard, each of the excised terms zero, three, six, nine is notably anomalous. None fit, even among themselves. They disrupt the ordinal gradient. The effect of their inclusion is something like an infection of alien numerical principles. Zero ordinal gradient arrives with one pair (0, 9), negative gradient with the other (3, 6). The outsideness of these numbers is thus primitively manifest. They destroy pattern, vividly.
Subtraction of the trinomic tetrad, decimal regularization, or hexation, is decisive. The importance of this operation on the path of decimal investigation is difficult to over-estimate. The hexated decimal residuum, or simple hex, repeatedly demonstrates its numerical consistency.
Every prime number above (or indeed other than) 3 decimates to 1, 2, 4, 5, 7, or 8.
The number 142,857 rotates under multiplication.
Powers of five cycle 5, 7, 8, 4, 2, 1 (which, as expected under decimal, is symmetrical with the powers of two).
Division (of any non-multiple of seven) by seven produces the same digital output, but in scrambled order. The recurring segment in each such case is 142857. The six-step cycle is preserved. The diplo-triadic turbine thus exhibits a distinct, consistent phase.
Ten minus four reverses the final phase of the tetraktys. It has the Pythagorean sense, then, of a terminal regression – a retreat from the end. Hexation is framed as a restoration, in which the largest and most recent stage of development (alone) is undone.
§07 — The Numogram was already ancient for the Sumerians, even to the point of amnesic oblivion. This is evident from their number system. Specifically, the Sumerian Ideal Year (of 360 days) is susceptible to zygonomic derivation. Its gnosis is typically entered through the Gate of Five Angles (90°, 81°, 72°, 63°, and 54°). The demonstration proceeds: 90 + 81 + 72 + 63 + 54 = 360. Sexagesimal thus arises spontaneously from a deep decimalism. Nothing beyond pairing-off within ten is required to generate the second power of six. Counting in sixes is an implicit decimal sub-function, in numerous ways.
Ten yields six through excision of the trinomic tetrad. This operation is hexation.
In addition, beginning from the tetraktys, the six-module can be decimally accessed through the method of descending triangles. Cumulative decimation makes the series ‘1, 3, 6, 10’ into a complete circuit. This is only the Pythagorean insight restated.