Integrating the Lindy Effect

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 Your intelligence increases when your learn things and decreases as the world changes and the things you know go out-of-date.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 and inversely proportional to the typical lifetime of things you know . Let’s use to denote your rate of learning. Put this together and we get a equation.

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

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

We must solve for . For convenience let’s declare that your intelligence is at time . Then must equal . That gives us a tidy solution.

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

Time and Lifetime

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

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


  • 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 is much greater than or much less than .

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

Suppose that is much greater than . Then . What used to be a constant function becomes an increasing linear function.

grows with respect to time while stays constant. Eventually, anyone on an trajectory will always become smarter than someone on an trajectory even if the person on the trajectory has higher .

In the long term, the lifetime of things you learn is far more important than how fast you learn . 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.