Orthogonality

TL;DR: I used to think the best way to get really good at skill $s_{i}$ was to specialize by investing lots of time $t_{i}$ into $s_{i}$ . I was wrong. Investing lots of time $t_{i}$ into $s_{i}$ works only as a first-order approximation. Once $t_{i}$ becomes large, investing in some other $t_{j \neq i}$ produces greater real-world performance $p_{i}$ than continued investment in $t_{i}$ .

I like to think of intelligence as a vector $s = {s_{1}, s_{2}, \dots, s_{n}}$ where each $s_{i} \geq 0$ is a skill level in a different skill. I think of general intelligence $g$ as the Euclidean norm $g = \sqrt{\sum_{i = 1}^{n} s_{i}^{2}}$ .

I use the Euclidean norm instead of the straight sum $\sum_{i = 1}^{n} s_{i}$ because generality of experience equals generality of transference. Suppose you are exposed to a novel situation requiring skill $s_{n + 1} = 0$ . You have no experience at $s_{n + 1}$ so you must borrow from your most similar skill. The wider a variety of skills you have, more similar your most similar skill will be to $s_{n + 1}$ .

The best way to increase your general intelligence $g$ is to invest time $t_{w}$ into your weakest skill $s_{w}$ . If your invested time $t_{s}$ for your strongest skill is already high then investments in $t_{w}$ can also increase the real world performance of your strongest skill faster than investments in $t_{s}$ .

Suppose you want to increase $p_{1}$ , your real world performance at $s_{1}$ . $\frac{\partial p_{1}}{\partial t_{1}} > 0$ . Investing time $t_{1}$ into $s_{1}$ always results in increasing $p_{1}$ . But eventually you will hit diminishing returns. For every $ϵ > 0$ there exists a $δ \in R$ such that if $t > δ$ then $\frac{\partial s_{1}}{\partial t_{1}} < ϵ$ .

$lim t_{1} \to \infty \frac{\partial p_{1}}{\partial t_{1}} = lim t_{1} \to \infty \frac{\partial s_{1}}{\partial t_{1}} = 0$

Here’s where things get interesting. “All non-trivial abstractions, to some degree, are leaky” and a system is only as secure as its weakest link; cracking a system tends to happen on an overlooked layer of abstraction. All real world applications of skill are non-trivial abstractions. Therefore performance in one skill occasionally leaks over to improve performance of adjacent skills. Your real-world performance at $p_{1}$ leaks over from adjacent skills $s_{j \neq 1} \in s$ on rungs above and below $s_{1}$ on the ladder of abstraction.

These adjacent skills increase your real world performance on $p_{1}$ by a quantity independent of $s_{1}$ . Since ${lim}_{t_{1} \to \infty} \frac{\partial s_{1}}{\partial t_{1}} = 0$ , there will inevitably come a time when increasing $t_{1}$ increases $s_{1}$ less than increasing $t_{i \neq 1}$ .

It follows that quantity of avocations correlates positively with winning Nobel Prizes, despite the time these hobbies take away from one’s specialization.

When I want to improve my ability to write machine learning algorithms, my first instinct is to study machine learning. But in practice, it’s often more profitable to do something seemingly unrelated, like learning about music theory. I find it hard to follow this strategy because it is so counterintuitive.