[Chinjut]

By "count of two sets", I don't mean to invoke set theory in any imposing, modern sense. Just the observation that, historically, we were adding counting numbers (in particular, non-negative ones with such properties as "The sum of x and y is always at least as large as x itself") long before we were adding integers.

Regardless, the point still stands: why would you allow the word "sum" to fit all those uses (disparate, but with a web of family resemblances), but not grant the same to "derivative"?

[AnimalMuppet]

Because it seems to me that the web of family resemblances for "derivative" should include the rate of change of one thing with respect to another, not just that the product rule is satisfied. That is, it seems to me that the attempts to extend "derivative" are extending it to the point that the web of family resemblances no longer fits.

[tel]

Unfortunately, all I can say to comfort that is that choosing "rate of change" as your centralizing analogy for "derivative" has been shown through the history of mathematics to be a great start, but a slow finish.

Frequently mathematics benefits a lot from abstracting to algebra because, at this point, it's purely about how to define elements and operations by their apparent behavior instead of by their metaphor or interaction with a larger idea (such as notions of space, continuity, rate, change... all of those require quite a lot of mechanics to get in place, while algebra is very light-weight).

As an example, there have been a lot of attempts to discretize calculus for computers. Usually, the goal here is to create a scheme of discretization which, in the limit, resembles the smooth computations we'd like to perform. This has been a successful program in practice, but it's known to be fraught with weird edge cases. It's easy to create discretized situations which violate intuition.

Much of the reason these failed is because they attempted to generalize from the notion of "rate of change".

There's also the idea of discrete calculus (not "discretized") which is what you get when you apply the algebraic laws alone to some very standard notions of discrete spaces (oriented simplicial complexes, in particular—the simplest discrete object which "has enough topology" to meaningfully have the algebraic laws of integration applied to it).

What you get in this case is a rich theory of discrete calculus which rederives half of manifold learning and graph theory as a special case. All of the laws follow precisely—and they must, as the entire construction was built to prevent such violations.

Finally, you can examine discrete calculus to find a notion of "rate of change" if you like. But it's alien from that which you might be familiar with from continuous domains. It would have been very difficult to arrive at this point trying to generalize that intuition.

But it's practically inevitable (not to say it's easy, just inevitable) to if you say that you want to take the algebraic structure of derivatives and integration and apply it to oriented simplicial complexes.

[Chinjut]

It seems to me a sum should be at least as large as each of its summands (or rather, it once did). The world paid no heed, and life trudged on. I don't see a need to pick one particular archetypal trait or another and say the word "derivative" (or any other bit of mathematical jargon) mustn't ever be extended by analogy to a situation no longer directly manifesting that trait. A web of family resemblances doesn't depend on any one fiber running through all of it.

It's not as though the similarity of terminology is chosen with intent to confuse; the intent is to illuminate. The name for the generalization is chosen to match its more familiar relative because it is often _useful_ to think in terms of the analogy, imperfect though it be. [It seems humans are such that we would never find our way to powerful abstractions without such overloading; the combinatorial explosion of names would be too great to comprehend.]