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.

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.