Integrating the Lindy Effect

[lsusr]

Suppose the following:

1. Your intelligence is directly proportional to how many useful things you know.

2. Your intelligence increases when your learn things and decreases as the world changes and the things you know go out-of-date.

How quickly the things you know become irrelevant is directly proportional to how many relevant things you know and therefore proportional to your intelligence $I$ and inversely proportional to the typical lifetime of things you know $L$. Let’s use $R$ to denote your rate of learning. Put this together and we get a equation.

$$\frac{I}{L}+\frac{dI}{dt}\propto R$$

If we measure intelligence in units of “facts you know” then the proportionality becomes an equality.

$$\frac{dI}{dt}=R-\frac{I}{L}$$

The solution to this first order differential equation is an exponential function.

$$I\left(t\right)=c{e}^{-t/L}+RL$$

We must solve for $c$. For convenience let’s declare that your intelligence is $0$ at time $t=0$. Then $c$ must equal $-RL$. That gives us a tidy solution.

$$I\left(t\right)=RL(1-e^{-t/L})$$

Our solution makes sense intuitively because your intelligence is directly proportional to $R$ and $L$. But wait a minute.$L$ isn’t just a coefficient. It’s in the exponential too.

Time $t$ and Lifetime $L$

Most human beings reading this article will be between 10 years and 100 years old. In other words, $t$ is measured in decades. In other other words, $t$ is on the order of 10 years.

$L$ values, on the other hand, are distributed exponentially across many orders of magnitude.

Order of days. (0.003 years)

• daily newspaper

Order of weeks (0.2 years)

• Sunday newspaper

• political story

• sports game outcome

Order of decades (10 years)

• programming languages

Order of centuries (100 years)

• classic literature

• most spoken languages

Order of millennia (1,000 years)

• cooking

• history

Order of 10,000 years

• human psychology

Order of gigaannum (billion years)

• biology

• physics

Forever

• math

The details of whether exactly each of these things fit on the scale is not important. What is important is that most things you can know have a useful lifetime at least one order of magnitude away from the human timescale of decades. In other words, we can assume that either $L$ is much greater than $t$ or much less than $t$.

Suppose $L$ is much less than $t$. Then the exponential vanishes and we’re left with $I=RL$. In other words, if $L\ll t$ then how long you have been learning for $t$ is irrelevant. I is constant with respect to time. Years and years of studying will not make you smarter over time.

Suppose that $L$ is much greater than $t$. Then $I=Rt$. What used to be a constant function becomes an increasing linear function.

$$I\left(t\right)=R\cdot \{\begin{array}{cc}L& \text{for}t\gg L\\ t& \text{for}t\ll L\end{array}$$

$I=Rt$ grows with respect to time while $I=RL$ stays constant. Eventually, anyone on an $I=Rt$ trajectory will always become smarter than someone on an $I=RL$ trajectory even if the person on the $I=RL$ trajectory has higher $R\ne 0$.

In the long term, the lifetime of things you learn $L$ is far more important than how fast you learn $R$. Over a lifetime of decades, someone who learns a few durable things slowly will eventually become smarter than someone who learns many transient ones quickly.

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[NicholasKross]

Say I’m deciding what to do with my time (not well-planned, although maybe it should be more deliberate). What’s a quick heuristic for deciding what to do?

“Focus on timeless knowledge” is relatively simple to do, and it started to affect how I view a lot of things.

Why dive deep into trivia that teaches me nothing? Why play the video games I used to play a lot? Why consume junk media (which lsusr wrote another post about)? With a bit of argument, this post nudged me further and faster, in the direction of actually prioritizing what I do with my time.

[Ben Pace]

I enjoyed this simple analysis.

[Raemon]

I had not read this post until just now. I think it is pretty great.

I also already had a vague belief that I should consume more timeless content. But, now I suddenly have a gearsy model that makes it a lot more intuitive why I might want to consume more timeless content. I also have a schema of how to think about “what is valuable to me?”

I bounced off this post the first couple times because, well, it opens with math and math makes my eyes glaze over and maybe it shouldn’t cause that but it is what it is. I suspect it would be worth rewriting this post (or writing an alternate version), that puts the entire model in verbal-english front and center.

[Raemon]

I just noticed that I still forgot what this post was about, while voting, and didn’t even remember that I gave it this glowing review. Which makes me think the title isn’t that great.

[Raemon]

A year later I still wish this post had a title that made it easier to remember the core point.

[lsusr]

Your wish is my command.

[philh]

(Note: your final equation has the << and >> swapped.)

[lsusr]

Fixed! Thank you.

[lsusr]

I’m not quite 100% sure if I can write $\frac{dI}{dt}=R-\frac{I}{L}$ or if I have to write $\frac{dI}{dt}\propto R-\frac{I}{L}$ or if there’s some other scaling factor I’m missing. Please check my math.

[Pattern]

Your writing suggests the second way is correct, not the first:

How quickly the things you know become irrelevant is directly proportional

[Emphasis mine.]

[lsusr]

Thanks. This helps. I’ve edited my post to fix.