Sorting Pebbles Into Correct Heaps

Once upon a time there was a strange lit­tle species—that might have been biolog­i­cal, or might have been syn­thetic, and per­haps were only a dream—whose pas­sion was sort­ing peb­bles into cor­rect heaps.

They couldn’t tell you why some heaps were cor­rect, and some in­cor­rect. But all of them agreed that the most im­por­tant thing in the world was to cre­ate cor­rect heaps, and scat­ter in­cor­rect ones.

Why the Peb­ble­sort­ing Peo­ple cared so much, is lost to this his­tory—maybe a Fish­e­rian run­away sex­ual se­lec­tion, started by sheer ac­ci­dent a mil­lion years ago? Or maybe a strange work of sen­tient art, cre­ated by more pow­er­ful minds and aban­doned?

But it mat­tered so dras­ti­cally to them, this sort­ing of peb­bles, that all the Peb­ble­sort­ing philoso­phers said in uni­son that peb­ble-heap-sort­ing was the very mean­ing of their lives: and held that the only jus­tified rea­son to eat was to sort peb­bles, the only jus­tified rea­son to mate was to sort peb­bles, the only jus­tified rea­son to par­ti­ci­pate in their world econ­omy was to effi­ciently sort peb­bles.

The Peb­ble­sort­ing Peo­ple all agreed on that, but they didn’t always agree on which heaps were cor­rect or in­cor­rect.

In the early days of Peb­ble­sort­ing civ­i­liza­tion, the heaps they made were mostly small, with counts like 23 or 29; they couldn’t tell if larger heaps were cor­rect or not. Three mil­len­nia ago, the Great Leader Biko made a heap of 91 peb­bles and pro­claimed it cor­rect, and his le­gions of ad­miring fol­low­ers made more heaps like­wise. But over a hand­ful of cen­turies, as the power of the Biko­ni­ans faded, an in­tu­ition be­gan to ac­cu­mu­late among the smartest and most ed­u­cated that a heap of 91 peb­bles was in­cor­rect. Un­til fi­nally they came to know what they had done: and they scat­tered all the heaps of 91 peb­bles. Not with­out flashes of re­gret, for some of those heaps were great works of art, but in­cor­rect. They even scat­tered Biko’s origi­nal heap, made of 91 pre­cious gem­stones each of a differ­ent type and color.

And no civ­i­liza­tion since has se­ri­ously doubted that a heap of 91 is in­cor­rect.

To­day, in these wiser times, the size of the heaps that Peb­ble­sorters dare at­tempt, has grown very much larger—which all agree would be a most great and ex­cel­lent thing, if only they could en­sure the heaps were re­ally cor­rect. Wars have been fought be­tween coun­tries that dis­agree on which heaps are cor­rect: the Peb­ble­sorters will never for­get the Great War of 1957, fought be­tween Y’ha-nth­lei and Y’not’ha-nth­lei, over heaps of size 1957. That war, which saw the first use of nu­clear weapons on the Peb­ble­sort­ing Planet, fi­nally ended when the Y’not’ha-nth­leian philoso­pher At’gra’len’ley ex­hibited a heap of 103 peb­bles and a heap of 19 peb­bles side-by-side. So per­sua­sive was this ar­gu­ment that even Y’not’ha-nth­lei re­luc­tantly con­ceded that it was best to stop build­ing heaps of 1957 peb­bles, at least for the time be­ing.

