by Neil Clarke
“Curious indeed. You think the heptapods’ idea of what’s simple doesn’t match ours?”
“Exactly, which is why I’m dying to see what their mathematical description of Fermat’s Principle looks like.” He paced as he talked. “If their version of the calculus of variations is simpler to them than their equivalent of algebra, that might explain why we’ve had so much trouble talking about physics; their entire system of mathematics may be topsy-turvy compared to ours.” He pointed to the physics primer. “You can be sure that we’re going to revise that.”
“So can you build from Fermat’s Principle to other areas of physics?”
“Probably. There are lots of physical principles just like Fermat’s.”
“What, like Louise’s principle of least closet space? When did physics become so minimalist?”
“Well, the word ‘least’ is misleading. You see, Fermat’s Principle of Least Time is incomplete; in certain situations light follows a path that takes more time than any of the other possibilities. It’s more accurate to say that light always follows an extreme path, either one that minimizes the time taken or one that maximizes it. A minimum and a maximum share certain mathematical properties, so both situations can be described with one equation. So to be precise, Fermat’s Principle isn’t a minimal principle; instead it’s what’s known as a ‘variational’ principle.”
“And there are more of these variational principles?”
He nodded. “In all branches of physics. Almost every physical law can be restated as a variational principle. The only difference between these principles is in which attribute is minimized or maximized.” He gestured as if the different branches of physics were arrayed before him on a table. “In optics, where Fermat’s Principle applies, time is the attribute that has to be an extreme. In mechanics, it’s a different attribute. In electromagnetism, it’s something else again. But all these principles are similar mathematically.”
“So once you get their mathematical description of Fermat’s Principle, you should be able to decode the other ones.”
“God, I hope so. I think this is the wedge that we’ve been looking for, the one that cracks open their formulation of physics. This calls for a celebration.” He stopped his pacing and turned to me. “Hey, Louise, want to go out for dinner? My treat.”
I was mildly surprised. “Sure,” I said.
It’ll be when you first learn to walk that I get daily demonstrations of the asymmetry in our relationship. You’ll be incessantly running off somewhere, and each time you walk into a door frame or scrape your knee, the pain feels like it’s my own. It’ll be like growing an errant limb, an extension of myself whose sensory nerves report pain just fine, but whose motor nerves don’t convey my commands at all. It’s so unfair: I’m going to give birth to an animated voodoo doll of myself. I didn’t see this in the contract when I signed up. Was this part of the deal?
And then there will be the times when I see you laughing. Like the time you’ll be playing with the neighbor’s puppy, poking your hands through the chain-link fence separating our back yards, and you’ll be laughing so hard you’ll start hiccupping. The puppy will run inside the neighbor’s house, and your laughter will gradually subside, letting you catch your breath. Then the puppy will come back to the fence to lick your fingers again, and you’ll shriek and start laughing again. It will be the most wonderful sound I could ever imagine, a sound that makes me feel like a fountain, or a wellspring.
Now if only I can remember that sound the next time your blithe disregard for self-preservation gives me a heart attack.
After the breakthrough with Fermat’s Principle, discussions of scientific concepts became more fruitful. It wasn’t as if all of heptapod physics was suddenly rendered transparent, but progress was steady. According to Gary, the heptapods’ formulation of physics was indeed topsy-turvy relative to ours. Physical attributes that humans defined using integral calculus were seen as fundamental by the heptapods. As an example, Gary described an attribute that, in physics jargon, bore the deceptively simple name “action,” which represented “the difference between kinetic and potential energy, integrated over time,” whatever that meant. Calculus for us; elementary to them.
Conversely, to define attributes that humans thought of as fundamental, like velocity, the heptapods employed mathematics that were, Gary assured me, “highly weird.” The physicists were ultimately able to prove the equivalence of heptapod mathematics and human mathematics; even though their approaches were almost the reverse of one another, both were systems of describing the same physical universe.
I tried following some of the equations that the physicists were coming up with, but it was no use. I couldn’t really grasp the significance of physical attributes like “action”; I couldn’t, with any confidence, ponder the significance of treating such an attribute as fundamental. Still, I tried to ponder questions formulated in terms more familiar to me: what kind of worldview did the heptapods have, that they would consider Fermat’s Principle the simplest explanation of light refraction? What kind of perception made a minimum or maximum readily apparent to them?
Your eyes will be blue like your dad’s, not mud brown like mine. Boys will stare into those eyes the way I did, and do, into your dad’s, surprised and enchanted, as I was and am, to find them in combination with black hair. You will have many suitors.
I remember when you are fifteen, coming home after a weekend at your dad’s, incredulous over the interrogation he’ll have put you through regarding the boy you’re currently dating. You’ll sprawl on the sofa, recounting your dad’s latest breach of common sense: “You know what he said? He said, ‘I know what teenage boys are like.’” Roll of the eyes. “Like I don’t?”
