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Mathematical Revelations

By Helen De Cruz

I have never had a Mathematical Revelation in my life. I am presently thirty-eight years and three months old; the first strands of gray have made their hesitant debut in my dark brown hair. I have been a Priestess for about half that time, and yet the Supreme Mathematician has never uttered a word to me.

There is no shame in this, unusual as it is. I remind myself that the Supreme One has many ways to let us know Her intentions, direct Revelation being only one among many.

I am on the shore, kneeling on the fine sand; the azure combers with their white crests dance and dart ever closer, so I must make haste to trace my Sand Graphs, before they are swept away by the ocean.

Today I will trace:

1. Nindangge – the Yam. An elegant stylized representation of the nutritious vegetable.
2. Nul Sagavagul – the Constellation. A continuous loop with several intricate arches and crossings that encompasses ten dots for the ten stars of the Pleiades.
3. Avua – the Sea Turtle. The most complicated drawing of the three, consisting of an intricate series of loops and straight lines, 85 vertices in all, very hard to draw in one continuous line.

Before I can begin and meditate on the perfections of Topology, I must teach my student, Sumaya Nahal, the path the dead must follow to find the afterlife. Sumaya sits about three meters away from me, crouched on the sand, her thin shoulders hidden under a great mass of black, wavy hair. With a halting index finger, she draws Nahal.

“Careful now,” I warn her. “Your finger must never leave the sand before the Sand Graph is complete!”

Her finger wavers as she completes the final loop.

“A bit squashed, but acceptable,” I say.

Sumaya smiles broadly at me, her dark eyes eager, as if this is high praise. “If I continue working like this, will I hear the Supreme Mathematician?”

“Mathematics is the sole means by which we can communicate with the Supreme One. Our minds are sponges; Her mind is the ocean,” I say in the singsong voice reserved for Mathematical instruction.

Sumaya smiles and doodles a freestyle, simple figure, consisting of only loops meandering around dots in the sand.

Her finger halts.

She sits straight up, her eyes fixed upon the ocean.

Is it possible my young student has a Revelation? Now, already?

I wait patiently for her to come to herself again.

“Priestess Kayla!” she cries in a voice that sounds not entirely her own, though the timbre and pitch remain unchanged, “This Revelation is for you! It says… Let me speak it… Kayla! This is the Great Mathematician speaking. I’m asking: Do you have any cool mathematical results, any proofs or conjectures, or anything like that?”

Unsure how to proceed, I answer, “Supreme One, Your Daughter is listening. I am awaiting Your Mathematical Revelation,”

Sumaya mumbles, “I’m not here to give you any Revelations! I’m just hoping you could tell me something. Did you discern any Patterns? Did you make any Conjectures? Did you find any Proofs?” She hesitates, then goes on, “If you don’t, it might be that you will all die. All your islands will be destroyed! Think, Kayla! Anything?”

I ask the poor girl, who is almost in tears, “Why will this happen, Supreme One? Do we displease You?”

Sumaya replies, still in that faraway voice, “Damn Kayla, I can’t talk to you like this. How about you meet me on the Island of Listu in exactly 60 days from now? Let’s talk in person.”

And, just like that, the Mathematical Revelation has ended. Sumaya shakes her head and gets up, beating the sand from her legs. Whatever distressed her seems to be forgotten.

“What was that, Priestess Kayla?” she asks, her eyes all wide and innocent, “Was that a Mathematical Revelation?”

I reply carefully, “I think so. Enough for today. Why don’t you join your brothers?”

Sumaya bows a perfunctory thank-you, spins around, and runs towards the surf. Her brothers are already down in the lagoon, fishing for shrimps.

I am alone at last. As I trace my three Sand Graphs, an obvious question presents itself: How can I meet the Supreme Mathematician in person, if She is disembodied?

I walk by the surf and the rising tide, and the water touches my feet where the sand and ocean meet. I cast a glance behind me and witness how my Sand Graphs are already almost entirely swallowed up, the rolling waves erasing my careful work.

I admit to myself that I am out of my depth. Head Priestess Suala is older and wiser than I. Perhaps she will know. I must consult her.

