Exploring the Idea Space Efficiently

Si­mon is writ­ing a calcu­lus text­book. Since there are a lot of text­books on the mar­ket, he wants to make his dis­tinc­tive by in­clud­ing a lot of origi­nal ex­am­ples. To do this, he de­cides to first check what sorts of ex­am­ples are in some of the other books, and then make sure to avoid those. Un­for­tu­nately, af­ter skim­ming through sev­eral other books, he finds him­self com­pletely un­able to think of origi­nal ex­am­ples—his mind keeps re­turn­ing to the ex­am­ples he’s just read in­stead of com­ing up with new ones.

What he’s ex­pe­rienc­ing here is an­other as­pect of prim­ing or an­chor­ing. The way it ap­pears to hap­pen in my brain is that it de­cides to an­chor on the ex­am­ples it’s already seen and ex­plore the idea-space from there, mov­ing from an idea only to ideas that are closely re­lated to it (similarly to a depth-first search)

At first, this search strat­egy might not seem so bad—in fact, it’s ideal if there is one best solu­tion and the closer you get to it the bet­ter. For ex­am­ple, if you were shoot­ing ar­rows at a tar­get, all you’d need to con­sider is how close to the cen­ter you can hit. Where we run into prob­lems, how­ever, is try­ing to come up with mul­ti­ple solu­tions (such as mul­ti­ple ex­am­ples of the ap­pli­ca­tions of calcu­lus), or try­ing to come up with the best solu­tion when there are many plau­si­ble solu­tions. In these cases, our brain’s de­fault search al­gorithm will of­ten grab the first idea it can think of and try to re­fine it, even if what we re­ally need is a com­pletely differ­ent idea.

Of course, the brain is not so stupid that it will choose the first idea it has and re­fine it for­ever. Even in the an­ces­tral en­vi­ron­ment, we would have run into prob­lems where depth-first search is not very effec­tive. Rather than spend­ing time on re­fin­ing your skills at chip­ping flint to make tools, for ex­am­ple, you may have been bet­ter served by learn­ing to pick a bet­ter type of flint to work with be­fore even start­ing. In mod­ern times, how­ever, these prob­lems have grown more challeng­ing and more nu­mer­ous. I’m sure all of us have had the ex­pe­rience of work­ing on some prob­lem for a long time, re­fin­ing our solu­tion, maybe even try­ing to make the prob­lem fit the solu­tion we came up with out of frus­tra­tion, only to give up, come back later, and then sud­denly have a com­pletely differ­ent and ob­vi­ous solu­tion come to mind. While part of this is prob­a­bly due to some periph­eral pro­cesses in our brain analysing the prob­lem while our con­scious thoughts were not fo­cused on it, I think the key com­po­nent is that by leav­ing the prob­lem alone we “for­got” our first solu­tion and were free to look for a bet­ter one.

Similarly, it is pos­si­ble that if Si­mon stops try­ing to come up with ex­am­ples when­ever he re­mem­bers the ex­am­ples he’s seen be­fore, and only re­turns to the task when his mind is rel­a­tively blank, he might be able to pro­duce some­thing origi­nal. Even if he does that, un­for­tu­nately, he might still find him­self com­ing up with only one ex­am­ple at a time, and then be­ing stuck think­ing only of ex­am­ples that are some­how similar to it. Either way, hav­ing to pause for sev­eral hours ev­ery time he finds him­self primed to think of some­thing is far from ideal, and there are bet­ter solu­tions.

The fun­da­men­tal thing you should do when ap­proach­ing a difficult, but tractable1, prob­lem is to avoid propos­ing a solu­tion im­me­di­ately. The mo­ment you pro­pose a solu­tion, your brain will be primed to to try to re­fine it or to look for similar solu­tions, even when it might be much more effi­cient to fur­ther analyse the prob­lem or to look for other, rad­i­cally differ­ent, solu­tions. Even when you have thor­oughly un­der­stood the prob­lem, how­ever, you should still wait be­fore propos­ing a solu­tion, un­less the prob­lem is fairly easy and one solu­tion is all you need.

