I recently read an article that uses the phrase "addition by subtraction", the idea that value may be gained by losing something of negative value. While I doubt this has literal meaning in mathematics, does this idea have any relevance in mathematics?
Well sure. If I'm trying to make a quantity $x−y$ bigger, one way is to make $x$ bigger. The other is to make $y$ smaller. More precisely, we say that $(x,y) \mapsto x-y$ is monotone in the first argument, and antitone in the second. (The monotone/antitone terminology is not completely standardized).