Beyond hard work

I average a little over 10 minutes for a Times cryptic crossword… Mark on the other hand averages a little over 5 minutes. But this means that if over the years we both spent an equal time on puzzles, Mark actually solves twice as many as I do. That’s a huge advantage on top of his natural speed because he sees more unusual words/more clues etc. So that gap in natural ability then gets amplified.

Simon, from Cracking the Cryptic

Two people can work just as hard at something for the same amount of time yet still come out with completely different levels of ability. Why is this the case? And what can you do when you find yourself on the wrong side of this phenomenon?

When it comes to sports there are genetic factors at play, but it is not entirely clear to me that this must be the case for intellectual pursuits. I’d believe that some level of “natural talent” is required, whatever that means, but beyond that I do not think it is too important. At the very least, in pretty much any intellectual field you pick, there are people at the top level who were not some child prodigy practicing for 6 hours a week starting at the age of 5.¹

Let’s go back to our crossword puzzle example. If the reason you are getting beat is because your competition sees more data points than you, and this is something you cannot change, then you have to process your data more efficiently. You have to learn more from less practice. This means you have to be more thoughtful about the practice you do.

If you are doing anything at a high level, your competition² is not going to be lazy. They will be working just as hard as you, if not harder, and you cannot (and do not) want to beat them in terms of hours spent. You do not want to study math for so long that you never see the sun. Nor do you want to be the trader who works 90 hours a week and never sees their family. So what is the only remaining way to beat them?

You have to reflect more than they do, and you have to be more introspective. Fortunately, this is not a hard thing to do. Think about the smartest people you know; how many of them are actively writing? How many of them think about the lessons life has taught them hard enough to put it into words? I’d wager not many (and if I am wrong, you are hanging around a really good crowd of people and should learn as much from them as possible).

Of course, this is not to say that smart people are not thoughtful; the vast majority of them are. They are definitely reflecting on things in their head, and a good number are on paper. So if you are not quite at that level yet, you have to reflect even harder than they are, because it is the only world in which you ever catch up.

Be intentional

At any point in time, most people do not even have a good answer to “what are you doing right now, what do you hope to get out of it, and what are you currently getting out of it?” If you are in college, your friends might tell you they are taking certain classes, but if they cannot even imagine how the things they learn might broadly impact their lives next semester, why on earth are they doing it?[^cat]

Can you answer these questions? As for myself, I know that I could not (and still cannot fully) give thoughtful responses. You only get to take so many classes in college, and more broadly, you only get so many hours in a day to live. And being introspective doesn’t merely allow you to do better things; it also allows you to do things better.

When I talk about introspection, I am referring to it at a more broad level. People are good at telling you the specifics of what they are doing, and very bad at telling you how it is broadly useful or important. As a specific example, a lot of my friends took Category Theory with me. One of the things they were very excited about was the Yoneda Lemma and what it told them about representable functors. They could not name a single real application of it.

It was very clear to me that they were very technically proficient with the language of categories. And it was equally clear to me that they still have no idea how any of it fits in the broader scheme of mathematics. In all fairness, I have done similar things in my mathematical education sinfully often! The answer is not to stop exploring new areas of math when you aren’t sure how it’ll fit in with what you already know or want to do. Rather, it is to occasionally stop and think, “it seems I have a leak in my understanding here, and I should fix it eventually”.

Hell, you ask the average math major how you actually get from $\mathbb{Q}$ to $\mathbb{R}$ and they will have no idea! And at many points in their lives, they implicitly used something like the Intermediate Value Theorem which relied on the fundamental properties of $\mathbb{R}$. For us it might’ve been earlier on in our lives, but we were no different.

So the question becomes: what else in our lives are we missing? What can you catch just by reflecting more thoroughly that other smart people might miss? And how can you let these kinds of gains compound until you catch up?