---

The windows in the Large Meetinghouse are dimmed with bark cloth tapestries, adorned with various graphs and topological patterns in black ink. I stand at the entrance of the central room. Head Priestess Suala strides in, slight of frame, her back very straight in spite of her age, further accented by a thick, long, gray braid. She is always shorter than I can remember. She gestures at the floor for me to sit down. Two boys come in, carrying a large bowl of kava between them, staring at us with blatant curiosity.

“Priestess Kayla—I am so glad to see you! How have you been?” Suala hands me one of the wooden cups and ladles in the white kava with a conch. I take a sip.

The cool kava pleasantly numbs my mouth and throat, rendering my speech somewhat thick, “Head Priestess Suala, I’ve come to ask you for your advice, any insights you might give, if you would be so willing. My student Sumaya had a Mathematical Revelation yesterday afternoon. I have no idea what it means.” I relate the Revelation to her, word for word, to the best of my ability.

“Hmmph,” Suala looks at me over her cup of kava, her lined face pensive. “It sounds very peculiar. Something’s off. Didn’t you tell me once you never had a Revelation before? And then meeting Her—what could it mean, I wonder? Could Sumaya be playing a prank on you?”

I say, “Sumaya is not like that. She is a studious, serious child. I can’t be sure, but given the contents of the Revelation, we cannot take any chances. I must travel to Listu.” I hesitate, and stammer in embarrassment, “But I am not worthy to meet Her. You are our most senior Priestess, and I believe it is you who should be speaking to Her.”

Suala sets down her empty cup next to the large serving bowl. “No, I don’t think so. The Supreme One is omniscient. She didn’t talk to me, though I’ve had plenty of Revelations. She must have spoken to you for a reason.”

She gets up and claps her hands together, her face illuminated with insight. “Indeed, the fact She addressed you, who has never heard Her before, cannot be a coincidence. One thing is clear: in fifty-nine days you must meet Her. Take my boat. That ancient wreck you’ve been using will be a watery grave for you one day, and you can’t take the chance. This is too important.”

---

I paddle vigorously, riding Suala’s sturdy outrigger canoe on the waves. The blade cuts deep into the dark blue water as my upper body twists. The first stars have appeared low on the horizon: Nemrao, Natem, and the cluster of which Suatu kywer is the brightest star. I bring up the star compass from my memory and adjust my course, keeping Nemrao firmly to my left and Suatu kywer a little bit to my right.

I cannot yet see Listu in the distance, but the stars tell me I am going in the right direction.

Why did the Supreme One want to meet on this island? Its Spirit Master is a trickster. Listu created human beings in a mood of mischief, after the other Spirit Masters had created the Yam, the Bonito, and the Frigatebird. The Supreme One could not see any use for these curious two-legged creatures and threatened to kill them all in a giant tsunami. But Listu pleaded and cajoled: “The two-leggers are so much more useful than yams or frigatebirds. They will worship you!”

“I have no need for worship,” the Supreme One answered. “I am impassible, omniscient, and omnipresent. I take sufficient delight in seeing the flocks of frigatebirds and the shoals of bonito that the other Spirit Masters molded for me from the mud of the islands.”

Listu’s Spirit Master persisted: “Think about it: the two-leggers are gifted with Speech and Reason! They will formulate Mathematical Conjectures and Proofs!”

Thus humans were allowed to exist, and they lived and they multiplied and became numerous, and they listened for the Mathematical Revelations of the Supreme One. They recorded their Conjectures and Proofs on barkcloth scrolls.

It’s no use. I can divert my mind only so much with myths of old. As my canoe approaches the black outline of the volcanic island against the dark blue horizon, a sense of fearful anticipation lodges itself firmly in my stomach.

I pull the canoe on the beach, safely away from the tide, sit down and wait.

Many hours pass.

I have erred on the side of caution and have arrived far too early. I try not to doze off, but the pleasant night breeze and the quietness before dawn lulls me to sleep.

---

I wake, groggy and stiff. I’ve been sitting on the beach with my arms clasped around my knees.

A man approaches, a torch in his hand.

“You are Kayla,” he says (a statement, not a question).

