by Nick Gevers
We bid farewell to Brother Francis, ascended to the first floor, and entered the library, sphere of Sister Margaret, whose calm demeanor and placid apple face belied the apocalyptic quality of her devotions. Two ornately carved jewel boxes rested before her on the reading table. I recognized the accoutrements of the Ruby Paradox, sometimes called Newcomb’s Paradox after the refinements wrought in modern times by William Newcomb of the Lawrence Livermore Laboratory. By the rules of the game, Sister Margaret was permitted to open both Box A and Box B, or else just Box B. She could not open only Box A. Whatever resided inside any box Margaret opened was hers to keep, but she could not retain the contents of any box she declined to open.
“Sister Margaret, please remove your wimple,” Articulis commanded.
Silently she complied, never lifting her eyes from the jewel boxes. Her caco jutted from her occipital area like a hair bun, emitting moans of psychosexual ecstasy and panegyrics to Vorg.
The abbot proceeded to expound on the Christianized iteration of the Ruby Paradox practiced at the Monastery of Tertullian. Heaven’s canniest angel, Gabriel, whose prophecies are almost always accurate, has placed in Box A a single gold coin, valuable enough to pay for refacing the monastery’s bell tower. If Gabriel has predicted that Sister Margaret will open only Box B, he has also deposited an enormous ruby, worth a thousand gold coins, in Box B. With such a sum, Sister Margaret could fulfill her lifelong dream of building an orphanage in Kabul. However, if Gabriel has predicted that Margaret will open both boxes, then he has put nothing in Box B.
Paradoxically, there was a decisive case for Margaret opening both receptacles, and an equally decisive case for her opening only Box B. Think about it. As I explained in chapter two of Adventures in Self-Reference, the classic Prisoner’s Dilemma turns on the same schizoid logic. The recently apprehended criminal will find himself constructing a sound argument for confessing and an equally sound argument for remaining silent.
Next Articulis took me upstairs to Brother Jonathan’s cell, a clean and ill-lighted place, spare as a crypt. His caco sat atop his cranium like a hideous brown yarmulke. Articulis explained that Brother Jonathan passed every waking hour meditating on the Liar Paradox, which, owing to its Scriptural heritage, enjoyed particular prestige within the Order. Speaking of the Jewish Cretans in his brief Epistle to Titus, Saint Paul warns his proxy, “One of themselves, even a prophet of their own, said, ‘The Cretans are always liars, evil beasts, slow bellies.’ This witness is true. Wherefore rebuke them sharply.” Beyond Paul’s usual anti-Semitism—not without reason have I become a Tillichean dialectical humanist—we find in these verses a cogent though evidently unconscious formulation of the Liar Paradox.
Not only was Jonathan engaged in an intense inward consideration of Russell’s Antinomy, he’d also written the basic formulation in an unbroken hundred-character line along the vertical surfaces of his cell, twenty-five characters per wall.
THE CLASS OF ALL CLASSES THAT ARE NOT MEMBERS OF THEMSELVES IS A MEMBER OF ITSELF IF AND ONLY IF IT IS NOT A MEMBER OF ITSELF.
Standing in the center of the room, the monk pivoted slowly on his heel, repeatedly pondering the paradox, reifying the vicious cycle by dint of his own continuous rotations. A buzz of sensual contentment arose from his caco. According to legend, puzzles of this sort had caused the death of at least one ancient logician, Philetas of Cos—a story I now believed: Philetas had indeed succumbed to his own thoughts.
“Alas, Russell’s discovery of set paradoxes did not mark the end of his troubles,” I noted.
“Gödel’s Incompleteness Theorem,” Articulis said with a knowing nod. “In any formal logical system predicated on the natural numbers, there exists at least one statement that can be neither proved nor disproved within the given structure.”
We exited the abbey and started across the courtyard, casting long El Greco shadows on the weed-choked grounds. A late afternoon breeze enveloped us, filling my nostrils with the heady scent of ripe grapes. Our destination, Articulis informed me, was the cloister, where we would meet Sister Ruth, keeper of the Lawyer Paradox.