What I would recom­mend that you do be­fore that is come up with a map of the idea-space, de­scribing where the pos­si­ble solu­tions might be found. For in­stance, be­fore look­ing at a sin­gle ex­am­ple of calcu­lus, Si­mon might have writ­ten down a list of idea ar­eas to ex­plore: “jobs, per­sonal life, the nat­u­ral world, en­g­ineer­ing, other”. He would then take the first broad cat­e­gory, “jobs” and ex­pand it into a longer list, per­haps “agri­cul­ture, teach­ing, cus­tomer ser­vice, man­u­fac­tur­ing, re­search, IT, other”. With this longer list, he can then fo­cus on each area in turn and ei­ther ex­pand it fur­ther if it seems es­pe­cially rife with ex­am­ples, or come up with an ex­am­ple from the area di­rectly. Once the area is de­pleted, which he might de­cide is the case if it takes him longer than one minute to come up with an ex­am­ple, he would move on to the next area.

There are two main ad­van­tages to this ap­proach. The ex­am­ples Si­mon finds should be fairly rep­re­sen­ta­tive of all the ex­am­ples he can think of, since he started with a map of all such ex­am­ples, and, bet­ter yet, he should be able to find ex­am­ples much faster be­cause he knows to stop look­ing in one small area when it be­comes de­pleted.

The same ap­proach is also use­ful when you’re try­ing to come up with the sin­gle best solu­tion. For ex­am­ple, if you’re try­ing to come up with a way to deal with cli­mate change, you might write down “re­duce car­bon emis­sions, en­g­ineer the cli­mate to be bet­ter, adapt to cli­mate change, other” and move on from ex­plor­ing one op­tion to the next when the op­tion runs into sig­nifi­cant difficul­ties. Note, how­ever, that in this case it is im­por­tant to ar­range your op­tions in or­der of how likely you think you are to find your best solu­tion within each of them to make sure you ex­plore the most solu­tion-rich ar­eas first.

In gen­eral, there are three main things2 to keep in mind when cre­at­ing a map of your idea-space:

  • Don’t get overly spe­cific with the ini­tial ar­eas, since you will re­fine them when you’re ex­pand­ing them.

  • Try to in­clude all the ar­eas that might con­tain a solu­tion and none that do not.

  • Try to pick ar­eas so each of them is equally likely to con­tain a solu­tion (or or­der the ar­eas by the num­ber of solu­tions and move on more quickly from solu­tion-poor ar­eas).

For in­stance, when Si­mon came up with his list of ar­eas for calcu­lus ex­am­ples, he cor­rectly did not in­clude “philos­o­phy” in the list, since it con­tains much fewer ex­am­ples than any of the other ar­eas.

In sum, when deal­ing with a challeng­ing prob­lem or com­ing with a lot of ex­am­ples: don’t jump to a solu­tion, in­stead care­fully con­sider the prob­lem, come up with a map of ar­eas of ideas where solu­tions might be found, and search the map un­til you get the solu­tion you’re look­ing for. If you use this ap­proach, solu­tions should come to you faster, be need­lessly com­plex less of­ten, and be a lot more cor­rect than the ones from your brain’s naïve search al­gorithm.

Footnotes

1: It’s im­por­tant to note that if the prob­lem is not tractable for you, in the sense that you can’t tell if you’re get­ting closer to a solu­tion or not, these recom­men­da­tions won’t do much. For ex­am­ple, if I asked you “What is the next num­ber in the se­quence 14, 15, 16, 17, 21, 23, 30, 33?” it will help only slightly to hold off on com­ing up with solu­tions, and your best bet might be to start do­ing a depth-first search (as long as you keep in mind that you should not look for overly com­pli­cated solu­tions).

2: This is some­what similar to Vladimir_Nesov’s post recom­mend­ing that we con­sider rep­re­sen­ta­tive data sets (in par­tic­u­lar the three mis­takes he lists, which are well worth read­ing).