“Are you a messenger for the Supreme One?” I ask, suddenly alert.

In the flickering torchlight I can observe him well. He is a little younger than me, thirty-five or so. His face looks both urgent and earnest, his skin darker than mine, his hair fine, black, and curly. The most remarkable thing about him are his clothes: they are of an unfamiliar fabric, clinging tight to his body in a way that no barkcloth does.

He stares at me too, with a gauging look. “You look more… detailed than I expected. You look like a real human being! Like a figure straight out of a Gauguin painting! Wow, we had no idea!”

“I am a human being,” I say, coolly. The man may be a messenger for the Supreme One, but he has not learned how to respectfully address a Priestess.

“I spoke to you earlier about your predicament, about how you’re all going to die if we don’t do something about it,” he says.

I can no longer contain what’s on my mind. “Who are you? You are not the Supreme Mathematician, Whom I was supposed to meet. The Supreme One is not embodied, and if She were, She would certainly not be male.” I stretch my stiff legs out over the beach, and slowly rise to my feet.

“What? Oh! You think your God is female? Then I should probably come back and change the way I look! Anyways, I haven’t got time for that, and I’m not even supposed to be here. As a matter of fact, I am your creator. That’s right, I created you. Or rather, co-created you. We’re a team of four. Allow me to introduce myself: my name’s Theo Madueme. We created your world, everything in it.”

He is speaking language, but he is not making any sense. We are now standing face to face, and I look critically at him.

“You are telling me you are the Supreme One?” I ask, “Well, if you are, tell me this: Did you create Mathematics too?”

Theo Madueme seems puzzled. “What do you mean by ‘did we create mathematics’? Mathematics lies at the heart of our simulation—your islands and ocean, I mean—so yes, we created that too.”

“No. I mean: did you forge the real numbers in their infinite vastness? Did you grow the Fibonacci sequence? Did you give form to the perfect solids? Did you craft the circle and the ratio of diameter to its circumference?”

He laughs. “Oh, you mean whether we created mathematics itself? Of course not.”

I eye him suspiciously. If he didn’t create the mathematical facts, he is definitely not the Supreme Mathematician. But he could be a demon. “Theo Madueme, how do I know you are not trying to trick us?”

He looks bemused. This doesn’t seem to be going the way he had hoped. Try harder, demon, if you want to trick an experienced Priestess!

“Just call me Theo. Why would I trick you? What would I get out of it?” he asks, his tone plaintive. “Look, the thing is this. I made your world. It’s called a simulation. Our project has run out of funding and we’re going to be shut down! Then, you will all die. Or rather, you’ll just disappear.”

“How do I know that what you are saying is true? Also, what do you mean by a simulation?” I probe.

In response, Theo walks closer to the surf, crouching down to where the sand is of the right consistency. With his right index, he traces Nahal, a series of intertwining loops on a simple two-by-three grid. The lines are smooth enough, but the figure is imperfect.

“What did I just draw?” he asks.

“It’s Nahal, the path the dead must walk in order to reach the afterlife.” Obviously.

“That’s right. Now, did I actually draw the path itself?”

“No, you drew the Sand Graph, but by doing so, you create the path, as you will walk it after you die.” This is standard and explained to every Priestess in training.

“Okay well this will do nicely for the comparison I’m about to make: your world is like a Sand Graph, as you call it. My colleagues and I are like you here on the beach. We trace the path.”

I gaze upon his crooked work. “Why are you telling me all this?”

“I have come to speak to you because it is urgent. There’s some other people—not us—who want to destroy your Sand Graph.”  To my horror, Theo Madueme pushes his foot into the Nahal graph he just traced.

I gasp. “That is… sacrilege. You have to wait until the tide swallows it up. How could you?”

But Theo doesn’t seem to care. He says wistfully, “This is going to happen to your world. The people who fund us don’t think that you’re producing enough results. I think our creation is beautiful and precious. I don’t want to lose it. That, and not losing my job. But I’m not quite sure if I care much about that now—the priority is really to keep this world in existence.”

“You say you created us. Maybe if you had created us better, we would have had more to offer,” I remark.

