Dora and Cassius

Dora went first, because Dora always went first. That was the honest thing to do.

"Watch," she said. "We're given two even numbers. Even means twice something. So the first is two-times-some-number — call it two-m. The second is twice some other number — two-n." She wrote them up, plain and forward. "Now I add them. Two-m plus two-n. And both of them have a two in them, so I can gather that two out front: it's two times the quantity m-plus-n."

She stepped back. "And two times a whole number is exactly what even means. So the sum is even. I started at the givens, I took true steps, and here I am, standing on the conclusion." She dusted her hands. "Front door. Straight through."

The claim on the board glowed a satisfied, settled gold. Proven.

"Now do it my way," said Cassius, stepping up, "which will feel, at first, completely backwards."

He did not start with the two even numbers. He started by assuming the exact opposite of what everyone wanted to be true.

"Suppose," he said, with the small wicked delight of someone opening a window, "that I'm wrong. Suppose the sum of our two even numbers comes out odd." He wrote the wrong idea on the board on purpose, and the whole proof-hall shivered a little, the way a room does when someone says something untrue out loud. "Just — go with me. We've assumed the sum is odd."

"But," a watching student squeaked, "it's not odd!"

"I know," Cassius said happily. "That's the whole plan. I'm going to follow this wrong idea, step by honest step, and watch what happens." He worked forward from the lie. Two-m plus two-n is two-times-(m-plus-n) — the same true step Dora took — "which is even. But three lines ago I assumed it was odd. So now I'm holding a number that has to be even and odd, at the very same time." He tapped the board where the two facts collided. "And no number can be both. The world cracks. Crrk."

The false assumption on the board split straight down the middle and crumbled.

"When the wrong idea breaks itself like that," Cassius said quietly, "the right idea is the only thing left standing. The sum can't be odd. So it's even. Same truth as Dora's — I just reached it by letting the opposite destroy itself."

The student who had squeaked, a small serious one named Cricket, looked back and forth between them.

"But you got the same answer," Cricket said. "You both proved the exact same thing. So why are there two ways? Isn't one of you just doing it the long way?"

Dora and Cassius looked at each other, and — this was the important part — neither of them looked smug.

"For this claim," Dora admitted, "my way's the neat one. Cassius took the scenic route."

"For this one, yes," Cassius agreed. "But come here." He wiped the board and wrote a new claim: The square root of two cannot be written as a neat fraction. "Try to prove that your way, Dora. Walk up to it from the front."

Dora looked at the front of it. She looked for a given to start from, a first true step to take. There wasn't one. The front of this claim was a smooth wall with no handle. "...I can't," she said slowly. "There's nowhere to start."

"There almost never is, with this kind," said Cassius gently. "But my door works. I assume it can be written as a neat fraction — and I follow that assumption until it contradicts itself, and shatters. Some truths only have a back door, Cricket. That's not a trick. That's just where their door happens to be."

Cricket's eyes went wide. "So you need both of you."

"You need both of us," Dora said.