1960: The Year The Singularity Was Cancelled

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[Epistemic sta­tus: Very spec­u­la­tive, es­pe­cially Parts 3 and 4. Like many good things, this post is based on a con­ver­sa­tion with Paul Chris­ti­ano; most of the good ideas are his, any er­rors are mine.]

I.

In the 1950s, an Aus­trian sci­en­tist dis­cov­ered a se­ries of equa­tions that he claimed could model his­tory. They matched past data with startling ac­cu­racy. But when ex­tended into the fu­ture, they pre­dicted the world would end on Novem­ber 13, 2026.

This sounds like the plot of a sci-fi book. But it’s also the story of Heinz von Fo­er­ster, a mid-cen­tury physi­cist, cy­ber­net­i­cian, cog­ni­tive sci­en­tist, and philoso­pher.

His prob­lems started when he be­came in­ter­ested in hu­man pop­u­la­tion dy­nam­ics.

(the rest of this sec­tion is loosely adapted from his Science pa­per “Dooms­day: Fri­day, 13 Novem­ber, A.D. 2026”)

As­sume a perfect par­a­disi­a­cal Gar­den of Eden with in­finite re­sources. Start with two peo­ple – Adam and Eve – and as­sume the pop­u­la­tion dou­bles ev­ery gen­er­a­tion. In the sec­ond gen­er­a­tion there are 4 peo­ple; in the third, 8. This is that old rid­dle about the grains of rice on the chess­board again. By the 64th gen­er­a­tion (ie af­ter about 1500 years) there will be 18,446,744,073,709,551,615 peo­ple – ie about about a billion times the num­ber of peo­ple who have ever lived in all the eons of hu­man his­tory. So one of our as­sump­tions must be wrong. Prob­a­bly it’s the one about the perfect par­adise with un­limited re­sources.

Okay, new plan. As­sume a limited world with a limited food sup­ply /​ limited car­ry­ing ca­pac­ity. If you want, imag­ine it as an is­land where ev­ery­one eats co­conuts. But there are only enough co­conuts to sup­port 100 peo­ple. If the pop­u­la­tion re­pro­duces be­yond 100 peo­ple, some of them will starve, un­til they’re back at 100 peo­ple. In the sec­ond gen­er­a­tion, there are 100 peo­ple. In the third gen­er­a­tion, still 100 peo­ple. And so on to in­finity. Here the pop­u­la­tion never grows at all. But that doesn’t match real life ei­ther.

But von Fo­er­ster knew that tech­nolog­i­cal ad­vance can change the car­ry­ing ca­pac­ity of an area of land. If our hy­po­thet­i­cal is­lan­ders dis­cover new co­conut-tree-farm­ing tech­niques, they may be able to get twice as much food, in­creas­ing the max­i­mum pop­u­la­tion to 200. If they learn to fish, they might open up en­tirely new realms of food pro­duc­tion, in­creas­ing pop­u­la­tion into the thou­sands.

So the rate of pop­u­la­tion growth is nei­ther the dou­ble-per-gen­er­a­tion of a perfect par­adise, nor the zero-per-gen­er­a­tion of a stag­nant is­land. Rather, it de­pends on the rate of eco­nomic and tech­nolog­i­cal growth. In par­tic­u­lar, in a closed sys­tem that is already at its car­ry­ing ca­pac­ity and with zero marginal re­turn to ex­tra la­bor, pop­u­la­tion growth equals pro­duc­tivity growth.

What causes pro­duc­tivity growth? Tech­nolog­i­cal ad­vance. What causes tech­nolog­i­cal ad­vance? Lots of things, but von Fo­er­ster’s model re­duced it to one: peo­ple. Each per­son has a cer­tain per­cent chance of com­ing up with a new dis­cov­ery that im­proves the econ­omy, so pro­duc­tivity growth will be a func­tion of pop­u­la­tion.

So in the model, the first gen­er­a­tion will come up with some small num­ber of tech­nolog­i­cal ad­vances. This al­lows them to spawn a slightly big­ger sec­ond gen­er­a­tion. This new slightly larger pop­u­la­tion will gen­er­ate slightly more tech­nolog­i­cal ad­vances. So each gen­er­a­tion, the pop­u­la­tion will grow at a slightly faster rate than the gen­er­a­tion be­fore.

