Professor Quillibrace was halfway through his tea when Mr Blottisham arrived.
“I have been reading the latest series,” said Blottisham.
Quillibrace looked up.
“That sounds ominous.”
“It is mathematics.”
“I see.”
Blottisham sat down heavily.
“I have one objection.”
“Only one?”
“One principal objection.”
Miss Elowen Stray, who had been reading by the window, looked over the top of her book.
“And what is the objection?”
Blottisham folded his arms.
“Mathematics is not interpersonal.”
Quillibrace put down his cup.
“Why not?”
“Because mathematics is mathematics.”
There was a short silence.
“That is not usually considered an argument,” said Quillibrace.
“It is in this case. Mathematics deals with quantities, relations, structures, proofs. Interpersonal meaning concerns interaction between people. Mathematics does not require people to interact. Therefore—”
“Therefore?”
“Therefore mathematics is ideational.”
Elowen closed her book.
“Does mathematics use propositions?”
“Obviously.”
“Can a proposition be asserted?”
“Certainly.”
“Can it be challenged?”
Blottisham hesitated.
“Well. Yes. But that doesn't make it interpersonal.”
“Why not?” asked Quillibrace.
“Because the challenge is external to the mathematics.”
Quillibrace raised an eyebrow.
“Is it?”
“Of course. Someone may object to a theorem. That is a social event. The mathematics itself remains unchanged.”
“Let us take something simpler.”
Quillibrace wrote on a sheet of paper:
x + 2 = 5
“Is that mathematics?”
“Obviously.”
“Is it a proposition?”
“Yes.”
“What does it say?”
Blottisham looked irritated.
“That x plus two equals five.”
“And if I write—”
Quillibrace added:
x = 5 − 2
“—what have I done?”
“Solved part of the equation.”
“Have I merely rearranged symbols?”
“Yes.”
“Does the second statement follow from the first?”
“Obviously.”
“Could I instead write—”
He added:
x = 5 + 2
Blottisham frowned.
“You could.”
“Would it follow?”
“No.”
“Why not?”
“Because it is wrong.”
Quillibrace smiled.
“There we are.”
“There we are where?”
“The proposition has acquired a status.”
Blottisham stared at him.
“It is either valid or it isn't.”
“Precisely.”
“That is mathematics.”
“Yes.”
“Which proves my point.”
“I am afraid it does not.”
Elowen smiled.
“You have just said that a mathematical proposition is not merely something that has meaning. It is something that can be judged with respect to what has already been established.”
Blottisham looked from one to the other.
“That is validity.”
“Yes,” said Elowen.
“Validity is mathematical.”
“Yes.”
“So?”
“So perhaps we should ask what kind of meaning validity belongs to.”
Blottisham leaned back.
“I still don't see the interpersonal connection.”
Quillibrace returned to the paper.
“Suppose I write:
He pointed to the second line.
“Is this statement independent of the first?”
“No.”
“Does it merely have its own meaning?”
“No.”
“Is its status partly determined by its relation to the preceding statement?”
“Yes.”
“And the third?”
“The same.”
“So each proposition is constrained by what has already been said.”
“Yes.”
“And open to what might subsequently be said about it.”
Blottisham paused.
“That sounds suspiciously like conversation.”
“It isn't conversation,” said Quillibrace.
“Exactly.”
“But it has something in common with conversation.”
“What?”
“The propositions have a status with respect to one another.”
Blottisham looked unconvinced.
“People have conversations. Equations don't.”
“Agreed.”
“People make claims. Equations don't.”
“Do they not?”
Blottisham pointed at the paper.
“That equation doesn't claim anything. It simply expresses a mathematical relation.”
Elowen spoke quietly.
“Then why can it be true or false?”
Blottisham turned towards her.
“Because of the relation it expresses.”
“And why does that matter?”
“Because—”
He stopped.
Quillibrace waited.
“Because it has to be correct.”
“Exactly.”
“But correctness isn't interpersonal.”
“Perhaps not by itself,” said Quillibrace. “But consider what happens when a proposition is challenged.”
Blottisham waved a hand.
“Someone says it is wrong.”
“And then?”
“We check.”
“And if the objection succeeds?”
“We reject the proposition.”
“And if it fails?”
“We retain it.”
Quillibrace nodded.
“Notice what has happened. The proposition has entered a space of possible acceptance, rejection, justification and challenge.”
Blottisham frowned.
“That still sounds like something mathematicians do to mathematics.”
“Does a proposition cease to have that status when nobody is currently objecting to it?”
“No.”
“Then the status belongs to the proposition, not merely to the social event of objection.”
Elowen looked up.
“That may be the important distinction.”
Blottisham sighed.
“Which distinction?”
“Between being challenged and being challengeable.”
There was a pause.
Quillibrace nodded slowly.
“Yes.”
Blottisham looked at the equation again.
“You mean that a proposition does not have to be challenged in order to be something that could be challenged.”
“Exactly.”
“That is a rather peculiar form of interpersonal meaning.”
“It may be,” said Quillibrace.
“There's no speaker.”
“There can be.”
“No addressee.”
“There can be.”
“No exchange of turns.”
“Certainly not necessarily.”
Blottisham looked triumphant.
“Then it isn't interpersonal.”
Elowen shook her head.
“Why must interpersonal meaning require a completed exchange?”
Blottisham opened his mouth.
Nothing came out.
Quillibrace took another sip of tea.
“Perhaps we have been looking in the wrong place.”
Blottisham frowned.
“Where should we have looked?”
“Not at whether mathematicians talk to one another.”
He tapped the paper.
“At what a mathematical proposition makes possible.”
Elowen nodded.
“A proposition can be stated. It can be supported. It can be questioned. It can be challenged. It can be accepted. It can be rejected.”
“And none of those possibilities requires a second person to be physically present,” said Quillibrace.
Blottisham looked thoughtful.
“But surely you are not suggesting that mathematics is a conversation.”
“No.”
“Good.”
“Not an ordinary one.”
“Better.”
“The point is rather that mathematics can organise relations between propositions in a way that gives them something resembling the status of conversational moves, without requiring conversational turns.”
Blottisham considered this.
“So the proof is answering the objection that has not yet been made?”
Quillibrace smiled.
“Now you are beginning to understand.”
“But that is absurd.”
“Is it?”
“Yes. A mathematician cannot answer an objection that nobody has made.”
Elowen closed her book.
“Perhaps a proof does not answer a particular objection. Perhaps it makes the proposition answerable to possible objections.”
Silence.
Blottisham looked down at the equation.
Then up at Elowen.
Then at Quillibrace.
“I don't like this.”
“I had suspected as much.”
“But I still have one objection.”
“Only one?”
“One principal objection.”
Quillibrace waited.
“If mathematics has interpersonal meaning without anyone taking turns, then what exactly are the turns for?”
Elowen smiled.
“They may be optional.”
Blottisham looked horrified.
“Optional turns?”
Quillibrace picked up his tea.
“Welcome to mathematics.”
Blottisham stared at the three lines on the paper.
After a moment he said:
“Very well. I concede that mathematics can apparently be challenged without anyone actually doing the challenging.”
Quillibrace nodded.
“I shall now go and challenge it.”
“You would need an argument.”
“I have one.”
“What is it?”
Blottisham stood.
“Mathematics is mathematics.”
Elowen laughed.
Quillibrace sighed.
Some conversations, it appeared, did require turns after all.
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