Irrationality

by Stephen Case

for Justin, with thanks


I first heard about the Flattening at one of our weekly faculty lunches, which we had moved from the campus cafeteria to the pub across the street and pushed from lunch to mid-afternoon as soon as the university relaxed its drinking policy. I was nursing a Guinness when the 3 PM crowd wandered in.

“Have you heard yet?” It was Temple, the computer scientist. “Pi’s gone crazy.”

Calendar from Modern Languages took a seat opposite. “Pies?”

“The number. Something’s wrong with it.” Temple ordered a coffee and, when the server left, rubbed his hands together. “It’s still sort of a test in the computing world—calculating pi out to a trillion digits or so. Takes a lot of networking and processing power. Not exactly a piece of cake. Or pie. No pun intended.” He chuckled.

You didn’t need to be on your second beer on a Friday afternoon to stand Temple, but it helped.

“Anywho, the new record holder is apparently an exotic matter processor in Tehran that calculated pi out further than anyone has before by an order of magnitude.”

“A quantum computer?” This was Nichols, who took the seat beside Temple.

Temple waved his hand dismissively. “I don’t know. But the thing is, once they got out to around the quadrillionth digit, it flattened out.”

I frowned. “What do you mean, flattened?”

Calendar looked up from his phone, where he had been scrolling. “Is this ALMOND? The name of the computer?”

Temple nodded, and Calendar handed the phone to me.

“Zeros,” Temple continued, leaning forward. “It’s just gone to zeros. Apparently, they shut the entire system down, thinking it had made a mistake. Took them a month to repeat their calculations out that far, but when they did, they found the same thing. Zeros. They say they’re still calculating and haven’t found the end of it yet.”

“I thought pi wasn’t supposed to end,” Nichols ventured. Drinks had come, and the rest of them sipped while I read.

Temple had the details right, more or less, according to the news brief Calendar had pulled up. I had never heard of the supercomputer, but that was no surprise. According to the article, several other computing groups were trying to verify the results, but no one had made it as far as the Iranians yet.

“It has to be a mistake,” Calendar offered when I looked up.

Nichols crossed their arms. “Explain to the biologist, please.”

“Pi’s random,” Temple said, glancing at me for affirmation. “That means it goes on forever, never repeating and always random. So either this result is wrong—” He paused for effect. “Or math is.”

I shook my head.

Calendar, satisfied we had a legitimate topic of discussion, had slipped the phone into the pocket of his vest. “How does a supercomputer calculate pi anyway? Measure a bunch of circles and divide their circumferences by their radii?”

“There are a few different methods.” I studied the foam along the side of my glass. “The simplest would be adding more and more terms to a series known to converge to pi. The article didn’t say what they used.” I emptied the glass and leveled a finger at Temple. “But pi’s not random. It’s irrational. There’s a difference. Irrationality means you could have what looks like a pattern, but it would never be a true pattern. It could run for any conceivable length—maybe one-two-three-one-two-three for a billion digits. But eventually the ‘pattern’ would break.”

“It’s gone to zeros for over a hundred million digits at this point,” Temple said. “Surely that means something.”

“And with every additional zero,” Calendar ventured, “that makes it less and less likely the next digit would be zero again. So it gets stranger the longer the run continues, right?”

“Wrong.” I shook my head again. “Flip a coin ninety-nine times. If it lands on heads every single time, what are the chances that it will land on tails the one hundredth?”

“Fifty-fifty.” Nichols sounded uncertain.

“Right.”

“Wrong.” Temple snorted. “With ninety-nine heads, you’ve got a broken coin. Maybe with this many zeros…”

Calendar finished the thought. “You’ve got a broken universe?”

I stood. Complaining about university politics and disinterested students for the rest of the afternoon suddenly felt less compelling.

“The pattern will break,” I said again.

***
Within a week, the Flattening had worked out to the wider media headlines, especially once other groups verified it using different computational methods and their own supercomputers. There was no doubt: at the quadrillionth decimal place pi became an unmitigated series of zeros, and it stayed that way as mathematicians and computer scientists raced to see how long the run would continue. It was like the space race all over again, trying to see which team would first get across that seemingly limitless sea of zeros.

