An Educational Singularity

Knowl­edge com­pounds. Paint­ing is good prac­tice for ar­chi­tec­ture. Stand-up com­edy is good train­ing for stage magic.

The knowl­edge trans­fer­ence be­tween math and physics might be as high as 75%. Un­for­tu­nately, the knowl­edge trans­fer­ence be­tween two ran­dom fields tends to be small. Between math and draw­ing it might be as low as 1%. In my per­sonal ex­pe­rience it’s hard to find any pair of broadly ap­pli­ca­ble sub­jects that don’t have at least 1% over­lap. Usu­ally the num­ber will be be­tween 1% and 75%.

Let’s sup­pose mas­ter­ing a new field of knowl­edge gives you a 5% dis­count on av­er­age on ev­ery sub­se­quent field. Sup­pose it takes time for some­one who knows noth­ing to mas­ter a new field. Then the amount of time it takes to mas­ter a new field is a func­tion of how many fields you have already mas­tered .

How much time does it take to learn fields in­stead of just the field ?

This is a ge­o­met­ric se­ries.

ap­pears to con­verge.

con­verges for ev­ery pos­i­tive trans­fer­ence rate. If we use 1% in­stead of 5% we just get .

What does this mean?

The ed­u­ca­tional phase transition

Ob­vi­ously, some­one who has hit is not go­ing to pos­sess all of hu­man knowl­edge. No mat­ter how much you know it’s still go­ing to take you some min­i­mum time to learn the di­alec­ti­cal quirks of, say, He­jazi Ara­bic.

What re­ally means is you’ve hit a cer­tain endgame. The pro­cess of learn­ing has un­der­gone a phase tran­si­tion. All the broad con­cep­tual ma­chin­ery and widely-ap­pli­ca­ble facts are there. Pick­ing up any­thing new is just a mat­ter of plug­ging new data into pre­ex­ist­ing sock­ets.

More in­ter­est­ing than “what hap­pens at this phase tran­si­tion” is the idea that “there is a phase tran­si­tion” and we can reach it in finite time. Per­haps even within a hu­man life­time.

Much like stream en­try, I sus­pect any­one who achieves this phase tran­si­tion is bet­ter off keep­ing his/​her mouth shut about it in po­lite so­ciety.

An al­ter­na­tive model

The con­cept of a sin­gu­lar­ity in the above model re­lies on a dou­ble-pos­i­tive feed­back loop. It as­sumes “Each sub­ject you know con­veys a com­pound­ing 5% dis­count on learn­ing each sub­se­quent sub­ject.” If we tweak this as­sump­tion into “Each of study time con­veys a com­pound­ing 5% dis­count on learn­ing each sub­se­quent sub­ject” then never con­verges as .

How­ever, this al­ter­na­tive model still breaks down at high . It just breaks down grad­u­ally. For ex­am­ple, at the ex­po­nen­tial model pre­dicts an ab­surd learn­ing rate times that of a be­gin­ner. In the hu­man world such a high rate of learn­ing is in­dis­t­in­guish­able from in­finity. The model has bro­ken down.