This matches re­al­ity. The world pop­u­la­tion barely in­creased at all in the mil­len­nium from 2000 BC to 1000 BC. But it dou­bled in the fifty years from 1910 to 1960. In fact, us­ing his model, von Fo­er­ster was able to come up with an equa­tion that pre­dicted the pop­u­la­tion near-perfectly from the Stone Age un­til his own day.

But his equa­tions cor­re­sponded to some­thing called hy­per­bolic growth. In hy­per­bolic growth, a feed­back cy­cle – in this case pop­u­la­tion causes tech­nol­ogy causes more pop­u­la­tion causes more tech­nol­ogy – leads to growth in­creas­ing rapidly and fi­nally shoot­ing to in­finity. Imag­ine a sim­plified ver­sion of Fo­er­ster’s sys­tem where the world starts with 100 mil­lion peo­ple in 1 AD and a dou­bling time of 1000 years, and the dou­bling time de­creases by half af­ter each dou­bling. It might pre­dict some­thing like this:

1 AD: 100 mil­lion peo­ple
1000 AD: 200 mil­lion peo­ple
1500 AD: 400 mil­lion peo­ple
1750 AD: 800 mil­lion peo­ple
1875 AD: 1600 mil­lion people

…and so on. This sys­tem reaches in­finite pop­u­la­tion in finite time (ie be­fore the year 2000). The real model that von Fo­er­ster got af­ter an­a­lyz­ing real pop­u­la­tion growth was pretty similar to this, ex­cept that it reached in­finite pop­u­la­tion in 2026, give or take a few years (his pin­point­ing of Fri­day Novem­ber 13 was mostly a joke; the equa­tions were not re­ally that pre­cise).

What went wrong? Two things.

First, as von Fo­er­ster knew (again, it was kind of a joke) the tech­nolog­i­cal ad­vance model isn’t liter­ally true. His hy­per­bolic model just op­er­ates as an up­per bound on the Gar­den of Eden sce­nario. Even in the Gar­den of Eden, pop­u­la­tion can’t do more than dou­ble ev­ery gen­er­a­tion.

Se­cond, con­tra all pre­vi­ous his­tory, peo­ple in the 1900s started to have fewer kids than their re­sources could sup­port (the de­mo­graphic tran­si­tion). Cou­ples started con­sid­er­ing the cost of col­lege, and the difficulty of ma­ter­nity leave, and all that, and de­cided that maybe they should stop at 2.5 kids (or just get a puppy in­stead).

Von Fo­er­ster pub­lished has pa­per in 1960, which iron­i­cally was the last year that his equa­tions held true. Start­ing in 1961, pop­u­la­tion left its hy­per­bolic growth path. It is now ex­pected to sta­bi­lize by the end of the 21st cen­tury.

II.

But no­body re­ally ex­pected the pop­u­la­tion to reach in­finity. Armed with this story, let’s look at some­thing more in­ter­est­ing.

This might be the most de­press­ing graph ever:

The hori­zon­tal axis is years be­fore 2020, a ran­dom year cho­sen so that we can put this in log scale with­out nega­tive val­ues screw­ing ev­ery­thing up.

The ver­ti­cal axis is the amount of time it took the world econ­omy to dou­ble from that year, ac­cord­ing to this pa­per. So for ex­am­ple, if at some point the econ­omy dou­bled ev­ery twenty years, the dot for that point is at twenty. The dou­bling time de­creases through­out most of the pe­riod be­ing ex­am­ined, in­di­cat­ing hy­per­bolic growth.

Hyper­bolic growth, as men­tioned be­fore, shoots to in­finity at some spe­cific point. On this graph, that point is rep­re­sented by the dou­bling time reach­ing zero. Once the econ­omy dou­bles ev­ery zero years, you might as well call it in­finite.

For all of hu­man his­tory, eco­nomic progress formed a near-perfect straight line pointed at the early 21st cen­tury. Its des­ti­na­tion varied by a cen­tury or two now and then, but never more than that. If an an­cient Egyp­tian economist had mod­ern tech­niques and method­olo­gies, he could have made a graph like this and pre­dicted it would reach in­finity around the early 21st cen­tury. If a Ro­man had done the same thing, us­ing the eco­nomic data available in his own time, he would have pre­dicted the early 21st cen­tury too. A me­dieval Bu­ru­gun­dian? Early 21st cen­tury. A Vic­to­rian English­man? Early 21st cen­tury. A Stal­inist Rus­sian? Early 21st cen­tury. The trend was re­ally re­silient.

