In mathematical logic class, I've learned that the only isomorphism on $\mathbb R$ is only the identity since isomorphism must preserve the order relation of reals.
But in my abstract algebra textbook, problem says:
Prove that there exists an isomorphism of fields $f: \mathbb R\rightarrow \mathbb R$ that maps $\pi$ to $-\pi$.
Does it make sense?
If your text phrases the problem in that way, then it is indeed incorrect: since the ordering on $\mathbb{R}$ is definable from the field structure ($a\le b\iff\exists c(a+c^2=b)$), any field automorphism must preserve the ordering, and from this (as you well know) it's not hard to show that there are no nontrivial field automorphisms at all.