How the Evolution of Mathematics Parallels Civilizational History
William Irwin Thompson:
"The beginning of mathematics, according to Whitehead, was in the recognition of set and periodicity. The first hunter who observed that three fish and three bears were both instances of threeness took the first step toward the observation of periodicity. Elsewhere I have argued at greater length that the first observations of periodicity had to have been involved with the menstrual cycle and that the primordial mathematician was probably not a hunter, but a gatherer. The Neolithic stick of computation, christened le haton de commandement by the Abbe Breuil, was probably no such male thing at all, but rather a midwife's tally stick for the lunar calender of "women's mysteries." 17 Menstruation and mensuration are related, and the lunar cosmologies that Alexander Thom has shown to be expressed in the megalithic stone circles of Britain speak of a cosmology that is not military, masculine, and Bronze Age.
The observation of periodicity in Woman and moon establishes a mentality that becomes developed in the Paleolithic systems of knowledge in midwifery and some form of lunar astrology. But enumeration is not simply counting; it is relating. Therefore the recital of the relationships of humans and animals, of offspring and parents, is a form of relating humans to a cosmology. Relating genealogy is relating the individual to the class, and it is so important and valued a form of organizing the universe that the mentality of enumeration survives up into the historic period. The enumeration of all the me's taken by the goddess Inanna from Eridu to Erecb is one of the earliest recorded performances of this mentality, but, so basic is it, that it survives from the Riverine up into the foundations of the Mediterranean epoch. In the catalogue of the ships in Book Two of the Iliad, in the recital of the shades who come forth to speak with Odysseus in Book Eleven of the Odyssey, and in hp recital of the lineages of the gods in Hesiod's Theogony, we have three classical performances of the world view implicitly organized by the mentality of enumeration.
To appreciate just what a transformation of world view it is to move from enumerating to geometrizing, we have only to compare the mentality of Hesiod with Pythagoras or Plato. Enumeration is a fairly straightforward way of relating
humanity to divinity, but when the line folds into triangles and squares, the pattern becomes more complex. One can begin to see the unconscious emergence of the geometrizing mentality in the Iliad, for there the lines of descent are beginning to cross over to create patterns. Leda and Tyndareus give birth to the twins Castor and Clytaemnestra; Leda and Zeus give birth to the twins Helen and Pollux. Then the two sets of twins cross, and Castor and Pollux are raised up into heaven; but Clytaemnestra and Helen remain on earth to become the sources of eros and thanatos in the world of passionate conflict. When, from another line of descent from Zeus, through Tantalus and Atreus, the brothers Agamemnon and Menelaus are wed to Clytaemnestra and Helen respectively, the lines of descent create the outlines of the battlefield of Troy.
When the line becomes the outline of a form, the metaphor that begins to obsess the ancient imagination is the wall, for the wall is the line seen as container. The Gilgamesh Epic opens and closes with a celebration of the wall of the city of Uruk. Book Twelve of the Iliad focuses on the wall the Greeks build to protect their invading ships. The wall is the limit, but when Patroclus dares to go beyond the limit, and when he dares to go beyond the limits of his own identity by putting on the armor of Achilles, he is cut down. With the concept of the limit, the mentality of enumeration begins to pass over into the mentality of geometry, for the limit is the form of a thing's existence in time as well as space. In the first thirty-three lines of Book Twelve, Homer explores the idea of the wall as a limit of the Greeks' presence in Troy, the limit of the length of time of Achilles's anger, and the limit of duration against entropy. The forces of chaos raging at the edges of order are personified as the gods Poseidon and Apollo, who take counsel together on how to destroy the wall through the eroding force of rivers, but it is clear that what is being described through gods and immortal spirits of rivers are the ideas of entropy and order. A genius such as Homer, possessed by his Daimon, maintains a permeable membrane between unconscious and conscious, and his ideas have such power because they are neither unconscious nor overrationalized. In that vibrant state they provide vital material for thought for generations to come, for when Thucydides portrays the Athenian fleet of Alcibiades proudly sailing off to disaster at Syracuse, he is performing the idea of Patroclus donning the armor of Achilles to go beyond the limit to his destruction; and when Anaximander explores the idea of the edge of things, the wall of definition that separates the limited from the nonlimited, he, too, is making explicit what was poetically expressed by Homer in Book Twelve:
The Non-Limited is the original material of existing things; further, the source from which things derive their existence is also that to which they return at their destruction, according to necessity; for they give justice and make reparation to one another for their injustice, according to the arrangement of Time.
The wall is the archetypal image of the limit, the edge between life and death, civilization and savagery, and the poetic metaphor of the wall marks the transition in the cultural evolution of consciousness from the mentality of enumerating to geometrizing. In the Babylonian creation myth, the Enuma Elish (circa 1000 B.C.), Ea puts a magic circle around the younger gods to protect them from the god of the underground water, Apsu, The older gods are restloving, but the younger gods throw noisy parties, and so the Great Mother of the saltwaters wishes to destroy them to return to her primordial rest. The thermodynamic activity of the youthful and newly emergent gods disturbs the condition of rest and entropy preferred by the Great Mother, and so the battle of the male god, Marduk, is no longer the old Neolithic cosmology of the male as the symbol of vanishing and the female as the symbol of continuity; it is a battle of form versus entropy, of civilized, military patriarchy versus prehistoric matriarchy, of the enduring and the changeless versus transformation. All the ideas that we have since rearticulated into the Second Law of Thermodynamics have their origin in this matrix of myth.