In 2005, in­ven­tor Ray Kurzweil pub­lished The Sin­gu­lar­ity Is Near, claiming there would be a tech­nolog­i­cal sin­gu­lar­ity in the early 21st cen­tury. He didn’t re­fer to this graph speci­fi­cally, but he high­lighted this same trend of ev­ery­thing get­ting faster, in­clud­ing rates of change. Kurzweil took the in­finity at the end of this graph very se­ri­ously; he thought that some event would hap­pen that re­ally would cat­a­pult the econ­omy to in­finity. Why not? Every data point from the Stone Age to the Atomic Age agreed on this.

This graph shows the Sin­gu­lar­ity get­ting can­cel­led.

Around 1960, dou­bling times stopped de­creas­ing. The econ­omy kept grow­ing. But now it grows at a flat rate. It shows no signs of reach­ing in­finity; not soon, not ever. Just con­stant, bor­ing 2% GDP growth for the rest of time.

Why?

Here von Fo­er­ster has a ready an­swer pre­pared for us: pop­u­la­tion!

Eco­nomic growth is a func­tion of pop­u­la­tion and pro­duc­tivity. And pro­duc­tivity de­pends on tech­nolog­i­cal ad­vance­ment and tech­nolog­i­cal ad­vance­ment de­pends on pop­u­la­tion, so it all bot­toms out in pop­u­la­tion in the end. And pop­u­la­tion looked like it was go­ing to grow hy­per­bol­i­cally un­til 1960, af­ter which it stopped. That’s why hy­per­bolic eco­nomic growth, ie progress to­wards an eco­nomic sin­gu­lar­ity, stopped then too.

In fact…

This is a re­ally sketchy of per cap­ita in­come dou­bling times. It’s sketchy be­cause un­til 1650, per cap­ita in­come wasn’t re­ally in­creas­ing at all. It was fol­low­ing a one-step-for­ward one-step-back pat­tern. But if you take out all the steps back and just watch how quickly it took the steps for­ward, you get some­thing like this.

Even though per cap­ita in­come tries to ab­stract out pop­u­la­tion, it dis­plays the same pat­tern. Un­til 1960, we were on track for a sin­gu­lar­ity where ev­ery­one earned in­finite money. After 1960, the graph “bounces back” and growth rates sta­bi­lize or even de­crease.

Again, von Fo­er­ster can ex­plain this to us. Per cap­ita in­come grows when tech­nol­ogy grows, and tech­nol­ogy grows when the pop­u­la­tion grows. The sig­nal from the end of hy­per­bolic pop­u­la­tion growth shows up here too.

To make this re­ally work, we prob­a­bly have to zoom in a lit­tle bit and look at con­crete re­al­ity. Most tech­nolog­i­cal ad­vances come from a few ad­vanced coun­tries whose pop­u­la­tion sta­bi­lized a lit­tle ear­lier than the world pop­u­la­tion. Of the con­stant pop­u­la­tion, an in­creas­ing frac­tion are be­com­ing re­searchers each year (on the other hand, the low-hang­ing fruit gets picked off and tech­nolog­i­cal ad­vance be­comes harder with time). All of these fac­tors mean we shouldn’t ex­pect pro­duc­tivity growth/​GWP per cap­ita growth/​tech­nolog­i­cal growth to ex­actly track pop­u­la­tion growth. But on the sort of or­ders-of-mag­ni­tude scale you can see on log­a­r­ith­mic graphs like the ones above, it should be pretty close.

So it looks like past pre­dic­tions of a techno-eco­nomic sin­gu­lar­ity for the early 21st cen­tury were based on ex­trap­o­la­tions of a hy­per­bolic trend in tech­nol­ogy/​econ­omy that de­pended on a hy­per­bolic trend in pop­u­la­tion. Since the pop­u­la­tion sin­gu­lar­ity didn’t pan out, we shouldn’t ex­pect the techno-eco­nomic sin­gu­lar­ity to pan out ei­ther. In fact, since pop­u­la­tion in ad­vanced coun­tries is start­ing to “stag­nate” rel­a­tive to ear­lier eras, we should ex­pect a rel­a­tive techno-eco­nomic stag­na­tion too.

