This is really just an extension of James D. Taylor's comment on the question, but recognizing the value of definitions, and the inherent ambiguity without them, is ridiculously helpful.

Define your terms!

I recently saw a talk on how to teach students to write mathematics well. The advice "think like a lawyer" was given, which I totally agree with when writing mathematics. Anyone here who has read a quality legal document will know the similarities which this analogy is getting at. Both mathematicians and lawyers define their terms clearly at the beginning (of a debate/proof/court case) in order to eliminate as much ambiguity as possible from the words being used.

This is a great skill to have in order to cut through the BS in lots of other situations. E.g. Time and time again you see opinion pieces in which writer has no real point, but just trivially exploitations the lack of a definition for a certain word. This really annoys me. Or you can be much better at seeing when an argument is a genuine difference of opinion or just a confusion arising from two people having different definitions.

If you take this too far, you end up as a bit of nihilist in that respect though: There's nothing to argue about because either people have different opinions (and there will always be people with different opinions) or people have different definitions... and arguments end trivially and inconclusively (I know I've done this many times to end boring arguments:) "If we accept your definition of [e.g.] feminism, then you're right and if we accept mine, then I'm right". (I suppose seeing to the core of an argument like this is similar to David White's first point).

On the other hand, you can debate quite freely things you have no clue about, just by deciding on a few axioms/vaguely reasonable assumptions and working from the definitions! (this is sort of the skill of debating competitions).

This is really just an extension of James D. Taylor's comment on the question, but recognizing the value of definitions, and the inherent ambiguity without them, is ridiculously helpful.

Define your terms!

I recently saw a talk on how to teach students to write mathematics well. The advice "think like a lawyer" was given, which I totally agree with when writing mathematics. Anyone here who has read a quality legal document will know the similarities which this analogy is getting at. Both mathematicians and lawyers define their terms clearly at the beginning (of a debate/proof/court case) in order to eliminate as much ambiguity as possible from the words being used.

This is a great skill to have in order to cut through the BS in lots of other situations. E.g. Time and time again you see opinion pieces in which writer has no real point, but just trivially exploits the lack of a definition for a certain word. This really annoys me. Or you can be much better at seeing when an argument is a genuine difference of opinion or just a confusion arising from two people having different definitions.

If you take this too far, you end up as a bit of nihilist in that respect though: There's nothing to argue about because either people have different opinions (and there will always be people with different opinions) or people have different definitions... and arguments end trivially and inconclusively (I know I've done this many times to end boring arguments:) "If we accept your definition of [e.g.] feminism, then you're right and if we accept mine, then I'm right". (I suppose seeing to the core of an argument like this is similar to David White's first point).

On the other hand, you can debate quite freely things you have no clue about, just by deciding on a few axioms/vaguely reasonable assumptions and working from the definitions! (this is sort of the skill of debating competitions).