“Don’t hold it against him,” I’ll say. “He’s a father; he can’t help it.” Having seen you interact with your friends, I won’t worry much about a boy taking advantage of you; if anything, the opposite will be more likely. I’ll worry about that.
“He wishes I were still a kid. He hasn’t known how to act toward me since I grew breasts.”
“Well, that development was a shock for him. Give him time to recover.” “It’s been years, Mom. How long is it gonna take?”
“I’ll let you know when my father has come to terms with mine.”
During one of the videoconferences for the linguists, Cisneros from the Massachusetts looking glass had raised an interesting question: was there a particular order in which semagrams were written in a Heptapod B sentence? It was clear that word order meant next to nothing when speaking in Heptapod A; when asked to repeat what it had just said, a heptapod would likely as not use a different word order unless we specifically asked them not to. Was word order similarly unimportant when writing in Heptapod B?
Previously, we had only focused our attention on how a sentence in Heptapod B looked once it was complete. As far as anyone could tell, there was no preferred order when reading the semagrams in a sentence; you could start almost anywhere in the nest, then follow the branching clauses until you’d read the whole thing. But that was reading; was the same true about writing?
During my most recent session with Flapper and Raspberry I had asked them if, instead of displaying a semagram only after it was completed, they could show it to us while it was being written. They had agreed. I inserted the videotape of the session into the VCR, and on my computer I consulted the session transcript.
I picked one of the longer utterances from the conversation. What Flapper had said was that the heptapods’ planet had two moons, one significantly larger than the other; the three primary constituents of the planet’s atmosphere were nitrogen, argon, and oxygen; and fifteen twenty-eights of the planet’s surface was covered by water. The first words of the spoken utterance translated literally as “inequality-of-size rocky-orbiter rocky-orbiters related-as-primary-to-secondary.”
Then I rewound the videotape until the time signature matched the one in the transcription. I started play
ing the tape, and watched the web of semagrams being spun out of inky spider’s silk. I rewound it and played it several times. Finally I froze the video right after the first stroke was completed and before the second one was begun; all that was visible onscreen was a single sinuous line.
Comparing that initial stroke with the completed sentence, I realized that the stroke participated in several different clauses of the message. It began in the semagram for ‘oxygen,’ as the determinant that distinguished it from certain other elements; then it slid down to become the morpheme of comparison in the description of the two moons’ sizes; and lastly it flared out as the arched backbone of the semagram for ‘ocean.’ Yet this stroke was a single continuous line, and it was the first one that Flapper wrote. That meant the heptapod had to know how the entire sentence would be laid out before it could write the very first stroke.
The other strokes in the sentence also traversed several clauses, making them so interconnected that none could be removed without redesigning the entire sentence. The heptapods didn’t write a sentence one semagram at a time; they built it out of strokes irrespective of individual semagrams. I had seen a similarly high degree of integration before in calligraphic designs, particularly those employing the Arabic alphabet. But those designs had required careful planning by expert calligraphers. No one could lay out such an intricate design at the speed needed for holding a conversation. At least, no human could.
There’s a joke that I once heard a comedienne tell. It goes like this: “I’m not sure if I’m ready to have children. I asked a friend of mine who has children, ‘Suppose I do have kids. What if when they grow up, they blame me for everything that’s wrong with their lives?’ She laughed and said, ‘What do you mean, if?’”
That’s my favorite joke.
Gary and I were at a little Chinese restaurant, one of the local places we had taken to patronizing to get away from the encampment. We sat eating the appetizers: potstickers, redolent of pork and sesame oil. My favorite.
I dipped one in soy sauce and vinegar. “So how are you doing with your Heptapod B practice?” I asked.
Gary looked obliquely at the ceiling. I tried to meet his gaze, but he kept shifting it.
“You’ve given up, haven’t you?” I said. “You’re not even trying any more.”
He did a wonderful hangdog expression. “I’m just no good at languages,” he confessed. “I thought learning Heptapod B might be more like learning mathematics than trying to speak another language, but it’s not. It’s too foreign for me.”
“It would help you discuss physics with them.”
“Probably, but since we had our breakthrough, I can get by with just a few phrases.”
I sighed. “I suppose that’s fair; I have to admit, I’ve given up on trying to learn the mathematics.”
“So we’re even?”
“We’re even.” I sipped my tea. “Though I did want to ask you about Fermat’s Principle. Something about it feels odd to me, but I can’t put my finger on it. It just doesn’t sound like a law of physics.”
A twinkle appeared in Gary’s eyes. “I’ll bet I know what you’re talking about.” He snipped a potsticker in half with his chopsticks. “You’re used to thinking of refraction in terms of cause and effect: reaching the water’s surface is the cause, and the change in direction is the effect. But Fermat’s Principle sounds weird because it describes light’s behavior in goal-oriented terms. It sounds like a commandment to a light beam: ‘Thou shalt minimize or maximize the time taken to reach thy destination.’”
I considered it. “Go on.”