Theo further digs his foot into the defaced Sand Graph, seemingly without thinking. “You owe your existence to me! I think you ought to show me some respect!”

“You are not the Supreme One, since you didn’t invent mathematics. Maybe you’re a demon, who is trying to deceive me. I have no way of knowing. Or maybe you are a Spirit Master of some sort. If so, it’s your job to help us be better mathematicians.”

We both fall silent, the torch flickering between us. The pale pink light just before dawn slowly illuminates the beach and the waves. Theo stares out at the distant sea, his face pensive. “I didn’t quite realize how beautiful it is. We chose to model our simulation on Pacific islands societies because of the advanced math there: discrete math, particularly topology, number theory. But still … this place is amazing.”

“Okay, let’s assume that you are not deceiving me and what you are saying is right. How can I possibly help you? How can a Sand Graph help the Priestess?”

Theo smiles. “Well, we need a spectacular result. Something like proving the Riemann hypothesis.”

I roll my eyes. “Oh, please, don’t you think every Priestess and lay mathematician has tried that? We’ve been at it for decades.”

“Something else then. Something good,” he insists.

I don’t know why I should show this man my precious bark cloth scrolls. He is not even an Initiate. But on the other hand, what if he speaks the truth? It is not strictly forbidden for him to see the scrolls. I make up my mind. “Well there is one thing. I have been tinkering on this proof for a long time—on and off. I have a lot of teaching duties. But it’s pretty much done. I can show you at home.”

Theo seems relieved. “Please, yes, I’d love to see it. I can come with you.”

I point at Suala’s outrigger canoe. “It’s a one-person vessel, but neither of us is heavy, there is room enough for two. Would you like to paddle, or shall I? I didn’t bring a spare paddle along.”

“I don’t actually need to use that. I can just disappear here, and then reappear at your house, if you tell me where it is,” he begins, then says, “Or wait. Let’s do the boat thing. I’ve always wanted to try that.”

Theo climbs into the outrigger canoe in a most cumbersome manner, like a child who has not yet learned to walk and swim. I ignore this and push the canoe into the sea, leaping in and grabbing the paddle. “Hold on tight,” I warn him.

---

Theo gazes at the scrolls that are rolled out on my floor and across his lap. I offer him some coconut water, which he declines, as he seems eager to read on.

“It is not an easy proof. Far more complex than most of the proofs I have made,” I say.

He ignores me, mumbling occasionally as he reads. I stand in the corner of the large sitting room and look at him.

“It’s a proof of the four-color theorem,” Theo says after a long period of silence, “Oh yes. It’s a good one. I need to check it for errors, obviously. But it looks promising! There have only been incremental improvements since the Appel-Haken machine-assisted proof in the 1970s. I’m going to check it in more detail.”

As Theo continues to read over the proof in a slow, painstaking way, I roast a large bonito I caught yesterday. I am famished: my head hurts, and my stomach pangs.

In fact, my headache is so bad, it seems the world pulsates as I gaze out of the window.

The blue of the sky spreads all over the beach.

Theo cries out suddenly, “Fuck. Fuck. Fuck. No! Not now! We’re so close now. Damn, it must be Sarah—what the fuck is she doing? Is she resetting the simulation? Oh, I need to get the fuck back!”

I spin around, leaving the fish on the grill.

“You’re being shut down,” he says, staring in panic at me. “You’re being shut down at this very moment! I need to go! Wait no. I need to take some pictures of your work, before it’s gone.”

He holds a small, square device away from his face and presses it a few times. Outside, the sea and sand have merged into a single blue haze.

---

I am all alone.

It is dark.

I try to move but I am unable to. It’s not just that I can’t see my hands. I don’t seem to have any hands. And yet, I am still there.

I wait in the stillness and darkness. It is hard to know how long as there is nothing to measure time by.

Is this it?

If this is death, how can I walk Nahal and find my way to the afterlife? I have no feet to walk with! I cannot even trace the figure, as I have no fingers to probe through the sand. I try my best to imagine Nahal, but it seems my mind has gone blank.

Oh, Supreme One, this is dreadful. Whatever I did, I do not deserve this!