Consider sly Protagoras, who trains attorneys and each year makes an unusual contract with his pupils. “You may attend all my lectures without any money changing hands,” he tells the class, “but you must pay me ten drachmas if you win your first case.” One particular pupil, Euathlus, takes Protagoras to court with the aim of getting free tuition, arguing that if he wins the case, then the teacher’s fee must be waived, since that is the object of the suit. If Euathlus loses, however, Protagoras’s fee must still be waived, since this is the pupil’s first case. So far, so good.
Reaching the arcade, we beheld a diminutive nun circumnavigating the quadrangle, east to west, north to south, west to east, south to north, her glassy eyes contemplating some unseen and ever receding focal point. Articulis told Sister Ruth to remove her wimple. Without breaking stride, she did as instructed, revealing a parasite suckling obscenely on her left parietal. The beast, an albino, emitted the usual sounds of cacodaemonic satisfaction.
As Ruth passed me for the third time, I realized that she was muttering, over and over, now and forever, Protagoras’s argument to the judge.
“If you find in favor of Euathlus, he owes me ten drachmas, as this is his first case, but if you resolve the dispute to my advantage, Euathlus still owes me ten drachmas, since that is the content of your decision . . . If you find in favor of Euathlus, he owes me ten drachmas, as this is his first case, but if you resolve the dispute to my advantage, Euathlus still owes me ten drachmas, since that is the content of your decision . . . If you find in favor of Euathlus . . .”
Our final stop was the vineyard. The student pickers had gone home for the day, leaving the labyrinth to Brother Thomas, a blowsy, roly-poly man, his enormous eyes resting in their sockets like cupped eggs. The monk wandered aimlessly through the maze, enacting his assigned conundrum, a version of the Sorites Paradox, sorites being Greek for “heap.” Stained with red grape juice and affixed to Thomas’s right temple like an ancillary brain, his parasite feasted greedily.
Consult chapter six of Adventures in Self-Reference, and you’ll learn that the Sorites Paradox belongs to a class of antinomies predicated on the pervasiveness of vagueness in the world. Consider a sand dune. Take away one grain. Is what remains still a heap of sand? Almost certainly. As a general rule, we may assert that if two sand heaps differ in number by just one grain, then both are dunes—or else neither is a dune. Alas, this seemingly innocuous principle leads to the unacceptable conclusion that all collections of sand, even single-grain collections, are heaps.
Not surprisingly, Brother Thomas employed the omnipresent grapes in contemplating the paradox. When is a bunch not a bunch? he wondered aloud. An arbor not an arbor? A vineyard not a vineyard? A harvest not a harvest?
“No doubt my philosopher has deduced the strategy by which I hope to defeat the cacos,” Articulis said as, abandoning Thomas, we meandered among the trellises.
“You intend to turn their appetites against them,” I said.
“Exactly.”
“This will require a baited trap.”
“Quite so,” Articulis said. “The snare in question is now walking beside you.” Reaching the core of the maze, we eased ourselves onto the wrought- iron bench. “Give me an impregnable paradox, Donald, something so tempting from a cacodaemonic perspective that they’ll all desert their hosts for me.”
My temples throbbed as if a pair of parasites had attached themselves to my head. “An impregnable paradox.”
“Impregnable, delectable, ironclad, and unique.”
A wave of nausea rolled through the whole of my digestive system. “I really want to help, but, you see—what am I trying to say?—I’ve never faced a challenge of this magnitude. I’m an academic. This smacks of the real world. I don’t like it.”
“And yet you’re clearly the man for the job.”
I reached toward a low-hanging cluster,
plucked a grape, and popped it in my mouth. “Do you think they’d be lured by Zeno’s quarrels with space, time, and motion?”
Quitting the bench, Articulis rose to full height and presented me with a tortuous smile. “The Confessions explicitly states, quote, ‘Zeno the Pagan will prove useless in healing the crack,’ and therefore I must assume a caco would find those puzzles equally unexciting. Tertullian is likewise unimpressed by the Grid Paradox, the Lottery Paradox, and the Paradox of the Ravens. He also anticipated and dismissed Moore’s Paradox—‘It’s raining outside but I don’t believe it is’—as lacking genuine eschatological muscle.”
“I won’t even ask about the Barber Paradox.”
In a remote Sicilian village, the barber shaves every man who does not shave himself. Who shaves the barber? Although the problem has some features in common with Russell’s Antinomy, it’s not especially difficult to resolve. The barber doesn’t shave. The barber is a woman. My favorite answer: the barber doesn’t exist.