Since the Great War of 1957, coun­tries have been re­luc­tant to openly en­dorse or con­demn heaps of large size, since this leads so eas­ily to war. In­deed, some Peb­ble­sort­ing philoso­phers—who seem to take a tan­gible delight in shock­ing oth­ers with their cyn­i­cism—have en­tirely de­nied the ex­is­tence of peb­ble-sort­ing progress; they sug­gest that opinions about peb­bles have sim­ply been a ran­dom walk over time, with no co­her­ence to them, the illu­sion of progress cre­ated by con­demn­ing all dis­similar pasts as in­cor­rect. The philoso­phers point to the dis­agree­ment over peb­bles of large size, as proof that there is noth­ing that makes a heap of size 91 re­ally in­cor­rect—that it was sim­ply fash­ion­able to build such heaps at one point in time, and then at an­other point, fash­ion­able to con­demn them. “But… 13!” car­ries no truck with them; for to re­gard “13!” as a per­sua­sive coun­ter­ar­gu­ment, is only an­other con­ven­tion, they say. The Heap Rel­a­tivists claim that their philos­o­phy may help pre­vent fu­ture dis­asters like the Great War of 1957, but it is widely con­sid­ered to be a philos­o­phy of de­spair.

Now the ques­tion of what makes a heap cor­rect or in­cor­rect, has taken on new ur­gency; for the Peb­ble­sorters may shortly em­bark on the cre­ation of self-im­prov­ing Ar­tifi­cial In­tel­li­gences. The Heap Rel­a­tivists have warned against this pro­ject: They say that AIs, not be­ing of the species Peb­ble­sorter sapi­ens, may form their own cul­ture with en­tirely differ­ent ideas of which heaps are cor­rect or in­cor­rect. “They could de­cide that heaps of 8 peb­bles are cor­rect,” say the Heap Rel­a­tivists, “and while ul­ti­mately they’d be no righter or wronger than us, still, our civ­i­liza­tion says we shouldn’t build such heaps. It is not in our in­ter­est to cre­ate AI, un­less all the com­put­ers have bombs strapped to them, so that even if the AI thinks a heap of 8 peb­bles is cor­rect, we can force it to build heaps of 7 peb­bles in­stead. Other­wise, KABOOM!”

But this, to most Peb­ble­sorters, seems ab­surd. Surely a suffi­ciently pow­er­ful AI—es­pe­cially the “su­per­in­tel­li­gence” some transpeb­ble­sor­ter­ists go on about—would be able to see at a glance which heaps were cor­rect or in­cor­rect! The thought of some­thing with a brain the size of a planet, think­ing that a heap of 8 peb­bles was cor­rect, is just too ab­surd to be worth talk­ing about.

In­deed, it is an ut­terly fu­tile pro­ject to con­strain how a su­per­in­tel­li­gence sorts peb­bles into heaps. Sup­pose that Great Leader Biko had been able, in his prim­i­tive era, to con­struct a self-im­prov­ing AI; and he had built it as an ex­pected util­ity max­i­mizer whose util­ity func­tion told it to cre­ate as many heaps as pos­si­ble of size 91. Surely, when this AI im­proved it­self far enough, and be­came smart enough, then it would see at a glance that this util­ity func­tion was in­cor­rect; and, hav­ing the abil­ity to mod­ify its own source code, it would rewrite its util­ity func­tion to value more rea­son­able heap sizes, like 101 or 103.

And cer­tainly not heaps of size 8. That would just be stupid. Any mind that stupid is too dumb to be a threat.

Re­as­sured by such com­mon sense, the Peb­ble­sorters pour full speed ahead on their pro­ject to throw to­gether lots of al­gorithms at ran­dom on big com­put­ers un­til some kind of in­tel­li­gence emerges. The whole his­tory of civ­i­liza­tion has shown that richer, smarter, bet­ter ed­u­cated civ­i­liza­tions are likely to agree about heaps that their an­ces­tors once dis­puted. Sure, there are then larger heaps to ar­gue about—but the fur­ther tech­nol­ogy has ad­vanced, the larger the heaps that have been agreed upon and con­structed.

In­deed, in­tel­li­gence it­self has always cor­re­lated with mak­ing cor­rect heaps—the near­est evolu­tion­ary cous­ins to the Peb­ble­sorters, the Peb­panzees, make heaps of only size 2 or 3, and oc­ca­sion­ally stupid heaps like 9. And other, even less in­tel­li­gent crea­tures, like fish, make no heaps at all.

Smarter minds equal smarter heaps. Why would that trend break?