# Expecting Beauty

Fol­lowup to: Beau­tiful Math

If you looked at the se­quence {1, 4, 9, 16, 25, …} and didn’t rec­og­nize the square num­bers, you might still es­say an ex­cel­lent-seem­ing pre­dic­tion of fu­ture items in the se­quence by notic­ing that the table of first differ­ences is {3, 5, 7, 9, …}. In­deed, your pre­dic­tion would be perfect, though you have no way of know­ing this with­out peek­ing at the gen­er­a­tor. The cor­re­spon­dence can be shown alge­braically or even ge­o­met­ri­cally (see yes­ter­day’s post). It’s re­ally rather el­e­gant.

What­ever peo­ple praise, they tend to praise too much; and there are skep­tics who think that the pur­suit of el­e­gance is like unto a dis­ease, which pro­duces neat math­e­mat­ics in op­po­si­tion to the messi­ness of the real world. “You got lucky,” they say, “and you won’t always be lucky. If you ex­pect that kind of el­e­gance, you’ll dis­tort the world to match your ex­pec­ta­tions—chop off all the parts of Life that don’t fit into your nice lit­tle pat­tern.”

I mean, sup­pose Life hands you the se­quence {1, 8, 27, 64, 125, …}. When you take the first differ­ences, you get {7, 19, 37, 61, …}. All these num­bers have in com­mon is that they’re primes, and they aren’t even se­quen­tial primes. Clearly, there isn’t the neat or­der here that we saw in the squares.

You might try to im­pose or­der, by in­sist­ing that the first differ­ences must be evenly spaced, and any de­vi­a­tions are ex­per­i­men­tal er­rors—or bet­ter yet, we just won’t think about them. “You will say,” says the skep­tic, “that ‘The first differ­ences are spaced around 20 apart and land on prime num­bers, so that the next differ­ence is prob­a­bly 83, which makes the next num­ber 208.’ But re­al­ity comes back and says 216.”

Serves you right, ex­pect­ing neat­ness and el­e­gance when there isn’t any there. You were too ad­dicted to ab­solutes; you had too much need for clo­sure. Be­hold the per­ils of—gasp! - DUN DUN DUN—re­duc­tion­ism.

You can guess, from the ex­am­ple I chose, that I don’t think this is the best way to look at the prob­lem. Be­cause, in the ex­am­ple I chose, it’s not that no or­der ex­ists, but that you have to look a lit­tle deeper to find it. The se­quence {7, 19, 37, 61, …} doesn’t leap out at you—you might not rec­og­nize it, if you met it on the street—but take the sec­ond differ­ences and you find {12, 18, 24, …}. Take the third differ­ences and you find {6, 6, …}.

You had to dig deeper to find the sta­ble level, but it was still there—in the ex­am­ple I chose.

Some­one who grasped too quickly at or­der, who de­manded clo­sure right now, who forced the pat­tern, might never find the sta­ble level. If you tweak the table of first differ­ences to make them “more even”, fit your own con­cep­tion of aes­thet­ics be­fore you found the math’s own rhythm, then the sec­ond differ­ences and third differ­ences will come out wrong. Maybe you won’t even bother to take the sec­ond differ­ences and third differ­ences. Since, once you’ve forced the first differ­ences to con­form to your own sense of aes­thet­ics, you’ll be happy—or you’ll in­sist in a loud voice that you’re happy.

None of this says a word against—gasp! - re­duc­tion­ism. The or­der is there, it’s just bet­ter-hid­den. Is the moral of the tale (as I told it) to for­sake the search for beauty? Is the moral to take pride in the glo­ri­ous cos­mopoli­tan so­phis­ti­ca­tion of con­fess­ing, “It is ugly”? No, the moral is to re­duce at the right time, to wait for an open­ing be­fore you slice, to not pre­ma­turely ter­mi­nate the search for beauty. So long as you can re­fuse to see beauty that isn’t there, you have already taken the need­ful pre­cau­tion if it all turns out ugly.

But doesn’t it take—gasp! - faith to search for a beauty you haven’t found yet?

As I re­cently re­marked, if you say, “Many times I have wit­nessed the turn­ing of the sea­sons, and to­mor­row I ex­pect the Sun will rise in its ap­pointed place in the east,” then that is not a cer­tainty. And if you say, “I ex­pect a pur­ple polka-dot fairy to come out of my nose and give me a bag of money,” that is not a cer­tainty. But they are not the same shade of un­cer­tainty, and it seems in­suffi­ciently nar­row to call them both “faith”.

