November 27th, 2023
I recently read one of Paul Graham’s essays on personal growth and mindset, titled Superlinear Returns.
When I am asked to write a brief summary of what his main point is, it would be the following: if you do effective work that will build up exponentially, you will one day reach the summit where all of the best performers reside. As long as you are constantly learning with ambition and curiosity, you will reach the top. Work is nonlinear; the amount of effort to become good at something is much less for top performers with more resources and knowledge than for someone at the bottom. But if you keep at it, results will eventually show. One day.
Here are a few nice points that I would like to elaborate on. All of these are quoted directly from the above article – all credit for italicized words belongs to the author.
In all of these, the rich get richer.
I agree. I admire how he states this without any doubt. In any case in politics, economics, science or art, the rich indeed get richer because they have mastered the method of compounding work. A small percent of a large quantity is a large quantity, and it gets added to their skillset, which from there a yet larger lump can be added, and so on.
But the main advantage is that by focusing on growth rate you tend to get something that grows exponentially.
A week ago, I was still struggling with how I worked for about a year to see minimal increases in my AMC and college competition scores. The above statement answered my question: I was too distracted focusing on scores that it actually inhibited me from focusing on my growth rate! I admit that I have gotten quite sidetracked over the past few years, but these gentle reminders make me aware that putting more thought into the actual process over the result will get you, at least closer, to your goal. How can you get to that top 0.001% if you sit there and squander your time gaping at dramatic achievements and don’t think about getting good? Emphasize the process, and make it enjoyable, so you’re constantly expanding that brain of yours.
Whenever how well you do depends on how well you’ve done, you’ll get exponential growth.
There’s so much wisdom in this simple essay that I would spend hours and hours addressing each of these sage remarks!
The above is another way of suggesting patience. When you get a little better, you know that you’ve put in the work and that you have a steady growth rate. Furthermore, if you are maintaining good habits every day, you know that you’re on the road to exponential growth. It all takes patience to see the fruit.
And exponential growth helps you cross thresholds: in a market with network effects, a company that grows fast enough can shut out potential competitors.
Thresholds, secondary to exponential growth, are the second key factor of superlinear returns. No wonder there are so few people at the top of any field. In math contests, most of us see thresholds as qualifications to further tests, which lead to even more tests. And even getting to a level above the primary counts as an accolade. In the world of math contests, the AIME (American Invitational Mathematics Examination) counts as a threshold for approximately the top 10% of math contestants. The prestigious USAMO (United States American Mathematics Olympiad) is held for about five hundred people around the country and is used as a selection for the intensive Mathematics Olympiad Program (only about eighty to ninety people attend). Although these tests are ultimately designed to choose the USA IMO team, there is no doubt that these are merely thresholds. If you know more math and are able to solve more problems, you are able to learn even more concepts and solve even more problems etc. At first, this process may seem slow, but the people at the top have cultivated exponential growth, and as a result, you see them on the podiums.
Or work can compound by teaching you, since learning compounds. This second case is an interesting one because you may feel you’re doing badly as it’s happening. You may be failing to achieve your immediate goal. But if you’re learning a lot, then you’re getting exponential growth nonetheless.
Unbelievable. I thought I was going nowhere with all the math grind for the past few months, but I finally saw a light at the end of the tunnel. Of course, I had aimed to score high in my computational contests, but after the AMCs, I settled into a little web of despair. Oftentimes, when I do poorly, I question whether or not I’m using my time properly.
The answer is yes.
It’s hard to imagine, but every single little improvement, every single hard problem, every single minute thinking about a geometry configuration or a number-theoretic approach to a computer science problem is worth it. Why should I get bogged down by a score if it wasn’t really the important focus?
Learning is the most important thing.
You don’t do science to get the accolade or show off to your peers to get a boost of ego. In freshman year, I boasted a lot. In sophomore year, I was too stressed about scores and performing my best. And in junior year, I’ll decide to let all of it go and enjoy intense learning while it lasts. That’s what being a true nerd is.
Which yields another heuristic: always be learning. If you’re not learning, you’re probably not on a path that leads to superlinear returns.
Again: learning is the most important thing. The only way to guarantee compounding is if you are learning. If you’re not, you’re basically making your brain static, which does not lead to improvement.
But don’t overoptimize what you’re learning. Don’t limit yourself to learning things that are already known to be valuable. You’re learning; you don’t know for sure yet what’s going to be valuable, and if you’re too strict you’ll lop off the outliers.
This is one that I struggle with a lot, as do many of my friends. Whether to start the olympiad grind late AIME grind a lot of Codeforces or do USACO Silver problems, has made me scratch my head with confusion. Additionally, I struggled to learn the “right concepts” and “do the right problems”, especially for AIME and college computational contests. Unfortunately, for most people, there is no “good path”: it ultimately boils down to doing whatever you feel like doing with the time you have to study. In fact, this produces a dilemma I commonly forget: once you’ve learned what’s good and what’s not for entering a threshold, it’s hard to tell other people about it, because chances are that it won’t work for them. You know what to do since you’re there, but other people may need to discover it by themselves. Thus, just learning a lot by yourself is likely sufficient to propel you to the next level.
Last one for today!
So while both ambition and curiosity can get you into this territory, curiosity may be the more powerful of the two. Ambition tends to make you climb existing peaks, but if you stick close enough to an interesting enough question, it may grow into a mountain beneath you.
I strongly agree. Ambition can take you lots of places, but curiosity will take you much further. Let this be a self-reminder to do whatever problem or learn whatever concept I feel like doing.