“Tertullian says nothing about the Barber Paradox, but I can’t imagine it would beguile our parasites,” Articulis said, starting toward the abbey. “Dinner will be served at eight o’clock. If the cacos continue to feed for another month, or perhaps just another fortnight, I fear our community will die of depletion”—my client vanished, enshrouded by the dusk—“which means the Devil wins after all.”
As darkness crept over the Adirondacks, I headed for Sister Ruth’s domain, the cloister. Lost in thought, I shuffled along the arcade at the methodical pace of the hooded figures in Escher’s Ascending and Descending. Step by step, I boxed the courtyard, north, east, south, west. Every ten minutes or so, the nun and I passed like ships in the night, each of our preoccupied minds oblivious to its transient neighbor.
It came to me that I’d entered into a kind of contest with Tertullian. The two of us had become long-distance rivals, facing one another across the valley of the shadow of death. Whatever venerable absurdity I might put forward by way of helping Abbot Articulis and, by extension, the rest of homo sapiens, Tertullian had almost certainly anticipated the problem and ruled it out. What I needed was a conundrum that couldn’t possibly have occurred to even the smartest third-century theologian—a wholly contemporary paradox, notorious for having tried the patience and taxed the sanity of modernity’s shrewdest thinkers.
By some poetic coincidence, stars bloomed over Rhinebeck just as the required riddle took shape in my brain. Stepping into the quadrangle I scanned the celestial dome with its countless suns, each twinkling point a piece of my epiphany. The answer to our needs, in every possible meaning of that perplexing predicate, was heaven sent.
I repaired to the library, seeking to verify my revelation. The cosmology section was larger than I expected, featuring not only eccentric Christians like Teilhard and Bergson but the secular likes of Michael Hart, Stephen Hawking, Nikolai Kardashev, Freeman Dyson, and M. D. Papagiannis. I perused each relevant paragraph with the frustrated frenzy of Achilles attempting to catch the tortoise, madly scribbling notes on the flyleaf of Shklovskii and Sagan’s Intelligent Life in the Universe. Fixed on her jewel boxes, Sister Margaret ignored my presence. Even after the nun left for dinner, I continued my self-incarceration, until I finally had enough material for a credible presentation to Articulis.
“Shazam!” I shouted, striding into the gloomy, tenebrous, candle-lit refectory.
The abbot sat at the head of the table, monks to his left, nuns to his right. Not a caco was in sight, the infestation being entirely hidden beneath wimples and cowls. At that moment the parasites’ murmurings were redolent of discontent—a sensible enough reaction: although the five hosts were still cultivating paradoxes, they had to devote a modicum of mentation to eating their stew and drinking their wine, with a concomitant reduction in the quality of the cacos’ nourishment.
“The Fermi Paradox!” I cried.
“Tell me more,” Articulis said.
“Why is it that, in such a vast cosmos, with two hundred and fifty billion stars in our galaxy alone, of which one hundred billion may be orbited by Earth-like planets, we have found no evidence of intelligent alien life? Why this Great Silence?”
“Continue,” Articulis said.
Consulting my jottings on the flyleaf of Intelligent Life in the Universe, I told the abbot that astronomers had to date detected over 300 exoplanets—worlds circling sun-like stars outside our solar system—and the tally was increasing every year. Many astrobiologists believed that somewhere between one thousand and one million advanced civilizations may have already arisen in the Milky Way, for an average of 500,500. And yet: no extraterrestrial radio signals, no signs of galactic colonization, no plausible UFO narratives, no credible archaeological evidence of past visitations, no von Neumann probes, no Bracewell probes, no Dyson spheres, no Matrioshka brains. Nothing, nada, zero, zilch, bupkis.
“It’s a genuine paradox,” I continued, “complete with an absurd conclusion: five hundred thousand five hundred advanced civilizations have thus far eluded our senses because we should have detected them by now. As Enrico Fermi famously put it, ‘Where is everybody?’ ”
Dressed in a splotched white apron, a hulking member of Constantine’s staff appeared at my side holding a fissured ceramic tureen. He introduced himself as Jeremiah, then proceeded to serve me a steaming glob of boiled venison using a ladle the size of a caco.
