Induction Ida and Strong-Induction Sten

Ida looked at Sten. Sten looked at Ida. They both volunteered.

"All right. Both of you. But Ida, you go first — you need to bring up the new rope, and Sten will come up after with the splicer's tools. The thing is — you have to coordinate. Each step Sten takes has to be coordinated with where you already are. Otherwise the rope will get tangled."

"Got it," Ida said. She started up.

She used her rule. If I can take the first step, and if I can take the next step from whatever step I'm on, I can take all the steps. The new bell rope coiled in her arms. She went up. Step one. Step two. From step five, take step six. From step thirty-eight, take step thirty-nine. Her rule was simple: the previous step is all I need to think about. She didn't need to remember anything about steps one through thirty-seven; she just needed to remember step thirty-eight, and that was enough to take step thirty-nine.

She reached step seventy-nine and started splicing the new rope. She called down.

"All right, Sten! Come up!"

Sten started up with the splicer's tools.

He used his rule. If I can take the first step, and if my next step depends on EVERYTHING I've already done, I can take all the steps.

This was important.

At step thirty-eight, Sten paused. He thought: Ida passed through this step a few minutes ago. She left the rope coiled along the right wall. I know this because I saw her do it from below. So when I take step thirty-nine, I need to NOT step on the rope, which means I need to step on the LEFT side of step thirty-nine, which I know from remembering everything Ida did, not just from looking at step thirty-eight.

He stepped left on thirty-nine. No tangle.

At step fifty-five, he paused again. He thought: Ida turned around at step fifty-five to call down to me. So she shifted the rope to the inside of the spiral. So when I step on fifty-five, I need to step OUTSIDE. He stepped outside on fifty-five. No tangle.

At step seventy-nine, he arrived next to Ida. The rope was not tangled.

"How did you know?" Ida asked.

"I needed everything," Sten said.

Ida thought about this on the way back down. They were walking together, and the new bell rope was now safely installed, and the bell would ring at sunset.

"So your rule isn't WRONG," Ida said. "But it's heavier."

"It's heavier," Sten said.

"My rule is — only think about the step you're on. Take the next step from THAT step. Don't carry the whole history with you."

"Right."

"Your rule is — keep ALL of the history with you. Use everything to decide the next step."

"Right."

"And mine works for most things."

"For most things."

"But for some things — like coordinating with you and the rope — mine isn't enough. Because the rope's position depends on every step that came before. So I'd need to know the WHOLE history of where the rope went, not just where I'm standing now."

"Right. That's when mine helps."

Ida thought about this. "If we were just climbing the stairs alone, both rules would get us to the top. They'd give the same answer."

"They would."

"It's only when the next step needs to know something OLDER than just-the-previous-step that yours matters."

"That's when stronger induction earns its weight," Sten said.