…maybe. Be­fore com­ing back to this, let’s ex­plore some of the other im­pli­ca­tions of these mod­els.

III.

The first graph is the same one you saw in the last sec­tion, of ab­solute GWP dou­bling times. The sec­ond graph is the same, but limited to Bri­tain.

Where’s the In­dus­trial Revolu­tion?

It doesn’t show up at all. This may be a sur­prise if you’re used to the stan­dard nar­ra­tive where the In­dus­trial Revolu­tion was the most im­por­tant event in eco­nomic his­tory. Graphs like this make the case that the In­dus­trial Revolu­tion was an ex­plo­sive shift to a to­tally new growth regime:

It sure looks like the In­dus­trial Revolu­tion was a big deal. But Paul Chris­ti­ano ar­gues your eyes may be de­ceiv­ing you. That graph is a hy­per­bola, ie cor­re­sponds to a sin­gle sim­ple equa­tion. There is no break in the pat­tern at any point. If you trans­formed it to a log dou­bling time graph, you’d just get the graph above that looks like a straight line un­til 1960.

On this view, the In­du­stiral Revolu­tion didn’t change his­tor­i­cal GDP trends. It just shifted the world from a Malthu­sian regime where eco­nomic growth in­creased the pop­u­la­tion to a mod­ern regime where eco­nomic growth in­creased per cap­ita in­come.

For the en­tire his­tory of the world un­til 1000, GDP per cap­ita was the same for ev­ery­one ev­ery­where dur­ing all his­tor­i­cal eras. An Is­raelite shep­herd would have had about as much stuff as a Ro­man farmer or a me­dieval serf.

This was the Malthu­sian trap, where “pro­duc­tivity pro­duces peo­ple, not pros­per­ity”. Peo­ple re­pro­duce to fill the re­sources available to them. Every­one always lives at sub­sis­tence level. If pro­duc­tivity in­creases, peo­ple re­pro­duce, and now you have more peo­ple liv­ing at sub­sis­tence level. OurWor­ldInData has an awe­some graph of this:

As of 1500, places with higher pro­duc­tivity (usu­ally richer farm­land, but bet­ter tech­nol­ogy and so­cial or­ga­ni­za­tion also help) pop­u­la­tion den­sity is higher. But GDP per cap­ita was about the same ev­ery­where.

There were always oc­ca­sional wind­falls from ex­cit­ing dis­cov­er­ies or eco­nomic re­forms. For a cen­tury or two, GDP per cap­ita would rise. But pop­u­la­tion would always catch up again, and ev­ery­one would end up back at sub­sis­tence.

Some peo­ple ar­gue Europe broke out of the Malthu­sian trap around 1300. This is not quite right. 1300s Europe achieved above-sub­sis­tence GDP, but only be­cause the Black Plague kil­led so many peo­ple that the sur­vivors got a wind­fall by tak­ing their land.

Malthus pre­dicts that this should only last a lit­tle while, un­til the Euro­pean pop­u­la­tion bounces back to pre-Plague lev­els. This pre­dic­tion was ex­actly right for South­ern Europe. North­ern Europe didn’t bounce back. Why not?

Un­clear, but one an­swer is: fewer peo­ple, more plagues.

Broad­berry 2015 men­tions that North­ern Euro­pean cul­ture pro­moted later mar­riage and fewer chil­dren:

The North Sea Area had an ad­van­tage in this area be­cause of its ap­proach to mar­riage. Ha­j­nal (1965) ar­gued that north­west Europe had a differ­ent de­mo­graphic regime from the rest of the world, char­ac­ter­ised by later mar­riage and hence limited fer­til­ity. Although he origi­nally called this the Euro­pean Mar­riage Pat­tern, later work es­tab­lished that it ap­plied only to the north­west of the con­ti­nent. This can be linked to the availa­bil­ity of labour mar­ket op­por­tu­ni­ties for fe­males, who could en­gage in mar­ket ac­tivity be­fore mar­riage, thus in­creas­ing the age of first mar­riage for fe­males and re­duc­ing the num­ber of chil­dren con­ceived (de Moor and van Zan­den, 2010). Later mar­riage and fewer chil­dren are as­so­ci­ated with more in­vest­ment in hu­man cap­i­tal, since the wom­en­em­ployed in pro­duc­tive work can ac­cu­mu­late skills, and par­ents can af­ford to in­vest more in each of the smaller num­ber of chil­dren be­cause of the “quan­tity-qual­ity” trade-off (Voigtlän­der and Voth, 2010).

