> Math doesn’t prove the sky is blue on a sunny day.Yeah, it doesn't. So it is not a truth, but a belief. It is pretty strong belief, so strong that it almost indistinguishable from the truth. But nevertheless it is just a belief.
> My main point is that of falsifiability. You don’t need math to make a falsifiable argument. However, much of our discourse can’t even meet this basic qualification.
Even falsifiability is a too rigid property for some discourses. My main point is that attempts to find a silver bullet to shoot "bad beliefs" are no-go. Knowledge couldn't be bad or good, true or false. It is the way knowledge applied could be good or bad.
For example, even "the sky is blue on a sunny day" could be misapplied. Take the sky of a Moon or a Mars for example, which are not blue in a sunny day. Or let us imagine atmosphere of Earth with a lot of dust after collision of Earth and an asteroid: on such Earth sky could be orange, or gray or of some other color. I mean that even "true" statements have limits of their truthiness.
As an opposite example, a "false" statement could be applied in a "good" way. Earth is flat? Oh, yes, it is convenient to think so when dealing with a relatively small area of the Earth surface. Or when communicating with people, who believe that Earth is flat. You need only to remember the limits of this belief, to not misapply it accidentally.
In math the situation a little different. Math statements also could go wrong when assumptions change (the sum of angles of a triangle could be not equal to PI, if triangle drawn not on a plane but on some other surface), but mathematicians make sure to strictly state all the assumptions as axioms and definitions. It is possible in ideal conditions, but not possible in real ones.