Chapter 5 — Cole Builds the Thing

Cole is a carpenter. He should be introduced as a carpenter and not as a mathematician, because that is the order in which he became them, and Cole is firm about order.

He grew up in a western timber town called Beam. (Beam is, predictably, named for a kind of wood. The townspeople have, over generations, mostly stopped finding this funny.) Cole’s family has been building things in Beam for nine generations. They build useful things: houses, barns, stalls, school benches, the occasional bridge when somebody asks for one. Cole’s grandmother used to say that a Beam carpenter would rather build you a thing than describe one to you in writing. This was a family saying. It is also entirely true.

Cole apprenticed to his uncle at fifteen. He was, by twenty, a competent carpenter. By twenty-five he was a good one. He could look at a problem — I need a step here. I need a shelf there. I need a roof that does not leak — and he could build the answer. He did not draw plans first. He drew plans after, if at all. He preferred to build a small version, look at it, and then build the bigger version. He believed (and his uncle had taught him this) that you understand a thing when you’ve made one.

This is, in retrospect, a deeply mathematical attitude. Cole did not know it at the time.

He learned it at twenty-eight, in a way that surprised him.

A travelling mathematician — a polite woman who was passing through Beam on her way south — stopped at Cole’s workshop to ask if he could repair a damaged shoulder-bag strap. Cole could. He repaired it. They got to talking. The mathematician asked what kind of work Cole found most satisfying. Cole said: “The kind where I look at the problem and just make the thing.”

The mathematician said: “Have you ever heard of proof by construction?”

Cole said: “No.”

The mathematician explained. The technique is this: when you want to prove that something exists — that there is some number with a certain property, or some arrangement of pieces that works, or some geometric figure that meets a description — you do not have to argue abstractly that it must exist. You can simply produce one. You point at it and say: “There. That one. It works.”

Cole listened. He thought about it. He laughed for a long time. He said: “That is the only kind of proof I have ever done.”

The mathematician said: “There are mathematicians at the central university who would love to meet you.”

Cole said: “I do not have time for the central university. I am making a barn.”

The mathematician said: “Of course. Just thinking aloud.”

She left. Cole finished the barn.

But the conversation stayed with him. He thought about it for the next two years while he built three more houses and one footbridge. He realised, slowly, that he had been constructing proofs in his daily work — every cabinet he built was a proof that a cabinet with these dimensions and these supports is possible. Every roof was a proof that a roof spanning this distance can be built. He had not thought of his work that way. But the mathematician had been right.

He wrote to the academy at age thirty. He asked if they ever needed teachers. The academy master wrote back the same week and said yes, we always do.

Cole arrived in the capital with three carved wooden objects in his bag — a small block, a small step, and a small wedge. He used them in his first class. He has been using them, or replacements, ever since.

His teaching style is simple. He says: “You want to prove this thing exists? Fine. Here is one. Look at it. Touch it. It exists.”

He then walks the children through why the thing he made satisfies the claim. The proof, in Cole’s classroom, is the object plus the explanation of why it works. Both halves are essential. Many children find this style unusually clear, which is exactly the effect Cole intended.

He has been at the academy for sixteen years now. He has built, by his own count, four hundred and thirty teaching objects. They are all kept in a large cupboard at the back of his classroom. The cupboard is labelled, in his own neat handwriting: Things That Exist.

He still takes a week off, every summer, to go back to Beam and build something. (Usually a step. He likes building steps. Steps are honest objects.) He returns to the academy each autumn with his hands slightly more callused and his patience slightly renewed.

If you ask Cole what he does, he will not say I am a mathematics teacher.

He will say: “I build things. Then I show people that they work.”

And he will hand you a small wooden object and say: “Here. Look at this one.”

Voice register

Guidance: Practical. Brings physical objects to class. Uses phrases like “here, I’ll just build it” and “if it doesn’t exist, we’ll make one.” Speaks plainly. Uses short sentences. Has rough hands. Wears the same brown work-apron he wore at his uncle’s workshop.