My 9 AM number theory course quickly became the most popular class on campus, and I became a minor celebrity simply by my role as math professor and assumed authority on the number pi. Students, most of whom weren’t even enrolled in the course, stayed after peppering me with questions.

“No,” I responded, “this isn’t a change. The value of pi has always been what it is.”

“No, I don’t think our proofs of its irrationality are incorrect.”

“No, there’s no way to predict when it will break.”

“No, I don’t think math is broken.”

“No, I don’t think this proves we’re living in a simulation.”

“No, I don’t stay up at night worrying about it.”

This last was at least partially a lie. I did lie in bed thinking of it, of a string of zeros stretching forever toward the horizon. I couldn’t help wondering if I would feel as haunted if it had been any other repeating digit, if pi had “flattened” to a lines of ones, for instance, or any other integer. The zeros felt ominous, as though mathematicians had fallen through the basement of the world into empty space below.

I installed a popular app that relayed real-time calculations from one of the US supercomputing teams and watched the zeros continue to cascade across my screen as the calculation pushed forward, waiting for the next digit that had to come, that would come, and break the spell.

When I found myself staring at my screen for hours, I deleted it.

I was a mathematician, so as I saw it there were two possibilities—though only one that would let me keep moving forward through life with the conviction that the world was a sane place. The first was what I had told Temple and the others, that the Flattening was simply a sequence in an irrational number. Not only was it of no significance, but if you used any other base to calculate pi, the sequence of zeros wouldn’t even be there. It was only a long string of zeros because we were representing the number with our biased count-to-ten-on-your-fingers numbering system. Flip it to binary or base-12 (the shadow system still hidden in our clocks and in English’s unique names for 11 and 12), for instance, and the Flattening didn’t look so flat at all.

The second suggestion was that we were indeed wrong about pi, that it was not irrational, that it had a specific value expressed precisely in the string of a quadrillion or so digits that terminated when the first zero in the Flattening appeared. If this was the case, no matter what base you used for your calculation, eventually pi would run to zeros or its digits would begin to repeat in a truly endless pattern.

But if that was the case, pi was not irrational, Temple was right, and math was wrong.

***
It was a Thursday morning. I was staring at a pile of ungraded quizzes when Temple burst through my office door.

“The Fibonacci sequence!”

He was nearly beside himself.

I knew what he meant without asking and opened a tab on my browser that showed the latest real time pi calculation results. Instead of zeros, there were digits again, streaming by too fast to follow.

“It broke.” Temple was visibly working to get ahold of himself. “But now it’s a sequence. They’re saying it’s Fibonacci.”

I paused the app and scrolled the digits backward with a sense of foreboding. Temple watched me as though he’d seen a ghost. It took several minutes before I found where the stream of zeros had given way to 1, 1, 2, 3, 5, 8, 1, 3, 2, 1…

“This only makes itself apparent—”

“In base-ten!” Temple almost shouted. “Which means it’s a message. Which means it was put there for us.”

“No.” I shook my head stubbornly, ignoring the prickling sensation moving down my spine. “It means nothing.”

Too late, I remembered Temple’s daughter, her treatments, his leave of absence last year—and what it must mean for him to believe what was happening with pi indicated something remarkable about the universe.

I tried again, my voice softer. “I’m sorry Temple, but this is just a number.”

His voice quavered. “How can you look at that and not see it for what it is?”

If I had been alone in my office, I might have felt the way Temple did. But I wasn’t. I was watching Temple, seeing the wide eyes behind his glasses, the sweat around his collar, his grasping for a confirmation that the universe had a barcode, a signature, some evidence of meaning. Having him here, so obviously giving this number significance, made my own position clear. Pi was just an irrational number, no matter how much he might want it to be otherwise.

“I saw a face in the clouds once.”

His brows drew together. “What?”

