Why is uniqueness a requirement for well-posed problems?

Question

I understand that, by definition, a well-posed problem must have a unique solution, but why is it important? For example, the polynomial $$x^4-x^2(2a-1)+a(a-1)$$ has real roots $x=\sqrt{a},-\sqrt{a},\sqrt{a-1},-\sqrt{a-1}$ for $a>1$, and there doesn't seem to be uniqueness, because there are four solutions. However, you can just say $$\boldsymbol{x}=(\sqrt{a},-\sqrt{a},\sqrt{a-1},-\sqrt{a-1})$$ and this solution is unique. By this argument, you can make any problem have a unique solution, what am I missing?

Answer 1

I dispute your contention that you have given a unique solution to that quartic polynomial. A solution is a real number (or complex, or whatever domain is appropriate to the context), but you have given a tuple of solutions. As a tuple it is not unique since it is sensitive to order, so maybe your intention was to declare a set of solutions. In any case this is a sloppiness in notation or terminology, and mathematics finds great value in being precise with both.

To say that every equation has a unique solution set is vacuously true, so conflating with with uniqueness of solutions defeats the purpose of using the word “unique”: it’s completely superfluous in the context of solution sets.

But enough criticism of the question. At its heart is a genuine inquiry: what benefit do we get from having uniqueness as a requirement? I can think of at least two that are fundamental to the way we communicate mathematics.

1. Once we have proven that a solution is unique, then we can unambiguously define something as the unique solution to that equation. E.g. we often define $e^x$ to be the unique function that solves $f’=f, f(0)=1$. This is especially useful in differential equations where we often lack an explicit expression for the solution. Well-posedness means never having to specify “which solution” beyond “the”. This is arguably the primary sense in which we mean “well-posed”: it posits a single, specific function by means of a defining property.
2. Once we have proven the uniqueness of an object satisfying a given property, we can then easily prove equality of another object by verifying that the property holds. This is especially common in algebraic contexts. It’s substantially less powerful when the solution is non-unique, though even then it may have value (e.g. many Diophantine results post-Wiles boil down to proving that a hypothetical object must be equal to one of a finite number of modular forms, and then narrowing down the candidates by some further computation).

Answer 2

In principle, maybe you could "just say" that, but if you're trying to set up a practical solution method (perhaps a numerical algorithm), it opens up considerable difficulties. Now you don't just have to find one solution, you have to find all solutions. Maybe there are infinitely many, so you'll never actually find all of them one by one. Or even if there are finitely many, you'll need a way to tell that you have found all of them.