“I’m impressed,” Articulis said.
“I’ve been serving you this sludge for the past twenty years, and tonight you’re impressed?” the cook said.
Articulis scowled and squeezed my hand. “Give me your indulgence while I play the contrarian. Like Tertullian, I subscribe to the doctrine of Original Sin. Perhaps the answer to the Fermi Paradox is simply that intelligent life, upon discovering thermonuclear weapons, inevitably destroys itself. Or maybe all those hypothetical advanced civilizations, in their mad pursuit of absolute security, have succeeded in annihilating one another.”
“Line 1: The sentence written on Line 1 is nonsense,” Brother Francis said.
“Like many renegade Catholics, I’m a Darwinist,” Articulis continued. “A Teilhardian and teleological Darwinist to be sure, convinced that evolution is drawing us toward an Omega Point, but still a Darwinist. We can dispense with the Fermi Paradox simply by assuming that the biological processes found on Earth do not obtain elsewhere. Perhaps the odds are formidably stacked against the transition from prokaryotic cells to eukaryotic cells. Or maybe the move from single-celled to multicellular life is difficult in the extreme.”
“I wonder if I should open Box B only?” Sister Margaret said.
“Perhaps Earth has been deliberately sealed off as a kind of wildlife refuge,” Articulis persisted. “Perhaps we’re under a moral quarantine, doomed to isolation until we eliminate poverty or outlaw warfare. Perhaps the aliens are deliberately concealing themselves, knowing from bitter experience that extraterrestrial contact inevitably leads to disaster.”
“The class of all classes that are not members of themselves is a member of itself if and only if it is not a member of itself,” Brother Jonathan said.
“Maybe the aliens are so intelligent they have no more reason to communicate with us than they do with petunias,” Articulis went on. “Maybe their minds defy our notions of conscious rationality. Maybe they’re philosophically and spiritually advanced but don’t make machines and never will.”
“If you find in favor of Euathlus, he owes me ten drachmas, as this is his first case,” Sister Ruth said.
The abbot was up to speed now. With uncanny perspicacity he reeled off a litany of possible resolutions, most of which I’d already scrawled on the Intelligent Life flyleaf. The aliens are already here, but they’re keeping out of sight. We haven’t been searching long enough. We aren’t listening properly. Advanced civilizations broadcast radio signals for only brief intervals in their histories. Such societies are too distant in space and time to contact us using their existing techno
logies.
Now Jeremiah got into the act. “You know what I think? I think they’ve uploaded themselves into computers and don’t give a fig about space travel.”
“If there are sixty grapes in a bunch,” Brother Thomas said, “does that mean fifty-nine grapes likewise constitute a bunch?”
“My favorite refutation is pure science fiction,” I said. “Our universe is a simulation created by aliens who deliberately left it devoid of extraterrestrial intelligence.”
“There, you see—even you don’t believe it’s a serious contradiction,” Articulis said.
“No, I merely believe that the only substantive threat to Fermi’s Paradox is the simulated-universe hypothesis, an argument for which the burden of proof clearly lies with its devotees. The remaining challenges all leave the conundrum in place. You see, Bartholomew, for any given advanced alien civilization C1, that is, an extraterrestrial society whose unavailability to our empiricism is easily explained, there is always C2, a civilization that by any rational measure should have become manifest by now. If in our perversity we choose to dismiss C2 with a plausible unavailability scenario, we must still deal with C3, a different civilization that should have revealed itself already. If we take the trouble to build a case against C3, we have to cope with C4, not to mention C5, C6, C7, C8, C9, C10—all the way up to C500,500. I don’t know about you, but somewhere around C250,000 I would begin to admit there’s a problem here. It takes just one community of celestial eager beavers to break the Great Silence—just one, that’s all. What’s going on, Bartholomew? Are we truly alone? How could that be?”
The abbot knitted his brow, smacked his lips, and took a long swallow of Château Pelagius. “All right, Donald, let’s give it a try. You’ve not quite won me over, but for the moment you’ve allayed my doubts.”
“Excuse me, Dr. Kreigar, but aren’t you overlooking the most obvious solution of all?” Jeremiah asked. “Isn’t the best answer staring us in the face?”