Maybe you don’t think about multiplying negative numbers because you are no longer in high school. Lamentably, I do. I teach high schoolers the finer aspects of the ACT and SAT, so negative numbers (along with square roots, the Pythagorean Theorem, and the circumference of a circle) are my purview.
I expect my students to grasp the concepts and not push the logic — why would two negative numbers multiplied by each other equal a positive? But now I don’t have to avoid these logical discussions (I pass off why 1 isn’t a prime number with “look it up on Wikipedia”) anymore. Read, and enjoy, as I did, the blog post “The Enemy of My Enemy” in yesterday’s NYT and behold the logic of mathematics, international relations alliances, and unbalanced triangles. It’s actually fun and helpful.