Expertise ladders

I have a habit of regularly revisiting the advice pages I keep bookmarked. Most of this advice seems obvious to me, yet I still manage to occasionally forget to follow it. One of those pages is rather strange. It’s Terence Tao’s.

His suggestions are divided into five categories, from primary school to postdoctoral research, gradually shifting from “do not focus on tests and exams” to “do not obsess over famous problems.” I’m not a professional mathematician, and at this point, I probably never will be. Still, whenever I reread these mostly irrelevant tips, I find myself energized, driven to work, and my mood instantly improves, more so than from any other advice. It always felt odd.

Long ago, as a teenager interested in programming, I came across Donald Knuth’s¹ Concrete Mathematics. I opened the first chapter: he quickly solved the Josephus problem, but kept generalizing it until he ended up with a function that mapped a domain with a radix x to a range with a radix y, where x and y are variables. I was drooling. I’d never seen anything like that in a textbook before.

But the best part was yet to come. The exercises were grouped into six categories, quoted as follows:

• Warmups are exercises that EVERY READER should try to do when reading the material.
• Basics are exercises to develop facts that are best learned by trying one’s own derivation rather than by reading somebody else’s.
• Homework exercises are problems intended to deepen an understanding of material in the current chapter.
• Exam problems typically involve ideas from two or more chapters simultaneously; they are generally intended for use in take-home exams (not for in-class exams under time pressure).
• Bonus problems go beyond what an average student of concrete mathematics is expected to handle while taking a course based on this book; they extend the text in interesting ways.
• Research problems may or may not be humanly solvable, but the ones presented here seem to be worth a try.

Every chapter had both the “every reader should try” exercises and the “may not be humanly solvable” ones.

There are other books with excellent exercises, like Grimmett’s, whose problem statements I would read and think no way, I don’t believe this, I have to prove it. Even reading those books, normally I’m too lazy to do the problems marked with a star.

But Knuth’s problems were special. Inspiring. I absolutely tried my hardest (although I wasn’t that good, and rarely cracked level 4).

If I ever write a textbook myself, I want the exercises to be organized exactly like that.

I’m writing this because these two memories spontaneously came to mind, and I realized something: I implicitly assume a bottomless gulf between myself and people like Knuth and Tao (duh).

Yet seeing myself on a ladder, far away from them but slightly closer than a year ago, forces me into a different perspective: there is no gulf, just many steps. It’s hard to dismiss when you see it. It feels incredibly rewarding.

I am writing my first essay, struggling not to give it the tone of school homework, which seems to have permanently twisted my mind. But there has to be a conclusion, and so…

I also have a habit of scrolling through CVs of top research scientists. It usually causes soul-crushing envy. Ergo, not all ladders work the same way.

For a ladder to work well, I have to be on it, and I have to know where I am. The distance between steps should be manageable. Somebody very cool should be at the far end. It doesn’t have to be in my preferred field, but should be adjacent. It can be surprisingly short, like Knuth’s, but also very long².

It would be nice to see more of them. Please share yours.