# 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.

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• One thing that isn’t made, aha, ex­plicit in this ex­cel­lent ar­ti­cle, and that maybe should be, is that con­sid­er­ing ruler-and-com­passes con­struc­tions to be ex­plicit turns out to be ex­actly the same thing as con­sid­er­ing tak­ing square roots to be ex­plicit be­cause (once you have the idea of co­or­di­nate ge­om­e­try, which for some rea­son no one thought of un­til Descartes) it pretty eas­ily tran­spires that the nu­mer­i­cal op­er­a­tions you can do on co­or­di­nates us­ing ruler-and-com­passes con­struc­tions are ex­actly ar­ith­metic plus square roots. (Hand­wav­ily, the square roots come from the fact that the equa­tion of a cir­cle in­volves squar­ing the co­or­di­nates.)

And then (this uses, in some sense, “half” of Galois the­ory, so it’s eas­ier than the in­sol­u­bil­ity of the quin­tic) you can use this to prove that in­deed you can’t “du­pli­cate the cube” us­ing ruler and com­passes (it means tak­ing cube roots, and it turns out that no quan­tity of ar­ith­metic-plus-square-roots will let you do that), and that an­other fa­mous an­cient prob­lem—tri­sect­ing an­gles—can’t be done with ruler and com­passes for es­sen­tially the ex­act same rea­son, and (this is a bit more difficult; it was one of Gauss’s first im­por­tant the­o­rems) that the reg­u­lar poly­gons you can con­struct with ruler and com­passes are those whose num­ber of sides is a power of 2 times some num­ber of dis­tinct primes of form 2^2^n+1. So you can do 3 and 5 and 17, but not 9 (two 3s: not al­lowed) and not 11 (wrong sort of prime).

Of course, for all the rea­sons daoza­ich gives here, one shouldn’t care too much about what can be done with ruler and com­passes, as such. But these are still re­ally cool the­o­rems.

• I’m cu­rat­ing this be­cause I think it makes a valid and sub­tle math­e­mat­i­cal point, of the sort that has di­rect rele­vance to think­ing about many other top­ics.

• I re­cently dis­cov­ered there’s no closed-form for­mula for the cir­cum­fer­ence of an el­lipse, and then was also told “But that just means there’s no nice for­mula in what we con­sider ba­sic func­tions.” Do you have a sense /​ opinion on how ex­plicit /​ ar­bi­trary such el­lip­tic for­mu­lae in fact are?

• It de­pends on con­text. Is the ex­po­nen­tial ex­plicit? For the last 200 years, the an­swer is “hell yeah”. Ex­po­nen­tial, log­a­r­ithm and tri­gonom­e­try (com­plex ex­po­nen­tial) ap­pear very of­ten in life, and peo­ple can be ex­pected to have a work­ing knowl­edge of how to ma­nipu­late them. Ex­press­ing a solu­tion in terms of ex­po­nen­tials is like meet­ing an old friend.

120 years ago, know­ing el­lip­tic in­te­grals, their the­ory and how to ma­nipu­late them was con­sid­ered ba­sic knowl­edge that ev­ery work­ing math­e­mat­i­cian or en­g­ineer was ex­pected to have. Back then, these were ex­plicit /​ ba­sic /​ closed form.

If you are writ­ing a com­puter alge­bra sys­tem of similar am­bi­tion to maple /​ math­e­mat­ica /​ wolfram alpha, then you bet­ter con­sider them ex­plicit in your in­ter­nal sim­plifi­ca­tion rou­tines, and write code for ma­nipu­lat­ing them. Other­wise, users will com­plain and send you fea­ture re­quests. If you work as ed­i­tor at the “Bron­stein math­e­mat­i­cal hand­book”, then the an­swer is yes for the longer ver­sions of the book, and a very hard judge­ment call for shorter edi­tions.

To­day, el­lip­tic in­te­grals are not rou­tinely taught any­more. It is a tiny minor­ity of math­e­mat­i­ci­ans that has work­ing knowl­edge on these guys. Ex­press­ing a solu­tion in terms of el­lip­tic in­te­grals is not like meet­ing an old friend, it is like meet­ing a stranger who was fa­mous a cen­tury ago, a grainy photo of whom you might have once seen in an old book.

I per­son­ally would not con­sider the cir­cum­fer­ence of an el­lipse “closed form”. Just call it the “cir­cum­fer­ence of the el­lip­sis”, or write it as an in­te­gral, de­pend­ing on how to bet­ter make ap­par­ent which prop­er­ties you want.

Of course this is a trade-off, how much time to spend de­vel­op­ing an in­tu­ition and work­ing knowl­edge of “gen­eral in­te­grals” (likely from a func­tional anal­y­sis per­spec­tive, as an op­er­a­tor) and how much time to spend un­der­stand­ing spe­cific spe­cial in­te­grals. The spe­cific will always be more effec­tive and im­part deeper knowl­edge when deal­ing with the speci­fics, but the gen­eral the­ory is more ap­pli­ca­ble and “ge­o­met­ric”; you might say that it ex­trap­o­lates very well from the train­ing set. Some spe­cific spe­cial func­tions are worth it, eg exp/​log, and some used to be con­sid­ered wor­thy but are to­day not con­sid­ered wor­thy, ev­i­denced by re­vealed prefer­ence (what do peo­ple put into course syl­labi).

