I know what it is, I just don't see why I SHOULD!

This post is about something that's happened to me a lot in mathematics. Off the top of my head I can think of three examples:

Dot product of two vectors: Okay, so you basically take the component of vector B that is parallel to vector A, and then multiply their magnitudes. Tell me again why that deserves its own operation? In basic physics, this is used to define Work. But saying "you take the parallel component of the force and multiply by the distance it acted for" makes WAY more sense than saying "use this operation that we've suddenly invented for this one very specific situation". In math, you can use its coordinate form to do some neat tricks like verifying if vectors are perpendicular, but again, it makes no sense to me to call it an operation and give it a special name. Just state the coordinate formulae and call them theorems about vectors. They didn't invent a "pythagoras product" which takes two numbers and square-roots the sum of their squares, because that would have been stupid.

Polynomials:: ...why should I care about this seemingly arbitrarily defined class of functions? What makes them special enough to deserve their own name? This has never been explained to me (the fact that a function is polynomials iff one of its derivatives is zero is some small comfort, although no educator I know has ever introduced them by emphasizing that fact, which makes it look like another arbitrary definition).

Everything in basic statistics: Okay, so that's what standard deviation is. But what the hell is it supposed to represent? For all I can see it's just an arbitrarily decided upon metric for how "tight" a distribution is. It doesn't even make intuitive sense: first you use the average, then the median for some reason...

ITT: definitions that seem totally arbitrary / explanations from others as to why they aren't arbitrary

jack wrote:This post is about something that's happened to me a lot in mathematics. Off the top of my head I can think of three examples:

Dot product of two vectors:

Polynomials:

Everything in basic statistics:

ITT: definitions that seem totally arbitrary / explanations from others as to why they aren't arbitrary

I can't think of any right now, but I can probably answer the first two of yours. Although they are very conceptual questions so to answer them will take some work.