Sometimes you can apply math just by raiding it for vocabulary. You may not need to apply a single theorem.

This has been a surprise to me. I’m more used to creating a mathematical model so you can compute something or apply some theorem. But sometimes you can move a project along just by providing a name for a concept. A meandering discussion can snap into focus because someone has a name for an idea everyone vaguely understands.

Sometimes it may be clear that only part of a mathematical definition applies. In this case math can guide the discussion by asking whether the rest of the definition applies. “It sounds like we’ve got a widget here. A widget has to have these five properties and clearly we have the first three. Let’s think about whether the last two hold.” The answers don’t have to be positive to be useful. You might realize something important in the process of explaining why your thing is **not** a widget.

Sometimes a definition may not apply at all and still be useful! “This reminds me of a widget. It’s not a widget in any strict sense. But if it were, this is what we’d do next. I wonder whether we can do something like that.”