This low birth rate was hap­pen­ing at the same time plagues were rais­ing the death rate. Here’s an­other amaz­ing graph from OurWor­ldInData:

Bri­tish pop­u­la­tion maxes out around 1300 (?), de­clines sub­stan­tially dur­ing the Black Plague of 1348-49, but then keeps de­clin­ing. The List Of English Plagues says an­other plague hit in 1361, then an­other in 1369, then an­other in 1375, and so on. Some his­to­ri­ans call the whole pe­riod from 1348 to 1666 “the Plague Years”.

It looks like through the 1350 – 1450 pe­riod, pop­u­la­tion keeps de­clin­ing, and per cap­ita in­come keeps go­ing up, as Malthu­sian the­ory would pre­dict.

Between 1450 and 1550, pop­u­la­tion starts to re­cover, and per cap­ita in­comes start go­ing down, again as Malthus would pre­dict. Then around 1560, there’s a jump in in­comes; ac­cord­ing to the List Of Plagues, 1563 was “prob­a­bly the worst of the great metropoli­tan epi­demics, and then ex­tended as a ma­jor na­tional out­break”. After 1563, pop­u­la­tion in­creases again and per cap­ita in­comes de­cline again, all the way un­til 1650. Pop­u­la­tion does not in­crease in Bri­tain at all be­tween 1660 and 1700. Why? The List de­clares 1665 to be “The Great Plague”, the largest in England since 1348.

So from 1348 to 1650, North­ern Euro­pean per cap­ita in­comes di­verged from the rest of the world’s. But they didn’t “break out of the Malthu­sian trap” in a strict sense of be­ing able to di­rect pro­duc­tion to­ward pros­per­ity rather than pop­u­la­tion growth. They just had so many plagues that they couldn’t grow the pop­u­la­tion any­way.

But in 1650, England did start break­ing out of the Malthu­sian trap; pop­u­la­tion and per cap­ita in­comes grow to­gether. Why?

Paul the­o­rizes that tech­nolog­i­cal ad­vance fi­nally started mov­ing faster than max­i­mal pop­u­la­tion growth.

Re­mem­ber, in the von Fo­er­ster model, the growth rate in­creases with time, all the way un­til it reaches in­finity in 2026. The closer you are to 2026, the faster your econ­omy will grow. But pop­u­la­tion can only grow at a limited rate. In the ab­solute limit, women can only have one child per nine months. In re­al­ity, in­fant mor­tal­ity, in­fer­til­ity, and con­scious de­ci­sion to de­lay child­bear­ing mean the nat­u­ral limits are much lower than that. So there’s a the­o­ret­i­cal limit on how quickly the pop­u­la­tion can in­crease even with max­i­mal re­sources. If the econ­omy is grow­ing faster than that, Malthus can’t catch up.

Why would this hap­pen in England and Hol­land in 1650?

Lots of peo­ple have his­tor­i­cal ex­pla­na­tions for this. North­ern Euro­pean pop­u­la­tion growth was so low that peo­ple were forced to in­vent la­bor-sav­ing ma­chin­ery; even­tu­ally this reached a crit­i­cal mass, we got the In­dus­trial Revolu­tion, and eco­nomic growth sky­rock­eted. Or: the dis­cov­ery of Amer­ica led to a source of new riches and a con­ve­nient sink for ex­cess pop­u­la­tion. Or: some­thing some­thing Protes­tant work ethic print­ing press cap­i­tal­ism. Th­ese are all plau­si­ble. But how do they sync with the claim that ab­solute GDP never left its ex­pected tra­jec­tory?

