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 Your intelligence increases when your learn things and decreases as the world changes and the things you know go out-of-date.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{d I}{d t} \propto R

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

\frac{d I}{d t} = R - \frac{I}{L}

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

I (t) = c e^{- t / L} + R L

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

I (t) = R L (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 $I = R L$ . In other words, if $L ≪ t$ then how long you have been learning for $t$ is irrelevant. I is constant with respect to time. Years and years of s...