If mathematicians behaved simply as you describe, then those resultant extension theories would never have been developed, because everyone would have applied modus tollens regarding in a not-yet-proven-useful case. (Disclaimer: I know nothing about the actual historical reasons for the first explorations of complex numbers.)
Therefore, it’s best for mathematicians to always keep the M-T and M-P cases in mind when using a proof by contradiction. Of course, a lot of time the contradiction arises due to theorems already proven from axioms, and what happens if any one of the axioms in a theory is removed is usually well explored.
If mathematicians behaved simply as you describe, then those resultant extension theories would never have been developed, because everyone would have applied modus tollens regarding in a not-yet-proven-useful case. (Disclaimer: I know nothing about the actual historical reasons for the first explorations of complex numbers.)
Therefore, it’s best for mathematicians to always keep the M-T and M-P cases in mind when using a proof by contradiction. Of course, a lot of time the contradiction arises due to theorems already proven from axioms, and what happens if any one of the axioms in a theory is removed is usually well explored.