A lot of words to dodge around the main point...

When's the last time you proved a theorem in calc? I'm talking about "integral tanh^-1(x) dx", not the tedious problems they give you in the beginning of your homework.

When's the last time you played with the axioms of algebra?

When's the last time you determined isomorphisms between geometric shapes?

If you talk to a student in "mathematics", most don't talk about numbers, many don't even talk about symbols. My previous roommate used to describe organizations to me in terms of their fundamental axioms and theorems. A physicist friend who used to live across the hall from me once described jazz from the basis of musical keys and fractal patterns. The last conversation I had with the girl who shares my bathroom was about the properties of manifolds, functors, and automorphisms.

To them, "inconsistent" means a specific idea that can be disproved from set of codified rules. They ask for definitions and properties work out the rest out for themselves.

Not everyone needs to think this abstractly. Not everyone is practiced to perform it with the ease they do. But it's damn useful for rationalizing problems. Hell, finding the rules to turn a people problem into a graph theory problem is the sort of thing that the yuppies at Twitter get paid bank every year to do.

It's only one aspect of the larger problem of debate -- logos won't get you far without ethos and pathos -- but understanding what makes an argument rock-solid and how to see that another is full of holes is an important skill to have in life.