When we left off last time, class, the Tortoise was excited about a new revelation. In his experiment with Achilles, using the same total number of points and scoring rubric for individual plot points for Romeo and Juliet, he and his friend landed in the same spot along the number line.
Achilles noticed that the arrangement of their equations differed drastically, even though they arrived at the same total on the same side of the line. Since associative and commutative properties affirmed that each student could act out Romeo and Juliet according to their own interpretations, the Tortoise realized the outcome is less important than what happens inside the equation.
When the Tortoise shares his discovery with the teacher, he can see the look of pride on the teacher’s face.
“It’s an interesting hypothesis, and it makes me wonder if we are ready to return to our initial questions: ‘How do dualities interact with one another?’”
Particles like electrons or photons can exist in a state of probability, described by a "wave function," until they are observed or measured. When not observed, they don't have a definite position or state but exist in a superposition of possibilities. When measured or looked at, the wave function "collapses," and the particle takes on a definite position or state.
I have this seen this phenomenon in other parts of life—while teaching students who have lived their lives unnoticed, while walking through the woods, while saying thank you. Noticing renders the unobservable observable.
The year I was born, Doris Lessing published Prisons We Choose to Live Inside. The first chapter-“When In the Future They Look Back On Us”-Lessing is condemnatory of the state of a society in decline. Her criticisms describes a culture preceding this one by thirty-seven years and though strangely recidivist:
“This is a time when it is frightening to be alive, when it is hard to think of human beings as rational creatures. Everywhere we look we see brutality, stupidity, until it seems that there is nothing else to be seen” (3).
It’s hard to find against the pessimism that accompanies knowledge—the realization that humanity is a blip on an eternal timeline: easily, quickly, and self-destructively snuffed out, taking as much as it can with it. Although Lessing finds much to feel disheartened about around her, she’s still hopeful. Humans, she reminds us, have something working in their favor:
“Against these enormously powerful primitive instincts, we have this: the ability to observe ourselves from other viewpoints.”
Zoë Schlanger does exactly that in The Light Eaters, challenging the way we think about plant behavior, memory, communication, and intelligence. Schlanger reports on research that sets asides the limitations of human perception to consider these topics in a broader context. In her book, Schlanger explains how her work as a climate scientist caused her to lose touch with her own emotions and sense of self.
Plants offered a new lens, a non-human lens, to look through: “Plants are the very definition of creative becoming: they are in constant motion, albeit slow motion, probing the air and soil in a relentless quest for a livable future” (3). For Schlanger and Lessing to face their lives with hope, they needed to observe life with more than their human eyes.
Not unlike Pythagoras, Jean Jacques Rousseau was a terrible man obsessed with order and personal pride who had an occasional good idea or paid someone else to have a good idea for him. One of his many bad ones was his belief about harmony, something he complained about at length in Confessions. Rousseau argued that music acted in accordance with natural law; the introduction of harmony was the destruction of melody, the most perfect and absolute expression of Form.
The Pythagoreans also saw law and order as the pillars of cosmic harmony. They were singularly focused on uncovering examples of unity and harmony in math and science. Zero was a challenge to this system of belief, but irrational numbers were unthinkable.
Rational numbers can be expressed by integers in a precise fraction: 1/2 or 3/4. These fractions, composed of two integers, have perfect, non-repeating correlative decimals: .5 and .75. Irrational numbers, however, live in the space between rational numbers, messing up the works. When trying to determine the square root of 2, the Pythagoreans encountered a never-ending, never-repeating number. It was enough to shatter everything they knew to be true about the universe.
Rational numbers also correspond to music. An octave can be expressed as the ratio 2:1 and a perfect fifth 3:2. Rousseau saw music as elevated language with melody as its holy and perfecting expression. Harmony, he believed, muddied the clarity of the message. Both Rousseau and Pythagoras lacked an understanding of how their own perceptions, cultures, and biases shaped their understanding. Like the Pythagoreans, Rousseau was running from the space between notes but, as we now know, learning is a conversation with mystery.
I want to clarify here that you are not in the same class as the Tortoise and Achilles, although you have the same teacher. You wouldn’t want to be their classmates. Achilles is always trying to race everyone, and the Tortoise is a know-it-all. Plus, and don’t ask me how I know this, turtles don’t smell that great. But you’re doing well, I can tell.
Zeno told another story about Achilles and the Tortoise. In this one, the two friends are engaged in yet another race. The Tortoise gets a head start, but Achilles takes off soon after, not far behind the terrapin. In order for Achilles to reach the Tortoise, he must first cover half the distance between himself and his friend. However, when he reaches the midway point, the Tortoise has already moved, increasing the distance of Achilles’ next midway point.
Each time Achilles tries to catch up by crossing half the distance to the Tortoise, the process repeats. An infinite number of halfway points exists between Achilles and the finish line. This is Zeno’s Paradox, the idea that time and space can be divided in half an infinite number of times and, therefore, have no end point. It’s all circular, endless, looping.
A Turtle in Ecstasy, 2024
The teacher takes your class back to the gym or beach or snowy field. Once more, you act out the plot of Romeo and Juliet, this time according to the six affixed plot points. When you reach the third point—the secret marriage between Romeo and Juliet—the teacher stops everyone. Each student freezes in place. Students are scattered across the gym floor, different points on a line.
The teacher asks the students about where they are—whether they are in the process of moving toward free will or running away from it. Two students standing at the same point along the number line realize that they have both arrived there having made different decisions about all three plot points so far.
“I don’t understand how we can all be in such different places,” one student says. “It’s like we aren’t even reading the same story.”
“But what if we aren’t,” another student says. “What if a story is always impacted by our perception of it?”
“Then how do break free from perception?” the teacher asks.
“By looking at it,” you offer.
“So then tell me this, how do the positive and negative numbers interact, whether we are aware of our perception or not?” the teacher asks.
“They seem to move past one another, sliding into and through. Nothing seems fixed. Just because you assign a negative number, you could still end up on the positive point of the line. The purpose is the motion it creates, not the number itself,” a student says.
The teacher nods: “If each point on the line represents a movement up and down, positive or negative, rational and irrational, fate or free will, then what does that tell us about the points themselves?”
Your classmate offers, “That each point is infinite, that it inhabits both—all—parts of the duality. Only our perception gives the point itself meaning.”
The teacher nods. “So what should we look at first?”