Is this mathematics thing ambiguous or confusing?


There is an excellent chance it is! Mathematicians sometimes assert the object of their study is a universal truth, independent of all human culture. It may be. But the expression of that interest depends on the humans expressing it. And as with all human activities it picks up quirks. Patterns that don’t seem to make sense. Or that seem to conflict with other patterns. It’s not two days ago I most recently saw someone cross that 0 times anything is 0, but 0! is 1.

Mathematicians are not all of one mind. They notice different things that seem important and want to focus on that. They use ways that make sense to their culture. When they create new notation, or new definitions, they use the old ones to guide them. When a topic’s interesting enough for many people to notice, they bring many trails of notation to describe it. Usually a consensus emerges, that there are some notations that work well to describe these concepts, and the others fall away. But it’s difficult to get complete consistency. Particularly when there are several major fields that don’t need to interact much, but do have some overlap.

Christian Lawson-Perfect has started something that might be helpful for understanding this. WhyStartAt.xyz is to be a collection of “ambiguous, inconsistent, or just plain unpleasant conventions in mathematical notation”. There’s four major categories already: inconsistencies, ambiguities, unpleasantness, and conflicting definitions. And there’s a set of references useful for anyone curious why something is a convention. (Nobody knows why we use ‘m’ for the slope in the slope-intercept or point-slope equations describing a line. Sometimes a convention is arbitrary.) It’s already great reading, though, not just for this line from our friend Thomas Hobbes.

Why Call The Intercept b


Just because there are in principle uncountably many possible equations for any line doesn’t mean we ever actually see any of them. Actually, we just about always pick one of a handful of representations. They’re just the convenient ones. I’m going to say there’s four patterns that actually get used, because I can only think of three that turn up, as long as we’re sticking to Cartesian coordinate systems and aren’t doing something weird like parametric descriptions, and I want to leave some hedge room for when I realize I overlooked the obvious. The first one — that I want to talk about, anyway, and just about the first one anyone encounters — is called the slope-intercept form, and it’s probably what someone means if they do talk about “the” equation for a line.

Continue reading “Why Call The Intercept b”

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