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.

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).