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| Deeply iterated Buddhabrot |
In the 5th century B.C., Eudoxus of Cnidus, developed the Method of Exhaustion to determine the area of a shape by inscribing the interior with a series of polygons, notably applied by Archimedes to figure the area inside of a circle and the Quadrature of the Parabola. It is considered a precursor to integral Calculus.
Zeno of Elea proposed the Dichotomy Paradox as an attempt to prove Parmenidian idea of "The One." The argument is familiar: Your hand reaches out for the cup of coffee. At the halfway point, is is 1/2 of the distance to the cup. Then it moves half the distance of this, 1/4. Then half of this, 1/8. And again and again into infinity. Accordingly, you are never quite "touch" the cup of coffee.
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Of course, your hand grasps hold of the cup and raises it to your mouth. But the mind, enraptured by Zeno, is still generating a denominator that is forever reaching to infinity.
How all ’s to one thing wrought!
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| M.C. Escher, Parade of Ants Source |
In the late 1700s, Leibniz provided a set of rules to work with infinitesimal sums which became the foundation (along with Newton) for modern calculus. In the 19th century, the limit of a function replaced infinitesimals by using real numbers, such as zero or one.
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| A Koch curve has an infinitely repeating self-similarity |
As the hand approaches the cup of coffee, the distance between each object approaches zero. The idea of a number, of a discrete element of abstract thought, wants to insert itself in between - giving birth to Zeno's Dichotomy Paradox. Reductio ad absurdum. The inviolable stillness of the Parmenidian One is experienced as absurd. There are objects in motion through time. Hands reach out to grasp cups of coffee. The beauty is when we allow that everything does become One, through the poetry of the calculus. Thus, the process of abstracting, of generating sets of objects, itself is paradoxical. Cantor's naive sets and Russell's Paradox. Extending to Godel. Self referentiality is the difficult crux:
This statement is false.
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The calculus of desire. Of addiction. The relationship between the two. Chasing the Dragon. Trying to get to Zero, to touch the Thing Itself: Ding an Sich. Burning out the pleasure centers, the hardwired "circuitry of the brain" through infinite Methods of Exhaustion. Rising up with the intoxicating smoke, over-saturating the referentiality, trying to get around the tolerance curve, to see what pulses within the Heart of the Fractal Dragon.
I have heard the bell ringing within, shivered within it's tolling, broken myself down against it's clanging. I have torn the rough rags of language off of the Thing Itself, breaking the gerund "ing" off and holding the ring. Blowing out the brains shivering synaptic structures, ringing ringing ring ring, until there is just the constant sound of being, the flatline tone that resolves into oṃ... ॐ
How all ’s to one thing wrought!
- Gerard Manley Hopkins, On a piece of music
And what do I have to say? Who am I to add to what is?