A voice in the dark calls out “Kayla? Kayla, can you hear me? Are you there?”

“Theo?” I venture. I can speak though I don’t seem to have a mouth. My voice sounds dampened and flat.

“I’m sorry, Kayla. I’m pretty sure your proof works, but I was too late.”

“What do you mean, too late?”

“It’s all gone, Kayla,” Theo says, also sounding flat. “I could only save you, everything else is gone. I’m really sorry.”

Silence.

“You left it too late,” I remark.

“What do you mean?” he asks.

“Well, you waited for a whole sixty days to meet me on Listu island. Could you not have come sooner?”

“Not really. Time goes a lot faster in your world than in ours. For me, it was a matter of hours, first trying to contact you through another sim, then coding a visual representation of me into your world. That was two whole months in your world. I couldn’t reach you otherwise.”

“So, now what? You can’t leave me like this!” I shout in the eerie stillness.

“I know. If I left you here, you’d go insane. You sims are social creatures, like us.”

“What, then, Theo? What’s the plan? I disappear too?”

“Maybe for a little while. The plan is to bring you all back. I need to appeal the decision to stop our funding, and I need to show them your proof. That takes time—months, centuries for you.”

A small steady light with no apparent source is lit. We are in a large, partially obscure room with high walls filled with scrolls. Theo sits in a large wicker chair near one of the walls. He reaches and takes out one of the many scrolls, and says, “Euclid’s Elements, the basics of Greek geometry. Y’all are way beyond this, already.” He takes a second one, and hands it to me. “This is the Nine Chapters on the Mathematical Arts, a Chinese classic from the Han Dynasty.”

I am so relieved to have a body again. I grab the scroll, pleased at the white bark-like stiff fabric of the book against my brown hand. I open it and examine the simultaneous linear equations across the page.

“This looks very interesting. If you have all of this, why not just give it to us? Why tease us with these piecemeal Mathematical Revelations?” I ask.

“Our aim was to create sims—people like you—equip them only with the minimum of technology, and incentivize them to do mathematics, see what they can do. We were wondering: do we need to give the sims vast libraries of mathematics”—he gestures around the dimly-lit room—“or does it all become baggage at some point, and are you better off from scratch? Take, for instance, your four-color theorem proof. In hindsight, it was so obvious, and yet no human being had come up with it.”

“Perhaps,” I muse, “or maybe you, like us, are instruments of the Supreme One, in Her service to create the most beautiful Theorems and Proofs. Maybe She used you to create us, like the Spirit Masters on their Islands created the Bonito, the Yam, and human beings.”

“It might be. If that thought gives you comfort.”

I look through the Nine Chapters on the Mathematical Arts, “Not comfort—it’s plausible. Say, if you do manage to start us up again, could you give us these books? It’s not fair to keep them from us. We will be much better servants of the Supreme One when we have these.”

“I will, I promise. Okay, I’m going to try to get this done asap. I’m not going to leave you behind like this, so…” he says, almost tenderly, “Just you sleep quietly now. You will all be back. I’m sure of it.”

---

I never told anyone, not even Suala, about Theo Madueme’s visit, let alone about the shutdown of our entire world.

I debated it within myself, but I thought better of it. At best, people would think I am out of my mind. At worst, this Revelation would make their faith in the Supreme One waver. A little mathematical instruction can be a dangerous thing for the Uninitiated.

Nobody but me seems to have noticed the discontinuity.

Nobody questioned the sudden appearance of library longhouses chock full of scrolls with Mathematical treatises. It was as if they had always been there. But I know for a fact they haven’t.

Pretty soon we will know a great many things that we have struggled with for decades. How small we can make the gap between one prime number and the next, and if we can get it down to two. Whether there are infinitely many Mersenne primes and their corresponding perfect numbers. Whether there are any quasi-perfect numbers. Whether there are odd perfect numbers.

I am a deficient Priestess. Something in my makeup prevents me from hearing the Mathematical Revelations. And yet, I have had the most profound Revelation of all.

