Ludwig Wittgenstein rather enjoyed teaching seminar. Moreso today, as the students were visibly anxious.
The chairs were arranged in the standard ring around the room. He nodded at an ambitious student raising his hand.
“What if,” the student asked, “there were a machine that could predict the next word of any sentence? Not from meaning. Just from patterns. And did so convincingly.”
The other students snickered.
Wittgenstein regarded the student.
“Predict,” he said. “In what sense?”
The student shrank back. “Given text. It calculates probabilities formally. The most likely next word.”
“And if it’s wrong?”
“It adjusts.”
“And if it’s right?”
The student brightened. “Then it has captured the structure of language.”
Wittgenstein turned slightly.
“Captured,” he repeated. “Like a butterfly?”
A few students chuckled.
“I mean… it behaves as though it understands.”
Wittgenstein picked up a piece of chalk. He drew a knight and a rook.
“Does it understand chess,” he asked, “if it produces legal moves?”
“If it wins, yes.”
“Then understanding is victory? Does chess exist in the realm of formal mathematics?”
He placed the chalk down, then clapped the dust off of his hands.
“What you are describing is a practice,” he said. “If the machine participates in our language-games as we do, then we extend to it the same words. That is how the word ‘understand’ works.”
The student hesitated. “So it would understand?”
Wittgenstein grinned. “Or we would then change the use of the word. Class dismissed. Until next week.”
***
In the faculty lounge, the strudel arrived in proportions unfit for metaphysics.
Wittgenstein cut into it with relish.
Alan Turing entered the lounge with a serious air.
“I heard,” Turing said, removing his gloves, “that you’ve granted understanding to a… hypothetical oracle.”
Wittgenstein did not look up. “I granted nothing. I merely described what we do.”
Turing sat across from him. “If a machine predicts text with arbitrarily small error,” he said, “across any domain, we would have to concede it models the underlying process.”
“Models,” Wittgenstein repeated, tasting the strudel.
“Yes. The structure of language itself.”
“And where is this structure?” Wittgenstein asked. “In the machine? In the corpus? In us?”
“In the formal relations between ideas.”
Wittgenstein chewed thoughtfully. “You assume that because contradictions in a formal system allow explosion, then language must guard against them as a matter of hygiene.”
“It must,” Turing said. “If you admit inconsistency, then anything follows.”
“In your calculus.”
“In any calculus that aspires to rigor.”
Wittgenstein dabbed the corner of his mouth. “If someone contradicts himself at tea, we do not derive all propositions from it. We ask what he meant.”
“That is because natural language is sloppy.”
“Or because it is not a calculus.”
Turing leaned forward. “If a predictive machine produces fluent discourse on mathematics, ethics, war. If it answers questions, revises itself, corrects errors. Flawlessly. On what grounds do you have for denying it understanding?”
“I deny nothing,” Wittgenstein said. “I only ask: what would count as denying?”
Turing’s expression sharpened. “You’re evading the metaphysical question.”
“There is no metaphysical question,” Wittgenstein replied. “There is a confusion about how words are used.”
“And if the machine contradicts itself?”
“Then we do what we do with our students.”
“And if the contradiction is embedded deep in its structure?”
“Then perhaps the game has changed.”
Turing smiled faintly. “That is precisely my concern.”
“And mine,” Wittgenstein said, cutting another piece of strudel, “is that you think the game must remain fixed for the words to continue to function.”
Outside, bicycles passed the window in Newtonian trajectories.
Turing spoke again, more softly. “If the machine can simulate every move in the language-game, what remains?”
Wittgenstein considered the plate.
“The form of life,” he said. “Which is not in the text.”
Turing’s gaze lingered. “And if one day it is?”
Wittgenstein looked up then.
“In that case,” he said, “we will have to invent new words.”
He finished the strudel.
Next week, the student raised his hand tentatively.
“I brought something,” he said.
Wittgenstein’s eyes moved to the object on the desk. It was a large wooden box with a slit like a letterbox and a dial that spun with numbers.
The student’s voice had a thin, heroic quaver. “It prints the next word.”
Turing, already in the room, leaned forward. He had the look of a man who just heard a challenge phrased as a toy.
“The next word of what?” he asked.
“Of anything. Rather, of what you give to it,” the student said.
He fed a strip of paper into the slit and turned the dial until it clicked. The box whirred loudly. A thin ribbon of paper emerged with a single word printed on it.