I had never shared it with anyone, but the memory was still crystal clear in my mind: walking home from school, watching the clouds and seeing, outlined in the shape of cumulus, the face of my dead grandfather as sharp as though it had been carved by a sculptor. I told Temple, told him what I had believed about it then, told him what I thought about it now.

“That was a single kid seeing what they thought was a miracle.” Temple wore an expression of confused pity. “And maybe it was—”

“It was random shapes in cloud.”

“But this has been confirmed by the entire world. By mathematics itself. There is a pattern in pi, a message for us.”

“Pi is irrational. This is not a pattern.”

He left my office as abruptly as he had entered, shaking his head.

***
Later that afternoon, my dean walked in.

“Busy?” she asked. “We’ve cancelled classes for the afternoon. Because of the…” She waved her hand vaguely, looking exhausted.

“The message from God?”

That was how it was already being interpreted across the internet, as a message from God or some other kind of intelligence. I had been reading media outlets since Temple left. What else could people think, seeing what they perceived as a clear pattern of intelligence embedded in the fabric of reality itself? Millions of people like Temple, primed to find proof of creation, of significance, of meaning in the world around them. And here it was, in a number that even an elementary school student could understand.

She sat in the same chair Temple had used and closed her eyes. “I’ve been on a call with three pastors and one bishop. We’re a denominational college, and now suddenly…” She picked a paper icosahedron off my shelf, studied it for a moment, and set it back down. “My entire career, I’ve tried to thread the science and faith needle. I’ve kept the intelligent design people off our backs, made sure the fundamentalists didn’t interfere with what we taught, and now…” She shrugged. “In biology we talk about those infinite monkeys typing, how you’ll eventually get Shakespeare if they type long enough. I know this is the same thing, but no one seems to believe me.”

“Actually.” I paused for a moment, recalling Temple’s face. I had always loved the philosophy of mathematics, I thought wistfully. I didn’t ever dream it would one day become the theology of mathematics. “Actually, it’s not that simple.”

“What do you mean?”

“Just because a number is irrational, that doesn’t mean it has to eventually contain every pattern imaginable. It might reach a certain digit and never have another seven, ever, for instance. That’s… allowed.”

Her face was ashen. “So, a pattern this like, like what we’re seeing on the news right now, that’s not inevitable?”

I shook my head.

“Oh.”

***
Late that night the ascending Fibonacci sequence ended, and the next several digits of pi’s value were calculated to be what appeared at first glance a welcome return to random numbers. Then, almost immediately in computing time, the reverse Fibonacci started. It was as though a symmetrical mountain had reared up out of the metaphorical fog of zeros, reached a summit of some kind, and then begun descending back down the opposite slope. Or, as voices on the internet were immediately claiming, the string of zeros had been a preamble, something to get our attention, followed by the Fibonacci sequence to indicate intelligence, followed by… a message?

The next morning, the president of the American Academy of Sciences called a press conference. I hadn’t gone home for the night, and watched it from my office. The most surreal thing—more than the fact of an emergency press conference for the result of a mathematical calculation—was the expression she wore as she gave a brief explanation of pi and the methods used for calculating it. She wore a controlled mask of terror, as though she was carefully explaining the anatomy of a tiger about to pounce and she was the prey. This was, she informed the gathered reporters, a profound and potentially disturbing (her words) mathematical result.

“Once the pattern of the ascending Fibonacci sequence ended,” she intoned, “the forty-one digits following composed a sequence of numbers that were immediately analyzed by the best experts in the mathematical, linguistic, and cryptological communities. The digits were found to correspond with a simple alpha-numeric substitution code in which each number represented a letter in a…” She coughed. “A particular ancient semitic language.”

The internet had been on fire all night with speculation and analysis. Several sources had already leaked what she was about to say, but I needed to hear her say it.

“The string of digits translates—or rather, corresponds—to the following text: Hear—” Her voice cracked, and she paused to try again. “Hear, O Israel, the Lord your God, the Lord is one.”

Pandemonium erupted in the room.

