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!