17. Induction | 帰納法

Induction

2 minute read

Some things in life we simply don’t know. We only have limited information about the world, and we must extrapolate from said information—hence, inductive reasoning. And we rely on it more than we realize. Whether we make assumptions about an entire population based on a small sample, or trusting that our world will remain the way it always has been, our lives are built upon assumptions drawn from limited experience. Induction is comforting in the sense that it helps us feel like we understand more than we actually do.

But a small sample may not be representative of a whole population, and just because something has happened consistently doesn’t mean it will continue. This is the “problem of induction”, we can never be fully certain that the future will resemble the past, and making assumptions about the future may not be rational. Yet induction works well enough for most of us to navigate our daily lives.

Mathematics

Ironically, the other form of induction, mathematical induction, isn’t about uncertainty at all. In fact, it’s a form of deductive reasoning taking a base case and showing that “P(n) implies P(n+1)” .

Mathematical induction works like a chain of dominoes. You show that the first domino falls, then each domino knocks over the next one, all the way to infinity. If both parts hold true, then every domino will fall. This method lets you prove statements regarding every natural number or even every integer. Mathematical induction gives certainty where regular induction only gives chance.

Yet when I take a step back, I’ve found mathematical induction to be strangely satisfying. It embodies something sweeping and massive, contained within a few, tiny claims. Prove that something works at the beginning, prove that each step leads to the next, and you've proven something infinite using a finite number of steps.

Application

In life, we rarely have the certainty of mathematical induction. We can never guarantee that P(n) implies P(n+1) for all positive integers n. But maybe we can take inspiration from it. Maybe progress in life is reminiscent of setting up your own line of dominoes. Take care of the first step. Then build systems that encourage the next step to follow naturally. Success won’t be inevitable, but it becomes more likely. But in the meantime, we live with regular inductive reasoning: gathering pieces of evidence from our own lives, guessing patterns, and moving forward as best we can.

And if only an n’th day of workshop implied an (n+1)’th day, maybe we would be infinitely happier.

Life Update!

It's hard to believe that three weeks have already passed. The workshops that filled my mornings every day have concluded, and are to be replaced by maxis and minis for the second half. Once again, I'm reminded that in the moment, time passes so slowly, yet looking back, it all passed by in two blinks. I think everyone here is genuinely amazing, and I want to maintain friendships even after the program ends. But who knows what'll happen? Inductive reasoning doesn't guarantee anything.