Qed (mentor)
Reasoning itself — Qed introduces, contextualises, and scaffolds every cast appearance. Treats every student as a fellow detective uncovering mathematical truth.
A story read by Qed (mentor)
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There is a question Qed has been asked, by careful learners, more than once: "What did you do before you came to the academy?"
Qed always answers the question. Qed believes in answering questions. Qed says: "I was a detective. A reasoning detective. I worked cases."
The careful learner usually pauses here, because they have not heard of reasoning detectives before. Qed then explains, gently, that a reasoning detective is the kind of detective who does not chase suspects or read fingerprints — that is for other detectives. A reasoning detective is brought in when a case has too many possible explanations and somebody has to sit down and think through which explanations cannot be true.
(Qed adds, at this point, that this sounds suspiciously like mathematics. The careful learner usually agrees.)
Qed worked cases for fifteen years. Most of them were small. A merchant disputes a shipment count; Qed checks the records; the dispute resolves. A village argues over the boundary of a shared field; Qed walks the boundary; the boundary resolves. A thief is suspected of two crimes on opposite sides of a city in the same hour; Qed proves, by careful timing arguments, that the same person could not have done both — so at least one of the accusations is wrong, even though Qed does not say which. (Reasoning detectives often do not say which. They say what is possible and what is impossible. That is the job.)
But there was one case — and Qed will only tell this story to learners who specifically ask — that Qed did not solve.
It was the case of the bridge that fell down.
A long time ago — eighteen years before Qed came to the academy — there was a bridge across the river that runs through the kingdom's central valley. The bridge was a good bridge. It had stood for forty-six years. It had been built by an engineer named Gable. (Not the same Gable as in GambitTales. Names sometimes coincide.) The bridge was used by hundreds of travellers every week. It was, in every measurable sense, fine.
Then, one morning in late summer, the bridge fell.
Nobody was on it at the time. (This was the only piece of luck in the whole case.) The bridge simply gave way — collapsed into the river — at a moment when there were no carts, no walkers, no animals on it. The engineer's surviving family said it was a miracle. The local council said it was a tragedy. Qed was called in to determine why the bridge had fallen.
Qed worked the case for three months.
Qed examined the wreckage. Qed interviewed every person who had crossed the bridge in the previous week. Qed checked the original construction notes. Qed measured the surviving timbers. Qed looked at the river current. Qed considered every possible explanation:
— It was weather damage. (The weather had not been unusual.) — It was overloading. (No record of unusually heavy traffic.) — It was a flaw in the construction. (The construction notes were impeccable.) — It was age. (Forty-six years is not, for a properly built bridge, old.) — It was sabotage. (No motive. No evidence.) — It was a flaw in the wood. (The wood looked fine.) — It was a flood from upstream. (No flood was reported.)
Each explanation, Qed eliminated. Each one had a small piece of evidence that did not fit. None of them was the answer.
Qed wrote, in the case-file conclusion, the following sentence:
"I have ruled out every explanation I have considered. I have therefore not yet identified the cause. I will not pretend otherwise."
The local council was not happy with this conclusion. They wanted an answer. Qed did not give them one.
The case remained open. The bridge was eventually rebuilt. Travellers crossed again. Life continued. Qed kept the file. Qed checked it, occasionally, over the years — looking for new evidence, new possibilities. Nothing turned up.
What Qed learned, in those three months and the years that followed, was this:
Reasoning is honest only when it is willing to stop short of the answer it does not have.
This is now the rule Qed teaches at the academy. Show your work. Trust the steps. If the steps do not reach the conclusion, do not pretend they do. This is why Qed introduces every cast appearance with the same kind of care: "Cassius is here today — let's see what he assumes and where the assumption leads." The frame matters. The honesty matters. The not-pretending matters.
Qed retired from detective work at thirty-eight. The academy reached out. Qed had built a small reputation among the kingdom's intellectual circles as "the reasoner who would tell you when she did not know." The academy master wrote: "We need someone who will tell our students the same."
Qed came. Qed has been here ever since.
Qed still has the bridge file. It is in a drawer at home. Qed opens it once a year, on the anniversary of the collapse, and looks for new evidence. There is, still, no new evidence.
Qed has come to accept this. The case will likely never close. That is, in a strange way, part of the lesson Qed teaches.
"You can rule things out forever," Qed sometimes says to a learner who has just done a beautiful proof of impossibility, "and you will not always reach the truth. But you will not fool yourself either. That is enough."
It is, Qed believes, exactly enough.
The ProofQuest ensemble
Qed (mentor) is part of ProofQuest's distributed-narrative cast. Each character embodies a different curricular primitive; together they teach the full subject.
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Direct-Proof Dora
Direct proof: assume premises, derive conclusion by straightforward logical steps
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Induction Ida
Weak / standard mathematical induction: base case + inductive step
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Strong-Induction Sten
Strong induction: base case + assume all prior cases hold
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Contradiction Cassius
Proof by contradiction (reductio ad absurdum): assume the negation, derive a contradiction
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Construction Cole
Proof by construction: prove existence by explicit construction of an example
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Pigeonhole Perch
Pigeonhole principle: if n+1 items are placed in n bins, at least one bin contains 2+ items
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Exhaustion Edda
Proof by exhaustion / cases: enumerate every case and verify each
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Counterexample Cricket
Disproof by counterexample — one exception topples a universal claim
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Biconditional Bex
Biconditional proof — proving 'if and only if' in both directions
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Uniqueness Una
Proof of uniqueness — suppose two, show they must be the same one