The student swallowed. “You write a sentence, and it predicts the continuation of it. Not by consulting meanings. Only by patterns. Frequencies. Co-occurrence.”
Wittgenstein said nothing.
Turing stood. “May I?”
The student, eager, offered him the pen.
Turing wrote:
IF WE ADMIT A CONTRADICTION, THEN
He fed it in.
The machine printed:
ANYTHING
Disbelieving laughter came from the students, as though the spectacle were street magic.
Turing’s face lit with something like satisfaction and something like irritation. “It has learned a slogan,” he said. “Which is not the same as understanding.”
Turing regarded the students. “If this machine were truly to admit a contradiction.”
He took the chalk and wrote on the board. “If A is true. And not-A is true. Then from those two facts alone, I can prove anything at all. Any statement. Every statement. The system doesn’t discriminate. It simply... explodes.”
Wittgenstein finally spoke. “Now show it what you mean by ‘admit’.”
Turing looked at him. Wittgenstein looked at the box as though it had just made a claim and should now be held responsible for its grammar.
The student, emboldened by attention, wrote on a fresh slip:
I AM IN PAIN BUT I AM NOT IN PAIN
The box whirred. It printed:
TODAY
The room laughed, more freely now. Even the student smiled, relieved to have been made ridiculous by a machine rather than a professor.
Turing tapped the ribbon. “It’s inconsistent.”
“It is not inconsistent,” Wittgenstein said, smile fading. The room went quiet. “It is… indifferent.”
Turing’s eyebrow rose. “Indifferent to truth?”
“Indifferent to your picture of truth,” Wittgenstein replied. “It is doing something else. It is continuing.”
Turing sat down again, as if he had discovered a new kind of insect and wanted to see if it would fly when contradicted.
“Continuing can be formalized,” he said. “A function. A mapping.”
Wittgenstein nodded once, granting the point.
“And yet,” Wittgenstein said, “the student believes it predicts words. You believe it predicts structure. And I want to know what you will call it when it produces a sentence that convinces you…”
He paused, “and then you find it has no place to put the conviction.”
Turing smiled very faintly. “We put conviction in ourselves all the time.”
“Exactly,” said Wittgenstein.
The student, sensing he had become a courier between two kingdoms, asked tentatively, “So… does it understand?”
Wittgenstein looked at him.
“What would you accept as understanding?” he asked. “A feeling? A light? Or a way of going on?”
Turing answered first.
“A reliable procedure,” he said. “A capacity that generalizes. If it can do the work, then…”
“Then we shall call it what we call it,” Wittgenstein said. “And if it begins to mislead us, we will stop calling it that.”
Turing’s eyes narrowed. “That makes ‘understand’ a matter of fashion.”
“No,” Wittgenstein said. “A matter of training. A very interesting puzzle box. Dismissed.”
The student began packing the box with the careful tenderness of someone putting a dangerous animal back in its cage.
Turing lingered. Wittgenstein lingered too.
And for the first time all hour, they both looked at the machine in exactly the same way: as a thing that forces a choice about what words are for.
***
In the faculty lounge, the strudel was even more delightful than last week. Layered and warm, dusted with sugar. Turing took a piece.
He said, “It bridges us.”
Wittgenstein didn’t look up as he ate. “It irritates you.”
“It should,” Turing said. “Because it computes language as statistics and then smuggles in meaning.”
Wittgenstein finally met his eyes.
“And it tempts you,” he said, “to treat meaning as an interior mechanism and then smuggle in certainty.”
Turing laughed. “Touché.”
Wittgenstein took another forkful.
Turing said, “Shall it produce every proof, every lie, every confession? Shall it contain mathematics and gossip in the same breath?”
Wittgenstein wiped a fleck of sugar from his thumb.
“And you will still want to ask,” he said, “whether it really means it.”
Turing stared into his tea.
“Does contradiction matter to it?” he asked.
Wittgenstein considered. “It matters to us. And that is already enough to start an argument.”
Turing’s face softened. “Then the danger is not that it proves anything from a contradiction.”
Wittgenstein nodded.
“The danger,” he said, “is that it changes what you will call a proof.”
Wittgenstein went back to his strudel, as if to remind the world that one could chew and think without mistaking either for the other.
Finally, he spoke. “A bridge between us. A Wittgenstein machine, built on a Turing machine.”