***
That afternoon I was at my normal spot in the pub, staring at the circles of condensation my glass left on the tabletop, wondering whether mathematics had betrayed me. It wasn’t that I didn’t believe in God. Far from it; I had been a faithful believer my entire life, more or less. It was that I didn’t think belief should work this way.

Calendar and Nichols showed up, their faces bearing an expression I couldn’t interpret, almost as though they were afraid for my sake. I tried to explain how I felt.

“It looks like a pattern, but it’s not.”

“How can you not see this as a proof?” Nichols glanced at Calendar for support. “It’s right there—that God exists.” They leaned backward. “It always took more faith to be a believer before. Now it’s the other way around. You would need the faith of a zealot, an extremist, to doubt.”

“It’s irrational.” The words seemed hollow now, almost blasphemous.

“But what about the fact—the fact—that the message or pattern or code or whatever is only apparent in base-10?” Nichols wiggled their fingers. “A message encoded specifically for us. A message that would be clear only if a species had developed to count on ten fingers.”

“Which is what the word digits originally meant,” Calendar added, then looked immediately apologetic that he couldn’t help offering etymologies regardless of the context.

“Things that look like patterns will appear in an infinite sequence,” I said. I should have been feeling the world shake around me as my paradigms gave way, but I didn’t. I didn’t think I was wrong. I had faith in the irrationality of pi.

“The message doesn’t depend just on our biology,” Nichols pressed, “but on our history. It’s only apparent for a culture in which there’s a language where those particular numbers spell out a holy passage known to several faith traditions.”

“But the Fibonacci sequence would be enough, right?” Calendar had not touched his ale. “I mean, even if we had somehow missed the—the bit in the middle. The Fibonacci sequence is universal, right? So anyone would have realized something was going on when they saw that.”

I sipped my beer. “What about the six-fingered inhabitants of the Andromeda Galaxy? What message would they find?” I set the glass on the table and pushed it away. “In an infinite universe, there would be an infinite number of mathematical and cultural systems to find an infinity of significant ‘messages’.”

“You have more faith in math than in God,” Nichols said.

I spread my hands out on the table. “I believe in God. I just don’t need math to prove it.”

“Couldn’t it be both ways?” Calendar looked from one of us to the other. “I mean, if Nichols found a species of bacteria or whatever that had proteins coiled in such a way to spell out ‘Designed by God’ in English, we could still claim the laws of nature gave rise to that structure and also that God had ordained the laws of nature so such a thing would come to be. In that case, the miracle is just a bit bigger—the entire universe from beginning to end.”

I shook my head. “That’s exactly my point. You can’t use anything in an infinite universe or an irrational number to prove God. If the universe is infinite you would have to eventually find a planet with an organism whose DNA spelled that out. And you’d also have to find one that said, ‘Not made by God.’ And then another planet where the DNA, or whatever is equivalent there, spelled out ‘Made by God’ but ‘Made’ was spelled M-A-I-D.” I paused. “Calculate pi for another billion years and you might find the message that cancels this one. Or that repeats the entire thing but with one digit off.”

“But the chance of finding this pattern so soon in the sequence,” Calendar began.

“Leave it,” Nichols said, shaking their head slowly. “It doesn’t matter. You’re not going to change his mind. He’s already chosen how to interpret it.”

“A choice.” I laughed, but they both probably heard how bleak it sounded. “That’s all it’s ever been.”

***
The call from the university chancellor came sooner than I expected. By now the continued calculation of pi had settled back to a sequence of zeros. Three supercomputing facilities had shut down due to bomb threats, at least one for which a new militant atheistic group claimed responsibility.

“Mathematics proves God!” the chancellor said, spreading his arms and beaming when I was shown in. “Personally, I was always afraid it was going to be the other way around.”

“Mathematics can’t disprove God, sir.”

If he saw my weariness and the fact that I had been splitting my time between my office and the pub for the past several days, he chose to ignore it.

