Contradiction Cassius

Sometimes the lawyer would agree. Most of the time, they argued back. So Cassius would patiently walk them through their own story. He would show them the crack. Then they would see it. They would sit down. And Cassius would rule for the other side.

This way of winning has a special name. It’s called *proof by contradiction*.

Cassius didn't know it had a name in math. Not until he was forty-eight. His nephew, who was a student at the big university, came for dinner.

His nephew explained the idea. "It's a math trick, Uncle Cassius. You pretend the thing you want to prove is wrong. Then you follow the steps. If you end up with something impossible, like two plus two equals five, you've won! It means your first idea was a mistake. So the thing you wanted to prove must have been right all along."

Cassius set down his fork. "That," he said slowly, "is what I've been doing for twenty years."

"Yes, Uncle," his nephew said. "Lawyers and mathematicians are more alike than you think."

Cassius didn't quit his job right away. He thought about it for two more years. He still liked being a judge. He liked the heavy wooden gavel and his dark robe. He liked the quiet courtroom in the morning.

But something had changed. He couldn't stop thinking about proof. His nephew sent him math books. Cassius did the problems at night. He found that the listening he did in court was the same kind of listening he needed for math.

So, when he turned fifty, Cassius retired.

He gave his gavel to his clerk. (She became a judge later, too.) He gave his robe to the town's theater group. But he kept his best notebook and his favorite pen.

He walked to the ProofQuest academy. He arrived late in the afternoon, wearing his regular clothes. He carried a small bag with his notebook and pen. He asked the person at the gate if they needed a teacher.

The academy master looked at the older man. "What do you teach?" he asked.

"Contradiction," said Cassius.

"And where did you work before this?"

"In a courtroom," Cassius said. "For twenty years. I listened to arguments. I learned to hear the moment a story cracked."

The master was silent for a long moment. He looked Cassius right in the eye. "Mister Cassius," he said. "I believe we've been waiting for you."

He has taught at the academy for fourteen years now. He is a calm man. He always sits when he speaks. He says, "Suppose, for the sake of argument..." so much that the kids have started saying it too. He never, ever raises his voice.

He has a special nod. He does it right before a student's argument is about to break. It's a small, kind nod. It tells the student, "The crack is coming." They know it's coming even before they hear it. He learned that trick as a judge, not as a teacher.

He has a friendly argument going with another teacher, Direct-Proof Dora. She is his opposite. Dora thinks you prove something by walking straight down the right path. Cassius thinks you prove something by showing all the other paths are broken.

They have argued about this for years. They are always polite. Neither one has ever changed their mind. They still respect each other. They even sit together at dinner. They are both right, in their own way. Qed, the head of the academy, thinks their arguments are good for everyone.

Cassius still has the notebook from his first day. It's almost full now. On the last page he used, he wrote a little proof for himself.

"Suppose, for the sake of argument, that I did nothing for the last fourteen years.