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 Your in­tel­li­gence in­creases when your learn things and de­creases as the world changes and the things you know go out-of-date.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

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 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.