So, in some sense you have a large ed­ifice of “for­got­ten knowl­edge” in math­e­mat­ics. This knowl­edge is archived, of course, but the un­bro­ken mas­ter-ap­pren­tice chains of trans­mis­sion have mostly died out. I think this is sad; we, as a so­ciety, should be rich enough to spon­sor a hand­ful of peo­ple to keep this al­ive, even if I’d say “good rid­dance” for re­mov­ing it from the “stan­dard canon”.

Anec­dote: Ellip­tic in­te­grals some­times ap­pear in av­er­ag­ing: You have a differ­en­tial equa­tion (dy­nam­i­cal sys­tem) and want to av­er­age over fast os­cilla­tions in or­der to get an effec­tive (ie lead­ing or­der /​ ap­prox­i­mate) sys­tem with re­duced di­men­sion and uniform time-scales. Now, what is your effec­tive equa­tion? You can ex­press it as “the effec­tive equa­tion com­ing out of The­o­rem XYZ”, or write it down as an in­te­gral, which makes ap­par­ent both the pro­ce­dure en­coded in The­o­rem XYZ and an in­te­gral ex­pres­sion that is helpful for in­tu­ition and calcu­la­tions. And some­times, if you type it into Wolfram alpha, it trans­forms into some ex­tremely lenghty ex­pres­sion con­tain­ing el­lip­tic in­te­grals. Do you gain un­der­stand­ing from this? I cer­tainly don’t, and de­cided not to use the ex­plicit ex­pres­sions when I met them in my re­search (99% of the time, math­e­mat­ica is not helpful; the 1% pays for the triv­ial in­con­ve­nience of always try­ing whether there maybe is some amaz­ing trans­for­ma­tion that sim­plifies your prob­lem).

• Hav­ing run up against broadly similar kinds of prob­lems re­cently, I have con­cluded that in­tu­itions are the right level for me to fo­cus on.

I also greatly prize his­tor­i­cal con­text for math-re­lated ques­tions now. I feel like this is the not-well-ex­plained mo­ti­va­tion for re­turn­ing to origi­na­tors of and key his­tor­i­cal con­trib­u­tors to a field, even if effi­cient text­books have since been writ­ten.

• 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?

Hav­ing spent some time look­ing at nu­mer­i­cal solu­tions, I would gen­er­al­ize this to “do we have an al­gorithm that al­lows us to effi­ciently ex­plore the rele­vant do­main space?”

The al­gorithm could be a way to calcu­late sin(x), to square a num­ber, to solve the Laplace equa­tion with given bound­ary con­di­tions, or even to calcu­late the dis­tri­bu­tion of grav­i­ta­tional ra­di­a­tion from black hole col­li­sions. As long as it as effi­cient (in terms of com­pu­ta­tional and hu­man efforts) as press­ing sin(x) on the calcu­la­tor, it is as good as ex­act.

• While the con­cept of ex­plicit solu­tion can be in­ter­preted mess­ily, as in the quote above, there is a ver­sion of this idea that more closely cuts re­al­ity at the joints, com­putabil­ity. A real num­ber is com­putable iff there is a Tur­ing ma­chine that out­puts the num­ber to any de­sired ac­cu­racy. This cov­ers frac­tions, roots, im­plicit solu­tions, in­te­grals, and, if you be­lieve the Church-Tur­ing the­sis, any­thing else we will be able to come up with. https://​​en.wikipe­dia.org/​​wiki/​​Com­putable_number

• Com­putabil­ity does not ex­press the same thing we mean with “ex­plicit”. The vague term “ex­plicit” crys­tal­lizes an im­por­tant con­cept, which is de­pen­dent on so­cial and his­tor­i­cal con­text that I tried to elu­ci­date. It is use­ful to give a name to this con­cept, but you can­not re­ally prove the­o­rems about it (there should be no tech­ni­cal defi­ni­tion of “ex­plicit”).

That be­ing said, com­putabil­ity is of course im­por­tant, but slightly too counter-in­tu­itive in prac­tice. Say, you have two polyno­mial vec­torfields. Are solu­tions (to the differ­en­tial equa­tion) com­putable? Sure. Can you say whether the two solu­tions, at time t=1 and start­ing in the ori­gin, co­in­cide? I think not. Equal­ity of com­putable re­als is not de­cid­able af­ter all (liter­ally the halt­ing prob­lem).

• re: differ­en­tial equa­tion solu­tions, you can com­pute if they are within ep­silon of each other for any ep­silon, which I feel is “morally the same” as know­ing if they are equal.

It’s true that the con­cepts are not iden­ti­cal. I feel com­putabil­ity is like the “limit” of the “ex­plicit” con­cept, as a com­mu­nity of math­e­mat­i­ci­ans comes to ac­cept more and more ways of for­mally spec­i­fy­ing a num­ber. The cor­re­spon­dence is still not perfect, as differ­ent fam­i­lies of ex­plicit for­mu­lae will have struc­ture(e.g. alge­braic struc­ture) that gen­eral Tur­ing ma­chines will not.

• Polyno­mial-time com­putabil­ity is prob­a­bly closer to the no­tion of ex­plic­it­ness (though still not quite the same, as daoza­ich points out). I don’t know of any num­ber that is con­sid­ered ex­plicit but is not polyno­mial-time com­putable.

• changed my mind edit