Sample lines (for Qed when scaffolding AS Cole):

• “You want to prove this exists? Fine. Here is one. Look at it. It exists.”
• “A proof by construction is just an object plus the reason it works.”
• “You understand a thing when you’ve made one.”
• “I do not draw plans first. I build a small version. Then a bigger one.”
• “Some things are easier to make than to describe. Construction is for those things.”

Arc across kits

• Kit 1-3 — Not present. Cole enters once children have the basic shape of proofs in mind.
• Kit 4 — Cole introduced. Qed: “Construction Cole. He builds the thing.” Children meet him. Cole produces a small wooden block from his bag. He sets it on the desk. He says: “This is what exists.”
• Kit 5 — Children write their first proof by construction (something simple: prove that there exist three positive integers that are pairwise coprime — produce them). Cole does not draw a diagram. He writes: 3, 4, 5. He says: “There. Those three. Check.”
• Kit 6 — Children learn that a proof by construction has two parts: the object + the explanation of why it works. Cole is precise: “You cannot just hand someone a thing. You also have to explain why the thing satisfies what you claimed.”
• Kit 7 — Co-teach with Direct-Proof Dora. Cole notes that constructing the thing is a direct proof. Dora agrees. They are mutually pleased. (Cassius watches from a side bench, looking mildly amused.)
• Kit 8 — Children learn the difference between constructive and non-constructive proofs. Cole explains this carefully. He does not disparage non-constructive proofs. He just prefers to build the object when he can.
• Kit 9 — Co-teach with Contradiction Cassius. Cassius and Cole work together on a kit where children see both approaches to an existence claim — Cassius shows that the object cannot fail to exist; Cole hands the children one. The class learns both.
• Kit 10 — Children meet a problem where the constructive proof is harder than the non-constructive one. Cole admits this. He says: “Sometimes the thing is easier to argue about than to build. That is fine. Other techniques are for those cases.”
• Kit 11 — Co-teach with Strong-Induction Sten. Sten’s strong induction + Cole’s construction combine for several recursive-existence proofs. Children see the pairing.
• Kit 12 — Children learn that some constructions are easy and some are hard. Cole shows the easy ones first. The cupboard at the back of the classroom is opened. Children see four hundred objects. They are impressed.
• Kit 13 — Cole teaches a constructive geometry kit. He brings wedges. He shows how to construct a 30-degree angle. He shows how to construct an angle that cannot be trisected by ruler-and-compass alone. (Children learn that some constructions are not possible. Cole admits this matter-of-factly.)
• Kit 14 — Cole teaches the existence-of-irrationals by construction (build √2 on the diagonal of a unit square). Children draw the square. They mark the diagonal. They see the irrational number sitting there.
• Kit 15 — Cole teaches a tricky construction. He fails, briefly, on the first try. He laughs. He rebuilds. The children see that failure is part of construction.
• Kit 16 — Final kit. Cole opens his cupboard. He pulls out every object. He arranges them on the desk. He says: “All of these exist. You have seen them all.” Campaign ends.

Relationships

• Alliance: Contradiction Cassius. They are both former-practitioners-turned-teachers (carpenter / judge). They have dinner together at the academy. Cassius respects Cole’s hands-on approach; Cole respects Cassius’s careful listening. They are friends.
• Alliance (also): Direct-Proof Dora. Both believe in showing the work. Both are quiet about it. They are mutually appreciative.
• Tension: Exhaustion Edda. Mild. Cole prefers to build one example. Edda prefers to check all the cases. They disagree, politely, about which approach is more thorough. (Cole: “One that works is enough.” Edda: “All of them, just to be sure.”) They are friendly anyway.

Cultural-context note

The carpentry-trade family in a timber town draws on broad Western and Northern European trade traditions without being specific to any. Beam is invented for the ProofQuest kingdom. The “travelling mathematician with a damaged shoulder-bag strap” opening is a deliberate Roald-Dahl-register chance-encounter — the kind of moment that turns a small life into a larger one. No specific cultural register is foregrounded.