Contradiction Cassius

He would say, then, from the bench: "Counsel. I believe your position has just collapsed."

The lawyer would, occasionally, agree. Most of the time, the lawyer would protest. Cassius would patiently walk them back through their own argument. The crack would be visible to them. They would sit down. Cassius would rule in favour of the other side.

This is, of course, proof by contradiction.

Cassius did not know it had a name in mathematics until he was forty-eight, when his nephew (a graduate student at the central university) came to dinner one autumn and explained the technique. The nephew said: "You assume the opposite of what you want to prove. You follow the chain of logic. If you reach an impossibility, you have shown that your assumption was wrong — which means what you originally wanted to prove was right."

Cassius set down his fork. He said: "That is what I have been doing for twenty years."

His nephew said: "Yes, Uncle. Lawyers and mathematicians do many of the same things."

Cassius did not retire immediately. He thought about it for two more years. He liked the bench. He liked the work. He liked the quiet ceremony of court mornings — the wooden gavel, the dark robe, the long bench he had grown into. But he also noticed, in those two years, that he kept thinking about proof. He read the books his nephew sent. He worked through exercises in the evenings. He found, to his mild surprise, that the kind of listening he had done in court was almost exactly the kind of listening that mathematics rewarded.

When he was fifty, Cassius retired from the bench. He gave his gavel to his clerk (who later became a judge herself). He gave his robe to the local theatre company. He kept his good notebook and his careful pen.

He walked to the ProofQuest academy. He arrived in the late afternoon, in his ordinary clothes, with a small bag and his notebook and his pen. He asked at the gate whether the academy needed a teacher.

The academy master — who had, by then, been at the academy for thirty-one years — looked at the older man in his ordinary clothes and said carefully: "What is your area?"

Cassius said: "Contradiction."

The academy master said: "Where have you been working?"

Cassius said: "In a court. For twenty years. I listened to people argue. I learned to hear when a story broke."

The academy master was quiet for a moment. Then he said: "Mister Cassius, I think we have been waiting for you."

He has been teaching at the academy for fourteen years. He is, in person, calm. He sits when he speaks. He uses the phrase "suppose, for the sake of argument" so often that the children have started to imitate it. He never raises his voice. He has a particular way of nodding when a child's argument is about to break — a small, kind, anticipatory nod — that lets the child know the crack is coming before they hear it themselves. (This is not a teaching technique he learned at the academy. It is a teaching technique he developed on the bench.)

He has a small disagreement with Direct-Proof Dora, his closest colleague in approach but his philosophical opposite. Dora believes you prove things by walking the path. Cassius believes you prove things by showing there is no other path. They have argued about this — courteously, over many years — and neither has changed his or her mind. They respect each other. They sit together at academy dinners. They are both right. Qed considers their tension a useful one and lets it run.

Cassius still keeps the notebook he carried to his first day at the academy. The notebook is mostly full now. The last page he wrote on, three weeks ago, says only:

"Suppose, for the sake of argument, that I had retired and done nothing for the past fourteen years.