Look­ing for math­e­mat­i­cal beauty you haven’t found yet, is not so sure as ex­pect­ing the Sun to rise in the east. But nei­ther does it seem like the same shade of un­cer­tainty as ex­pect­ing a pur­ple polka-dot fairy—not af­ter you pon­der the last fifty-seven thou­sand cases where hu­man­ity found hid­den or­der.

And yet in math­e­mat­ics the premises and ax­ioms are closed sys­tems—can we ex­pect the messy real world to re­veal hid­den beauty? Tune in next time on Over­com­ing Bias to find out!

• Do­minic: How many bits does it take you to com­mu­ni­cated the for­mula you dis­cussed? How many for just the last term of that pat­tern?

• When you wrote “But nei­ther does it seem like the same shade of un­cer­tainty” I sup­pose you mean that it doesn’t seem that way, to you. Nor does it to me. But be­fore, as a think­ing per­son, I sug­gest that the differ­ence is mean­ingful, I need a con­text or a rea­son. You haven’t pro­vided one, and that’s why your ar­gu­ment has the fla­vor of re­li­gion, to my palette.

I’d love to see your an­swer to the ac­tual skep­ti­cal ar­gu­ment, rather than the straw man you offer, here. Here you are do­ing the equiv­a­lent of an­nounc­ing “I’m think­ing of a num­ber!..… 5!...… I’m right again! My quest for or­der is re­warded!”

If you use math­e­mat­ics to find or­der in the messy world, and you suc­ceed, does that amount to a proof that the or­der you found is the ac­tual or­der? Ke­pler would have ar­gued yes! So would have New­ton. Both were wrong. We know they were wrong. Wrong but their ideas are en­dur­ingly use­ful, as far as we know… so far… The skep­ti­cal po­si­tion is not one of deny­ing the value of ideas, but rather that of con­tin­u­ing the in­quiry.

When my in­quiry ceases, my be­liefs be­come hard­ened premises that define my world and pre­vents me from benefit­ing from ideas of peo­ple with differ­ent premises. That’s fine in a sim­ple world. A gamer’s world. I’ve be­come con­vinced that there is no sim­ple world, ex­cept in our fan­tasies. Over­com­ing bias is about find­ing our cen­ter in a messy world. It’s about over­com­ing fan­tasy.

• Isn’t the last value in the se­quence defi­ni­tion­ally zero for the first five terms, re­duc­ing the en­tire first term to zero as well and leav­ing only the n^2?

Does this mean that if we gave you (1, 4, 9, 16, 25, 36...) you would claim the next value in the se­quence was 6! plus 49??

• Michael: do you think we should de­cide that the sim­plest for­mula is the best?

But then how do we define sim­ple? What do you mean by ‘com­mu­ni­cat­ing’ and ‘bits’? Do we as­sign ar­bi­trary com­plex­ity points to the op­er­a­tors? What would be the rel­a­tive com­plex­ity of a power op­er­a­tion as com­pared to a mul­ti­pli­ca­tion? And what of my pet op­er­a­tor I just in­vented that lets me re­place “(n − 1) (n − 2) (n − 3) (n − 4) (n − 5)” with “5##” or some­thing similarly silly?

Ask your­self, how can we be sure we have the sim­plest ex­pla­na­tion? What is the sim­plest for­mula for the se­quence {1, 2, …}? Is it the pow­ers of two or the nat­u­ral num­bers? What about the se­quence {1, …}? Is it re­ally sen­si­ble to ask such ques­tions?

• I think you could use Kol­go­morov com­plex­ity to define sim­ple, for these pur­poses. That way re­plac­ing your for­mula with “5##” wouldn’t make it any sim­pler, be­cause the ma­chine would still have to ex­e­cute all those mul­ti­plica­tive op­er­a­tions.

How can we be sure we have the sim­plest ex­pla­na­tion? We can’t be sure, be­cause new data could come in to make us change our minds. But given a finite amount like {1, 2, …} we can still weight pos­si­ble for­mu­lae by Ko­go­morov com­plex­ity and pre­fer the sim­pler hy­poth­e­sis.

I think nat­u­ral num­bers is sim­pler in this case, be­cause n is sim­pler to calcu­late than 2^n. As for {1, …} I don’t think we have enough in­for­ma­tion to lo­cate a hy­poth­e­sis.