“It’s an old question in the philosophy of physics. People have been talking about it since Fermat first formulated it in the 1600s; Planck wrote volumes about it. The thing is, while the common formulation of physical laws is causal, a variational principle like Fermat’s is purposive, almost teleological.”
“Hmm, that’s an interesting way to put it. Let me think about that for a minute.” I pulled out a felt-tip pen and, on my paper napkin, drew a copy of the diagram that Gary had drawn on my blackboard. “Okay,” I said, thinking aloud, “so let’s say the goal of a ray of light is to take the fastest path. How does the light go about doing that?”
“Well, if I can speak anthropomorphic-projectionally, the light has to examine the possible paths and compute how long each one would take.” He plucked the last potsticker from the serving dish.
“And to do that,” I continued, “the ray of light has to know just where its destination is. If the destination were somewhere else, the fastest path would be different.”
Gary nodded again. “That’s right; the notion of a ‘fastest path’ is meaningless unless there’s a destination specified. And computing how long a given path takes also requires information about what lies along that path, like where the water’s surface is.”
I kept staring at the diagram on the napkin. “And the light ray has to know all that ahead of time, before it starts moving, right?”
“So to speak,” said Gary. “The light can’t start traveling in any old direction and make course corrections later on, because the path resulting from such behavior wouldn’t be the fastest possible one. The light has to do all its computations at the very beginning.”
I thought to myself, the ray of light has to know where it will ultimately end up before it can choose the direction to begin moving in. I knew what that reminded me of. I looked up at Gary. “That’s what was bugging me.”
I remember when you’re fourteen. You’ll come out of your bedroom, a graffiti-covered notebook computer in hand, working on a report for school.
“Mom, what do you call it when both sides can win?”
I’ll look up from my computer and the paper I’ll be writing. “What, you mean a win-win situation?”
“There’s some technical name for it, some math word. Remember that time Dad was here, and he was talking about the stock market? He used it then.”
“Hmm, that sounds familiar, but I can’t remember what he called it.”
“I need to know. I want to use that phrase in my social studies report. I can’t even search for information on it unless I know what it’s called.”
“I’m sorry, I don’t know it either. Why don’t you call your dad?”
Judging from your expression, that will be more effort than you want to make. At this point, you and your father won’t be getting along well. “Can you call Dad and ask him? But don’t tell him it’s for me.”
“I think you can call him yourself.”
You’ll fume, “Jesus, Mom, I can never get help with my homework since you and Dad split up.”
It’s amazing the diverse situations in which you can bring up the divorce. “I’ve helped you with your homework.”
“Like a million years ago, Mom.”
I’ll let that pass. “I’d help you with this if I could, but I don’t remember what it’s called.”
You’ll head back to your bedroom in a huff.
I practiced Heptapod B at every opportunity, both with the other linguists and by myself. The novelty of reading a semasiographic language made it compelling in a way that Heptapod A wasn’t, and my improvement in writing it excited me. Over time, the sentences I wrote grew shapelier, more cohesive. I had reached the point where it worked better when I didn’t think about it too much. Instead of carefully trying to design a sentence before writing, I could simply begin putting down strokes immediately; my initial strokes almost always turned out to be compatible with an elegant rendition of what I was trying to say. I was developing a faculty like that of the heptapods.
More interesting was the fact that Heptapod B was changing the way I thought. For me, thinking typically meant speaking in an internal voice; as we say in the trade, my thoughts were phonologically coded. My internal voice normally spoke in English, but that wasn’t a requirement. The summer after my senior year in high school, I attended a total immersion program for learning Russian; by the end of the summ
er, I was thinking and even dreaming in Russian. But it was always spoken Russian. Different language, same mode: a voice speaking silently aloud.
The idea of thinking in a linguistic yet nonphonological mode always intrigued me. I had a friend born of deaf parents; he grew up using American Sign Language, and he told me that he often thought in ASL instead of English. I used to wonder what it was like to have one’s thoughts be manually coded, to reason using an inner pair of hands instead of an inner voice.
With Heptapod B, I was experiencing something just as foreign: my thoughts were becoming graphically coded. There were trancelike moments during the day when my thoughts weren’t expressed with my internal voice; instead, I saw semagrams with my mind’s eye, sprouting like frost on a windowpane.
As I grew more fluent, semagraphic designs would appear fully formed, articulating even complex ideas all at once. My thought processes weren’t moving any faster as a result, though. Instead of racing forward, my mind hung balanced on the symmetry underlying the semagrams. The semagrams seemed to be something more than language; they were almost like mandalas. I found myself in a meditative state, contemplating the way in which premises and conclusions were interchangeable. There was no direction inherent in the way propositions were connected, no “train of thought” moving along a particular route; all the components in an act of reasoning were equally powerful, all having identical precedence.
A representative from the State Department named Hossner had the job of briefing the U.S. scientists on our agenda with the heptapods. We sat in the videoconference room, listening to him lecture. Our microphone was turned off, so Gary and I could exchange comments without interrupting Hossner. As we listened, I worried that Gary might harm his vision, rolling his eyes so often.