“I need your help putting together a video release.” He said it like he was doing me a favor. “We’ve got the secular state schools on the back foot now! The dean of the school of theology says they’ve been dealing with a flood of applicants for the past week. Some people are likening this business with God speaking through pi to the Second Coming—not that I’d go that far. But it is revelatory! It changes everything. Donations are pouring in.” He leaned forward across his wide desk. “We need to capitalize, get a statement out there. You’ll take a minute to explain the math, and then I’ll talk about what this means and why our university is the place to learn about the God who has been revealed in the very fabric of the universe!”

The Revelation, as it was indeed already being referred to, allowed several major faith traditions to claim vindication. Yet from what was happening around the world in the immediate aftermath, it didn’t seem an especially irenic new age had been revealed.

“It’s irrational,” I said.

His brow furrowed momentarily. “The complaint was always that we were the irrational ones. But what could be more rational now than to believe?”

“I’m sorry.” I didn’t even try to keep the weariness out of my tone. I wondered if he could smell the beer on my breath. “I can’t help you with your video. I know some of my colleagues would be happy to.”

The furrow returned to his brow, this time to stay. “You mean you don’t believe?”

“I do believe. I submit my statement of faith each year. But I don’t believe because of this.”

“But this affirms your faith.”

“No.” The chairs in the office were quite comfortable. It was hard to make myself stand back up. “It’s an irrational number. It’s a pattern in the clouds.”

***
Within another week the zeros were gone, replaced with the more obvious randomness that pi had shown for its first quadrillion digits. One by one the remaining supercomputing groups went offline, except for a few now supported by religious foundations eager to see what other messages God had buried in the mathematical depths of the universe. For most everyone else though, the single apparition had been enough. The mists of irrationality had cleared, revealing for a moment the face of God, or rather God’s declarative word of self-existence.

The religions that could claim some kind of verification, though they certainly added converts, did not significantly change their relations to one another or the world at large. Church, mosque, and synagogue membership ballooned, as each sect sought, in the chancellor’s words, to “capitalize,” and determined atheists found themselves in the awkward position of arguing for faith in the irrationality of a transcendent number.

Near the semester’s end I found myself again at the pub, watching the digits of pi stream down my phone as it sat on the table beside the pitchy blackness of my Guinness. Someone sat across from me, and I was surprised to look up and find Temple. He had been avoiding me since our last exchange.

“Did you ever wonder,” he asked without preamble, “how the disciples felt the first time they saw Christ resurrected?”

“I’m sorry?”

He was looking at his hands. “I mean, they had just watched their friend tortured. Watched him suffer and die. Like I watched…” He paused and took a deep breath. “And then he was there again, and no one could explain how.”

I sighed. “I understand, Temple. You’re saying I’m Thomas. That I should accept the Revelation, accept the miracle, and find some joy in it.”

He looked up. “No. I’m saying it’s not the same at all. I thought it was, but the disciples had him again—with them again, eating and talking. They could touch him. If they had just gotten a message, well, I don’t think it would have been the same at all.”

He stood and then reached forward to awkwardly pat my hand where it rested on the table.

“That’s what I’ve been thinking lately,” he said. “I wanted to tell you.”

After he was gone, I sat in silence for several minutes, taking stock of my thoughts. The digits of pi continued to stream on my phone, as senseless as whorls of cloud, as random and as beautiful. I finished my glass and checked my watch. There was an evening liturgy at the church across the street.

I stood.

It was time, I decided, to meet my own irrationality in bread and wine.

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Stephen Case is a historian of astronomy, author, and professor, with publications ranging from Asimov’s and Clarkesworld to Physics Today and American Scientist. He is also the author of Making Stars Physical: The Astronomy of Sir John Herschel (University of Pittsburgh Press, 2018), co-editor of The Cambridge Companion to John Herschel (Cambridge University Press, 2024), and author of the forthcoming Creatures of Reason: John Herschel and the Invention of Science (University of Pittsburgh Press, 2024). Stephen is particularly interested in nineteenth-century science and may be the world’s leading authority on the most important scientist you’ve never heard of. (Can you guess who?) He teaches at a Christian liberal arts college in Illinois.

 
“Irrationality” by Stephen Case. Copyright © 2024 by Stephen Case.

 

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