Beautiful Math

Con­sider the se­quence {1, 4, 9, 16, 25, …} You rec­og­nize these as the square num­bers, the se­quence Ak = k2. Sup­pose you did not rec­og­nize this se­quence at a first glance. Is there any way you could pre­dict the next item in the se­quence? Yes: You could take the first differ­ences, and end up with:

{4 − 1, 9 − 4, 16 − 9, 25 − 16, …} = {3, 5, 7, 9, …}

And if you don’t rec­og­nize these as suc­ces­sive odd num­bers, you are still not defeated; if you pro­duce a table of the sec­ond differ­ences, you will find:

{5 − 3, 7 − 5, 9 − 7, …} = {2, 2, 2, …}

If you can­not rec­og­nize this as the num­ber 2 re­peat­ing, then you’re hope­less.

But if you pre­dict that the next sec­ond differ­ence is also 2, then you can see the next first differ­ence must be 11, and the next item in the origi­nal se­quence must be 36 - which, you soon find out, is cor­rect.

Dig down far enough, and you dis­cover hid­den or­der, un­der­ly­ing struc­ture, sta­ble re­la­tions be­neath chang­ing sur­faces.

The origi­nal se­quence was gen­er­ated by squar­ing suc­ces­sive num­bers—yet we pre­dicted it us­ing what seems like a wholly differ­ent method, one that we could in prin­ci­ple use with­out ever re­al­iz­ing we were gen­er­at­ing the squares. Can you prove the two meth­ods are always equiv­a­lent? - for thus far we have not proven this, but only ven­tured an in­duc­tion. Can you sim­plify the proof so that you can you see it at a glance? - as Polya was fond of ask­ing.

This is a very sim­ple ex­am­ple by mod­ern stan­dards, but it is a very sim­ple ex­am­ple of the sort of thing that math­e­mat­i­ci­ans spend their whole lives look­ing for.

The joy of math­e­mat­ics is in­vent­ing math­e­mat­i­cal ob­jects, and then notic­ing that the math­e­mat­i­cal ob­jects that you just cre­ated have all sorts of won­der­ful prop­er­ties that you never in­ten­tion­ally built into them. It is like build­ing a toaster and then re­al­iz­ing that your in­ven­tion also, for some un­ex­plained rea­son, acts as a rocket jet­pack and MP3 player.

Num­bers, ac­cord­ing to our best guess at his­tory, have been in­vented and rein­vented over the course of time. (Ap­par­ently some ar­ti­facts from 30,000 BC have marks cut that look sus­pi­ciously like tally marks.) But I doubt that a sin­gle one of the hu­man be­ings who in­vented count­ing vi­su­al­ized the em­ploy­ment they would provide to gen­er­a­tions of math­e­mat­i­ci­ans. Or the ex­cite­ment that would some­day sur­round Fer­mat’s Last The­o­rem, or the fac­tor­ing prob­lem in RSA cryp­tog­ra­phy… and yet these are as im­plicit in the defi­ni­tion of the nat­u­ral num­bers, as are the first and sec­ond differ­ence ta­bles im­plicit in the se­quence of squares.

This is what cre­ates the im­pres­sion of a math­e­mat­i­cal uni­verse that is “out there” in Pla­to­nia, a uni­verse which hu­mans are ex­plor­ing rather than cre­at­ing. Our defi­ni­tions tele­port us to var­i­ous lo­ca­tions in Pla­to­nia, but we don’t cre­ate the sur­round­ing en­vi­ron­ment. It seems this way, at least, be­cause we don’t re­mem­ber cre­at­ing all the won­der­ful things we find. The in­ven­tors of the nat­u­ral num­bers tele­ported to Count­ingland, but did not cre­ate it, and later math­e­mat­i­ci­ans spent cen­turies ex­plor­ing Count­ingland and dis­cov­er­ing all sorts of things no one in 30,000 BC could be­gin to imag­ine.

To say that hu­man be­ings “in­vented num­bers”—or in­vented the struc­ture im­plicit in num­bers—seems like claiming that Neil Arm­strong hand-crafted the Moon. The uni­verse ex­isted be­fore there were any sen­tient be­ings to ob­serve it, which im­plies that physics pre­ceded physi­cists. This is a puz­zle, I know; but if you claim the physi­cists came first, it is even more con­fus­ing be­cause in­stan­ti­at­ing a physi­cist takes quite a lot of physics. Physics in­volves math, so math—or at least that por­tion of math which is con­tained in physics—must have pre­ceded math­e­mat­i­ci­ans. Other­wise, there would have no struc­tured uni­verse run­ning long enough for in­nu­mer­ate or­ganisms to evolve for the billions of years re­quired to pro­duce math­e­mat­i­ci­ans.

The amaz­ing thing is that math is a game with­out a de­signer, and yet it is em­i­nently playable.

Oh, and to prove that the pat­tern in the differ­ence ta­bles always holds:

(k + 1)2 = k2 + (2k + 1)

As for see­ing it at a glance:

Squares

Think the square prob­lem is too triv­ial to be worth your at­ten­tion? Think there’s noth­ing amaz­ing about the ta­bles of first and sec­ond differ­ences? Think it’s so ob­vi­ously im­plicit in the squares as to not count as a sep­a­rate dis­cov­ery? Then con­sider the cubes:

1, 8, 27, 64...

Now, with­out calcu­lat­ing it di­rectly, and with­out do­ing any alge­bra, can you see at a glance what the cubes’ third differ­ences must be?

And of course, when you know what the cubes’ third differ­ence is, you will re­al­ize that it could not pos­si­bly have been any­thing else...