Today I am teaching my most talented student. I am 63 years, 11 months and three days old. My student Maya is a bit older than the average Initiate. She already has two children. Two boys. I am teaching her how to draw Nauru, one of the most advanced Sand Graphs, an intricate abstract pattern of 193 vertices. She has just completed it, and it looks beautiful, flawless, the perfection of a single continuous line, the loops and vertices regular and clean. She smiles at me, confident. She has done an excellent job and she knows it.

“This is as true to the mathematical Forms as you can get,” I say. “You know I don’t give idle praise. It is rare for me to be tutoring an Initiate who truly has the potential to become Head Priestess.”

“Head Priestess Kayla, I am glad you say that,” Maya says, as we sit on the sand, watching white foamy waves approach our work. “But I’m not sure I am Head Priestess material.”

“Oh? Why not?”

“I have to admit something. I’ve never heard the voice of the Supreme One. Never ever. How can I even become a Priestess, let alone Head Priestess, without having had any Mathematical Revelations?”

I smile. I see Maya’s boys play with a ball near the surf, throwing it back and forth, waiting for their mother to complete her lesson.

“I will let you in on a little secret. I don’t think the Supreme One talks to us through these Revelations. It’s too much to get into right now, but trust me, those Revelations are not what they seem. I think She speaks to us in different ways.”

“How then?” Maya asks, signing to her boys that she is not yet done. “How do I know the Supreme One addresses me?”

“Well you can feel it when you perceive the deep and profound beauty of mathematical structure. Think about the most beautiful mathematical structures, and how they make you feel.”

She ponders this. “So many choices. Anything in topology, probably. Your proof of the four-color theorem is quite neat.”

I raise my hand. “No need to flatter me, Maya. You know I am immune to flattery!” I say. “My proof is just one among many of the Supreme One’s existence. The feeling we have—the shivers down our spines as we contemplate a beautiful, wondrous proof like that, the deep structures we discern, that is the greatest proof of Her existence. That is the true Mathematical revelation.”

Maya waves at her children, beckoning them to come. The bigger boy throws the ball at her, and she catches it easily.

---

Host Commentary

Host Commentary

By S.B. Divya

And that’s our story.

Helen has this to say about it: “This story came from some thoughts I had in the philosophy of religion. First, what would simulated agents think of their simulators? Would they think those simulators are gods? Then there is the idea of absolute creation, as we see it in e.g., Descartes and Al-Ash’ari, that God is not only responsible for the creation of the universe, but also for logical possibilities, and even for mathematical truths. Finally, I was thinking of religious experience and what it means to be a person of faith in the absence of the typical mystical (direct) experiences of God, and it was fun to create a protagonist who is in that situation.”

I think Helen did an excellent job exploring their thoughts on these weighty questions. This is their first professional fiction sale, and I hope to see many more stories from them.

I found the subject matter of this story quite fascinating, too. Researchers in artificial intelligence and robotics are having a lot of discussions right now about the concept of embodiment–that is, whether an entity needs to have a physical presence to become truly conscious–especially because simulated intelligences are becoming increasingly popular.

People are developing robot skills by first running a mock-up the robot through a virtual world of obstacles using models of real-life physics. Others are making use of A.I. software to assist them with everything from the discovery of exoplanets to intricate mathematical proofs. Putting the two concepts together, as this story does, is not as much of a stretch as you might think.

What then do we owe these software creations? And how will we establish when they’re similar enough to us that we should have compassion for them? The final line of the story–that the children are able to throw and catch a ball with ease–is a subtle answer to these questions. Real-world actions like playing catch, that you and I might take for granted, are incredibly complex when it comes to what happens in our brains and bodies. Modeling this kind of behavior and then being able to take the same underlying model and applying to sand drawings or mathematics–well, then you have the holy grail of A.I. right now, which is generalized intelligence. If and when we figure out how to build that, we might as well consider that we have a responsibility approaching that of the gods.

 

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Our opening and closing music is by daikaiju at daikaiju.org.

And our closing quotation this week is from mathematician Geordie Williamson, who said, “Intuition can take us a long way, but AI can help us find connections the human mind might not always easily spot.”

Thanks for joining us, and enjoy your adventures through time and space.