On exact mathematical formulae

This is in­spired by the re­view on “Lin­ear Alge­bra done right”. I de­cided to do a top-level post, be­cause it hits a mis­con­cep­tion that is pretty com­mon.

The start­ing point of this post is this quote from “Lin­ear Alge­bra done right”:

Re­mark­ably, math­e­mat­i­ci­ans have proved that no for­mula ex­ists for the ze­ros of polyno­mi­als of de­gree 5 or higher. But com­put­ers and calcu­la­tors can use clever nu­mer­i­cal meth­ods to find good ap­prox­i­ma­tions to the ze­ros of any polyno­mial, even when ex­act ze­ros can­not be found.
For ex­am­ple, no one will ever be able to give an ex­act for­mula for a zero of the polyno­mial p defined by .

The au­thors mis­rep­re­sent an im­por­tant point that is un­der­stood by most math­e­mat­i­ci­ans, but not prop­erly un­der­stood by many laypeo­ple.

What does it mean to solve a prob­lem? What does it mean to have an ex­act for­mula for the solu­tion of a prob­lem?

The an­swers to both are a so­cial con­ven­tion that has his­tor­i­cally changed and is ex­pected to con­tinue to evolve in the fu­ture.

Back in the days, peo­ple only con­sid­ered ra­tio­nal num­bers, ie frac­tions. Oh, but what about the pos­i­tive solu­tion to ? Ok, we can’t ex­press this as a ra­tio­nal num­ber (im­por­tant the­o­rem). Be­cause these kinds of prob­lems oc­cured quite of­ten, the math­e­mat­i­cal com­mu­nity ar­rived at the con­sen­sus that , or more gen­er­ally for non­nega­tive should be con­sid­ered an ex­plicit solu­tion. Amaz­ingly, this al­lows us to ex­press the solu­tion to any quadratic equa­tion ex­plic­itly, with our ex­panded no­tion of “ex­plicit”. From an alge­braic view­point it was nat­u­ral to bless the pos­i­tive solu­tion to as an “ex­plicit for­mula” next; his­tor­i­cally it was a more con­tentious thing, be­cause greek ge­om­e­try wanted num­bers to be con­structible us­ing a ruler and com­pass only. “Dou­bling the cube”, ie ex­press­ing the pos­i­tive solu­tion to as a ge­o­met­ric con­struc­tion was a fa­mous old prob­lem (proven im­pos­si­ble in 1837, af­ter hav­ing been a very promi­nent math­e­mat­i­cal re­search prob­lems for more than 2000 years).

Now, this ob­vi­ously says not a lot about the cube root of 2, but says a lot about “con­structible with ruler and com­pass”.

In other words: “Ex­plicit solu­tions” are a messy his­tor­i­cal map to math­e­mat­i­cal ter­ri­tory, noth­ing more.

The same holds if you ask for ex­plicit for­mu­las for ze­ros of polyno­mi­als af­ter hav­ing grudg­ingly ad­mit­ted nth roots as “ex­plicit”. The same holds if you ask about ex­plicit in­te­grals of ex­plicit func­tions (also af­ter hav­ing grudg­ingly ad­mit­ted eg el­lip­tic in­te­grals as “ex­plicit”). The same holds for solu­tions of differ­en­tial equa­tions.

In math­e­mat­ics, ask­ing about an “ex­plicit for­mula” for solu­tions to prob­lems means just: As­sum­ing a gen­eral back­ground in math­e­mat­ics, is the solu­tion some­thing I already have spent years of my life de­vel­op­ing an in­tu­ition for?

And if the an­swer hap­pens to be “yes, un­con­di­tion­ally”, then it is worth­while.

If the “ex­plicit” for­mula uses things that are not com­monly taught any­more (crazy “spe­cial func­tions” that 100 years ago con­sti­tuted a perfectly fine ex­plicit solu­tion), or is too lenghty/​com­pli­cated to in­form in­tu­itions, then it is func­tion­ally equiv­a­lent to “we don’t know”, which is func­tion­ally equiv­a­lent to “we can prove that no for­mula us­ing terms of type xyz ex­ists”.

So there is noth­ing sur­pris­ing or scary about prob­lems not hav­ing an “ex­plicit” solu­tion.

The true value of Galois the­ory is that it prop­erly elu­ci­dates the hid­den struc­ture of polyno­mial equa­tions, not that it tells us that no “ex­plicit solu­tion for­mula” ex­ists for de­gree 5 polyno­mi­als for this very his­tor­i­cal no­tion of “ex­plicit”. The “ex­plicit” de­gree 4 for­mula is noth­ing more than a cu­ri­os­ity with in­ter­est­ing his­tory, but ab­solutely worth­less from both an in­tu­itive and nu­mer­i­cal stand­point.

I most of­ten en­coun­tered the un­jus­tified bias to­wards “ex­plicit solu­tions” for im­plicit func­tions (the func­tion is defined by for some fixed , im­plicit func­tion the­o­rem + new­ton solver) and solu­tions to differ­en­tial equa­tions. In­te­grals are mostly con­sid­ered “ex­plicit” to­day.