The Enuma Elish and the Iliad are profound milestones in the cultural evolution of consciousness, for they sum up and finish an ancient mentality at the same time that they announce the mentality to come. In Hesiod's Theogony and in Homer's Iliad, the mentality of enumeration is consummated and finished. Homer brings us up to the edge of the geometrizing mentality, but it will be the work of Pythagoras and Plato to transform mythology into mathematics. And although C. M. Cornford taught us to see that transformation as the great rational leap "From Religion to Philosophy," we now can see what a mixed blessing abstraction is. Homer remains the greater genius, for he understood and expressed in a way that no subsequent writer has surpassed, the violations of order.
Throughout the Mediterranean epoch, this geometrizing mentality is dominant, both in its medieval Christian elaborations and in its Islamic variations that replace iconography with geometry. Perhaps the supreme expression of this geometrizing world view is in the circles of Dante's Paradiso, for at that peak of ecstatic visionary elaboration, Mediterranean humanity can go no further. The revolution for modern humanity will be to clear the landscape by calling all into doubt, and Descartes will sweep his mind clean of medieval geometries to create the grid against which to perceive Galileo's failing bodies.
From analytic geometry to calculus, the genius of modern humanity is focused, not on the static objects held in the geometry of a Platonic ideal realm, but on the dynamics of movement. Plato's circles become Kepler's ellipses. Motion, the narrative that was so inconceivable for Zeno, becomes the beloved of Galileo, Kepler, and Newton. For a few centuries, the notations of movement focus on billiard balls moving in a black space; but in the nineteenth century movement becomes generalized into process, and both thermodynamics and evolution extend the mentality into transformations.
Transformations, of course, bring one to the edge of conventional dimensions, and as the narratives of quantum mechanics flirt with objects of perception that can never be seen but only imagined, human beings begin to realize that there is more to consciousness than objects of perception held in three dimensions.
The end of modernism comes with the multidimensional topologies of mathematics and physics. At first this finish to modernism is elitist and experienced by only a few physicists like Heisenberg or poets like Yeats, but the rise of electronic forms of communication in our generation has democratized this change of mentality, With the ability to express complex geometries in cathode tubes, computer graphics is beginning to stimulate the processes of visual thinking. There was only so much one could do with chalkboard and chalk, or pencil and paper, but now combinations of music and computer graphics begin to permit new forms of play with multidimensional topologies and ancient yantras. As these forms begin to dance in the imagination, they conspire against materialism by whispering in the scientist's ear, "All this is disguised autobiography, for these crystals are the intelligible bodies of angels and the soul." Like the slave in Plato's Meno, who could reason geometrically because of anamnesis, postmodern humans discover mysteries of consciousness where they least expect them.
Even so groping a comparison of mathematical modes of articulation and literary modes of narrative shows us that Lord Snow's famous remark about the "two cultures" of the sciences and the humanities is not helpful in understanding history. Mathematics is relating, genealogy is the logic of one's relations; and both are performances of narrative.
Narrative itself is a human response to time, for it is an attempt to escape the infinity of the present as duration by reifying time into a past. Ex-isting means "standing out," "arising out of the indeterminate," or "setting up." Consciousness without an object, without either a sensory construction or a spatial-temporal horizon, would be so maddeningly disorienting as to constitute a condition of absolute terror. Our response to this terror would be to project immediately a spatial-temporal horizon, to project a world.
Something like this consciousness without an object happens every night in dreamless sleep, but since the ego is not there to get in the way with its interpretation of terror, the experience is not remembered. Upon slipping out of this state of undifferentiated Being (described as returning to Brahman in the Upanishads), consciousness gathers like a dust cloud collecting in density, and dreams begin to project the world of psyche, that shore between the ocean of Being and the island of the ego. Conscious becomes so enamored with these projections that its attention becomes fixed, and it Wakes up into the projection. First consciousness fixes itself in the psychic world, then it falls asleep and dreams what are memories of the psychic experiences, and then it wakes up into the world of the ego to remember the dreams that themselves are memories of psychic experiences. If consciousness were to move without a transition from the fixed attention of the ego to the undifferentiated Being, it would be interpreted as an experience of terror, a death. But this kind of conscious dying, this mystic death, is precisely what the practioners of meditation strive for. Saint Paul said, "I die daily." But the experience of conscious dying is not exclusively a Christian crucifixion, for students of zazen are awakened at four in the morning so that meditation can begin to wear away the membrane between sleeping and waking, and so that as one is awake in one's dreams and dreaming while meditating, the background to consciousness becomes the foreground as all horizons drop and the ground becomes an open space.
Existence is literally a setup, and so our mathematical and literary narratives are repetition compulsions that move back and forth across the threshold of the infinitely extended present. We do the same thing when we scratch an itch or make love: back and forth across the sensitive spot, touching and withdrawing, to enjoy the sense of difference that is, as Bateson told us, the experience of information. Narratives leave the present to touch the present, to explain it, to know it. And whether the narratives say f =ma, or e=mc2, or "In the beginning was the Word," they go back and forth across the erotic threshold that separates eternity and time.
And so narratives are not merely about time, they are performances of time: incarnations in miniature that seek to re-mind us literally. As the bard performs his story, so the mind performs its story, the ego. Since the tongue cannot taste itself and the being cannot know itself, we must come at things through reflection and indirection. We tell stories, but the stories are not always directly about what they tell. Hesiod's Theogony, that great climactic work of the mentality of enumeration, is about the evolution of Mind, from the indeterminate, through the psychic realm of gods, and down to the most limited incarnation, the shepherd poet himself.
All narratives, whether they are artistic, religious, or scientific, are at their deepest level disguised autobiographies of the human race. At the level of the root idea, the Enuma Elish and the Second Law of Thermodynamics are mythopoeic. And when science tells us who we are, where we come from, and where we are going (as Darwin and Freud tried to do), it is inescapably mythic."