I find the idea that the In­dus­trial Revolu­tion wasn’t a de­vi­a­tion from trend fas­ci­nat­ing and provoca­tive. But it de­pends on eye­bal­ling a lot of graphs that have had a lot of weird trans­for­ma­tions done to them, plus writ­ing off a lot of out­liers. Here’s an­other way of pre­sent­ing Bri­tain’s GDP and GDP per cap­ita data:

Here it’s a lot less ob­vi­ous that the In­dus­trial Revolu­tion rep­re­sented a de­vi­a­tion from trend for GDP per cap­ita but not for GDP.

Th­ese Bri­tish graphs show less of a sin­gu­lar­ity sig­na­ture than the wor­ld­wide graphs do, prob­a­bly be­cause we’re look­ing at them on a shorter timeline, and be­cause the Plague Years screwed ev­ery­thing up. If we in­sisted on fit­ting them to a hy­per­bola, it would look like this:

Like the rest of the world, Bri­tain was only on a hy­per­bolic growth tra­jec­tory when eco­nomic growth was trans­lat­ing into pop­u­la­tion growth. That wasn’t true be­fore about 1650, be­cause of the plague. And it wasn’t true af­ter about 1850, be­cause of the De­mo­graphic Tran­si­tion. We see a sort of fit to a hy­per­bola be­tween those points, and then the trend just sort of wan­ders off.

It seems pos­si­ble that the In­dus­trial Revolu­tion was not a time of ab­nor­mally fast tech­nolog­i­cal ad­vance or eco­nomic growth. Rather, it was a time when eco­nomic growth out­paced pop­u­la­tion growth, caus­ing a shift from a Malthu­sian regime where pro­duc­tivity growth always in­creased pop­u­la­tion at sub­sis­tence level, to a mod­ern regime where pro­duc­tivity growth in­creases GDP per cap­ita. The world re­mained on the same hy­per­bolic growth tra­jec­tory through­out, un­til the tra­jec­tory pe­tered out around 1900 in Bri­tain and around 1960 in the world as a whole.

IV.

So just how can­cel­led is the sin­gu­lar­ity?

To re­view: pop­u­la­tion growth in­creases tech­nolog­i­cal growth, which feeds back into the pop­u­la­tion growth rate in a cy­cle that reaches in­finity in finite time.

But since pop­u­la­tion can’t grow in­finitely fast, this pat­tern breaks off af­ter a while.

The In­dus­trial Revolu­tion tried hard to com­pen­sate for the “miss­ing” pop­u­la­tion; it in­vented ma­chines. Us­ing ma­chines, an in­di­vi­d­ual could do an in­creas­ing amount of work. We can imag­ine mak­ing eg trac­tors as an at­tempt to in­crease the effec­tive pop­u­la­tion faster than the hu­man uterus can man­age. It partly worked.

But the in­dus­trial growth mode had one ma­jor dis­ad­van­tage over the Malthu­sian mode: trac­tors can’t in­vent things. The pop­u­la­tion wasn’t just there to grow the pop­u­la­tion, it was there to in­crease the rate of tech­nolog­i­cal ad­vance and thus pop­u­la­tion growth. When we shifted (in part) from mak­ing peo­ple to mak­ing trac­tors, that pro­cess broke down, and growth (in peo­ple and trac­tors) be­came sub-hy­per­bolic.

If the pop­u­la­tion stays the same (and by “the same”, I just mean “not grow­ing hy­per­bol­i­cally”) we should ex­pect the growth rate to stay the same too, in­stead of in­creas­ing the way it did for thou­sands of years of in­creas­ing pop­u­la­tion, mod­ulo other con­cerns.

In other words, the sin­gu­lar­ity got can­cel­led be­cause we no longer have a sure­fire way to con­vert money into re­searchers. The old way was more money = more food = more pop­u­la­tion = more re­searchers. The new way is just more money = send more peo­ple to col­lege, and screw all that.

But AI po­ten­tially offers a way to con­vert money into re­searchers. Money = build more AIs = more re­search.

If this were true, then once AI comes around – even if it isn’t much smarter than hu­mans – then as long as the com­pu­ta­tional power you can in­vest into re­search­ing a given field in­creases with the amount of money you have, hy­per­bolic growth is back on. Faster growth rates means more money means more AIs re­search­ing new tech­nol­ogy means even faster growth rates, and so on to in­finity.

Pre­sum­ably you would even­tu­ally hit some other bot­tle­neck, but things could get very strange be­fore that hap­pens.