> No, "both sidesing" is not at all how philosophers or mathematicians train to establish the truth

In math, proof by counterexample is pretty directly what you'd call "both-sidesing" above; you make a thorough, detailed, and compelling argument for a case against your hypothesis, then demonstrate why it can't be true.

If you can't do the work of the making such a detailed counterexample, then the proof of your actual hypothesis falls apart too.

Being able to make the strongest argument against your beliefs is necessary for you to hold your own beliefs most strongly.