1.) Dimensions in Math
In order to express addition and subtraction, you need a number line, and that is all. Therefore I would argue that addition is a one dimensional procedure. Moving upward, we have multiplication which is itself two dimensional, you need a square to express multiplication. The multiplication table is itself a square. Moving further up we get exponentiation which is three dimensional, which is a recursive function where the last output becomes the next input, and then tetration, the hyper four, is n^n, is a four dimensional expression of the recursive function.
Other examples of recursive functions are the factorial:
4! = 4*3*2*1
It is recursive because the last output gets fed back into the function to become the next input. Note that recursiveness is a looped iterative process and we can unlock the mechanism of texts based on this mere insight.
2.) The Prevalence of the Four
At the dawn of philosophy, the ancients believed that four elements existed in all things: Earth, Air, Water, Fire. Hofstadter mentions the prevalence of the four in his critique of Bach, and Blake is working with four Eternals. The Taoists have a fourfold dimensionality:
“The Way begets the one; one begets two; two begets three; and three begets the 10,000 things.” Lao Tzu.
The fourfoldedness is numeric in and of itself, moving from one to two to three to the world of individuation and separation; although it literally stops at three, a dimensionality known well by Christians in the form of the Holy Trinity, Father, Son, and Holy Spirit. There are four articulations of a solidifying process that goes from infinite mutability to the temporal existence of creatures here on earth. Thus the Way that becomes a way is not the eternal way. The Way cannot be explained because language will always attempt to freeze what it describes. The Way is infinitely mutable, like water, taking the form and function of that which it moves through. To attempt to make the Way the object of language is immediately to lose it (Sproul). It can never be the object, it is the eternal subject. To stare at the Way is to become blinded; to stare from the Way is to be enlightened. It settles the earth, gives the gods their powers, fills all valleys, and leads kingdoms.
Now, like the Taoists, the Dogon employ water as their paradigm metaphor for the creative principle. Like Christians, Jews, and Muslims, the Nummo, the first beings created with speech.
“[The making of clothing] manifested on earth the first act in the ordering of the universe and the revelation of the helicoid sign in the form of an undulating broken line.”
The wavelike fashion of the line echoes the way in which actual light travels, as well as the way water moves about. The importance of the principle of water as a paradigm metaphor for the Dogon's idea of the sign has already been stated, but the line is about to get an additional circular property, so what we are going to be talking about here is a spiral, “like the movement of a reptile,” rectilinear locomotion. Therefore, the numerical system cannot be divorced from the geometrical system which is employed by it.
When the Dogon are making their clothing they are reliving this divine myth of the first Nummo who are attempting to protect their mother from the Jackyl who is born of rape; and from a merely pragmatic perspective, it adds a religious depth to an activity which is fairly mundane. But the notion of weaving is a paradigm symbol for language in Derrida's work as well, at least insofar as a web is woven. For the Dogon, the metaphor of language is tied up in water, being the life blood of the earth itself. For myself it has always been light. The spiritual instruction is embedded within the mundane act of weaving, so the act itself connects the Dogon with a timeless mystery and a sublime revelation. The making of the skirt is understood as the creation of plant life here in this realm.
The first act of creation in the Dogon mythological system is an unrequited sexual advance, or in other words, Amma raped the earth. The result of this unrequited union was the meaner things of nature, ie: the jackal or the pale fox. The pale fox did not possess the power of speech being himself imbalanced and born out of an act of violence. So he must force himself on his mother (which he does) reliving the original act of violence committed by Amma himself.
On the surface these myths appear as simple etiological myths, just-so stories. However, they are structured in such a way as to imply a mathematical structuring which is in and of itself more complex than many of the western myths (not to demean the accomplishments of the Bible, merely to point out the amazing complexity of this system).
Now jackal, who is the child of rape, goes unto his mother, and rapes her, so that he could possess the power of speech, which to this point he does not possess. The result of this is menstrual blood staining the fibres of the sacred skirt, ie: the web of life.
So at this point in the story, we have two Nummo who were born in an acceptable manner, and the jackal-rapist whose birth was singular being a birth born with the consent of only one party. This echoes not only the Sophia myth of the birth Ialtobaoth, but also the sky's impregnation of Gaia in Hesiod's Theogeny.
Now every male and every female is born into this world with both feminine and masculine aspects, one of which will get removed during puberty. Females lose the male aspect of their sexual nature, the clitoris, which protrudes, and males lose the female aspect of their sexual nature, the foreskin, which envelops. This is part of the Dogon right of passage and should not be confused with how female circumcision is practiced in Islamic culture, which has to do with the male fear of not being special. Instead of a right of passage, it ensures that women will receive no pleasure during sex and thus the male won't have to worry about being cheated on. In Dogon culture you are fully a man once you lose the feminine aspect of your sexuality, and children are simply not sexed, or not considered sexually.
Now, the first man and the first woman have regular sex and bear eight children, which of course is two cubed. The twin births which were the norm, metamorphize into dual souls, and thus the need for circumcision. Dual souls are a danger to the world system.
So from the two Nummo, come the eight ancestors, and the original Nummo are hermaphroditic as are their offspring being capable of producing without the need of a consort. They had one dominant and one recessive sex, but their souls were dual in nature. They were defined in terms of a gendered contradiction. Also, the pervasiveness of the spiral defines the creative principle:
“The Spoken Word entered into her and wound itself around her womb in a spiral of eight turns.”
This concept is echoed again and again, the prevalence of 8 is the prevalence of four beings of each gender or two raised to the third power and then halved. The Nummo themselves are conceived as being made of water, and Ogotemmeli uses water to describe them interchangeably with their own name.
It is clear that in the beginning normal unions produce twins, and unrequited unions produce single births. The seed of all life is contained in the Nummo/water and the Nummo beings are half human and half serpent, the serpentine force being associated with the eight-fold coil that turns the earth and exists inside the sun. Eight is the symbol of speech, and all life is derived from speech/water/Nummo who “excrete” light because they are light.
The divine revelation of skirt-making is the Nummo's attempt to protect their mother from Jackyl who is mimicking the crime of his father. He wants to possess the creative power and the Earth is incapable of resisting him. The giving of the garment is the first act in the ordering of creation and leads to the birth of vegetable life. In response to the assault, Mother buries herself in her own womb (inside the ant hill). Etiologically this myth explains why there is menstrual blood.
So, the human soul, which is by nature schizoid, possessing both the feminine and masculine principles must be cut off from the recessive principality, therefore the need for the circumcision during the time of puberty, men lose their foreskins, women their clitorises.
The word is conceived also in a dual fashion being at once synonymous with water, and the eightfold spiral.
So after their great labors the Nummo-ancestors transform into purely spiritual beings and Ogotommeli stops at the seventh to pay special attention. The seventh represents perfection because the sum of the female element is four, and the male element is three. So seven represents a completion of the series, ie: a Gestalt or unity. What is left over is an imperfect knowledge of the Word, destined as it was for usage by man and nature. The Jackyl still had knowledge of the first word, which he could use to manipulate mankind, who was to be given the potency of the second Word. All in all, the first Word, or God's original script had run its course with the transformation of the Nummo into spiritual beings. Over time mankind takes over the generative power of the word, his mouth and teeth fusing with the ant-hill. He becomes the feminine aspect of the creative impulse, supplanting the Jackyl, and forever raging against his imposition.
3.) Magic Trick
Take a deck of cards and arrange the Ace, King, Queen, Jack of each suit in decending order and set the rest of the deck aside. Cut the cards and lay them out again in order. What do you see? Do it again. No matter which way you cut the deck, it always comes out the same way.