“I had another dream.”
Professor Quillibrace looked up.
Miss Stray smiled.
“Who was in it this time?”
“Turing and Gödel.”
Quillibrace put down his book.
“Ah.”
“I knew you would say that.”
“What happened?”
“I was in a room with two men and a machine.”
“What sort of machine?”
“That was the trouble. Nobody would tell me.”
“Was it working?”
“Yes.”
“What was it doing?”
“Thinking.”
Quillibrace raised an eyebrow.
Blottisham continued.
“Turing was standing beside it.
Gödel was sitting at a desk.
There was a piece of paper on the desk.”
“What did it say?”
Blottisham frowned.
“I don't know.”
“You couldn't read it?”
“I could read it perfectly.”
“What did it say?”
“I don't know.”
Miss Stray smiled.
“Go on.”
“I asked Turing what the machine did.
He said, ‘It computes.’
I asked what it computed.
He said, ‘Whatever can be computed.’
I asked what couldn't be computed.
He said, ‘That depends on the question.’
“So I asked Gödel.
He said, ‘Some questions cannot be decided within the system in which they are asked.’
“I said, ‘Then ask them somewhere else.’
“Gödel said, ‘Where?’
“I said, ‘Outside the system.’
“He said, ‘You have not told me what that is.’”
Quillibrace nodded.
“That sounds rather like Gödel.”
“I thought so too.”
“What happened next?”
“I asked Turing to build a machine that could decide everything.”
“And?”
“He said no.”
“Why?”
“He said that some problems have no general algorithm that will always give the answer.
I asked him to build a machine that would recognise those problems.
He said that was another problem.”
Miss Stray laughed.
“I became annoyed.”
“Understandably.”
“I said, ‘Why don't you just make the machine do what it needs to do?’
“Turing said, ‘Because then we would need to know what it needs to do.’
“Gödel said, ‘And you may not be able to prove that you know.’
“I said, ‘This is getting ridiculous.’”
“What did they say?”
“Nothing.”
“Why?”
“They were looking at the machine.”
Blottisham paused.
“It had started writing.”
“What?”
“I don't know.”
“You still couldn't read it?”
“I could read every symbol.”
“But?”
“But the symbols were asking questions.”
Quillibrace leaned forward.
“What sort of questions?”
“Questions about the machine.”
Miss Stray nodded.
“Self-reference.”
“Yes.”
“The machine asked whether it could answer its own question.”
“And could it?”
“That was the question.”
Quillibrace smiled.
“What did you do?”
“I switched it off.”
There was a silence.
“You can't do that,” said Quillibrace.
“I did in the dream.”
“What happened?”
“The machine continued.”
Blottisham looked offended by the memory.
“I said, ‘It's switched off.’
“Turing said, ‘That is a description of its state.’
“Gödel said, ‘Not necessarily of what follows from it.’
“I said, ‘If it's switched off, it isn't doing anything.’
“Turing said, ‘You are assuming that the machine is the only system involved.’
“I said, ‘What other system is there?’
“They both looked at me.”
“What did you realise?”
“I realised I was in it.”
Miss Stray smiled.
“The system?”
“The dream.”
“Ah.”
“I tried to leave.”
He paused.
“There was a door.”
“Was it locked?”
“No.”
“Then why didn't you leave?”
“It wasn't a computable door.”
Quillibrace closed his eyes.
“What does that mean?”
“I don't know. That's why I couldn't open it.”
Miss Stray laughed.
“So what happened?”
“I asked Turing whether there was an algorithm for opening the door.
“He said, ‘There may be.’
“I asked whether he knew it.
“He said, ‘No.’
“I asked Gödel whether he could prove that the door could be opened.
“He said, ‘Not from here.’
“I said, ‘Then you're both useless.’”
“And?”
“Turing smiled.
Gödel looked at the piece of paper.”
Blottisham became quieter.
“The paper was now on the door.”
“What did it say?”
“I still don't know.”
“You've said that already.”
“Yes.”
“But you could read it.”
“Yes.”
“So what did it say?”
Blottisham looked at them.
“It said:
THIS SENTENCE CANNOT BE PROVED.”
Quillibrace nodded.
“And then?”
“I said, ‘That's obviously a contradiction.’”
“Was it?”
“I don't know.”
“What did Gödel say?”
“He said, ‘That depends on the system.’”
“And Turing?”
“He said, ‘And on what you mean by proved.’”
“I said, ‘But surely one of you must know whether it's true.’”
Gödel said:
“Truth and provability are not the same thing.”
Turing said:
“And knowing what a machine does is not the same as making it do everything.”
“I said, ‘So there are things you can know that you cannot prove?’”
Gödel nodded.
“And things you cannot compute?”
Turing nodded.
“I said, ‘That's terrible.’”
Gödel looked at me.
“No,” he said.
“It is a boundary.”
Turing looked at the machine.
“A boundary can be useful.”
Then the machine printed something.
This time I could read it.
It said:
WHAT ARE YOU DOING?
I said:
“Trying to wake up.”
The machine printed:
THAT IS NOT A COMPUTATION.
I said:
“Why not?”
Turing said:
“Because you haven't specified the procedure.”
Gödel said:
“Perhaps it cannot be specified from within the system.”
I shouted:
“Then how do I get out?”
There was a long silence.
Finally Turing said:
“You can try.”
Gödel said:
“But you cannot prove that you will.”
I woke up.
Quillibrace was staring at him.
Miss Stray was trying not to laugh.
“And what did you do?”
Blottisham looked at the table.
“I checked whether I was still in the dream.”
“How?”
“I asked myself a question.”
“What question?”
“‘Am I awake?’”
“And?”
“I didn't know.”
Quillibrace smiled.
“That proves nothing.”
“I know.”
“Good.”
Blottisham looked relieved.
Then he noticed a piece of paper lying beside his tea.
He picked it up.
It contained a single sentence.
He read it.
Then read it again.
“What does it say?” asked Miss Stray.
Blottisham handed it to her.
She looked at it.
So did Quillibrace.
The paper said:
THIS SENTENCE CANNOT BE PROVED IN THE SENIOR COMMON ROOM.
Nobody spoke.
Then Blottisham stood up.
“I think I'm going back to bed.”
“Why?”
“So I can wake up somewhere else.”
Quillibrace looked at the paper.
“That is not a proof.”
“I don't need a proof.”
Blottisham reached the door.
“I need a procedure.”
He opened it.
Beyond the door was another Senior Common Room.
Blottisham stared.
Miss Stray looked at Quillibrace.
Quillibrace looked at the paper.
From the other room came Blottisham's voice.
“Can somebody please tell me which one is the original?”
There was a pause.
Then another Blottisham said:
“That's the wrong question.”
The door closed.
Nobody saw which one closed it.