Building off of this point, consider the polynomial x^4 + 2x^2 + 2. Over the rationals Q, this is an irreducible polynomial. There is no way to distinguish the roots from each other. There is also no way to distinguish any pair of roots from any other pair.

But over the reals R, this polynomial is not irreducible. There we find that some pairs of roots have the same real value, and others don't. This leads to the idea of a "complex conjugate pair". And so some pairs of roots of the original polynomial are now different than other pairs.

That notion of a "complex conjugate pair of roots" is therefore not a purely algebraic concept. If you're trying to understand Galois theory, you have to forget about it. Because it will trip up your intuition and mislead you. But in other contexts that is a very meaningful and important idea.

And so we find that we don't just care about what concepts could be understood. We also care about what concepts we're currently choosing to ignore!

Exactly.

That is why the "forgetful functor" seems at first sight stupid and when you think a bit, it is genius.