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In gradeschool calculus I learnt that derivative = slope. That was a nice teacher’s lie (like the Bohr atom is a nice teacher’s lie) to get the essential point across. But “derivative = slope” isn’t ultimately helpful because in real life, functions aren’t drawn on a chalkboard. ℝ→ℝ drawings don’t always look like what they feel like (e.g. this parabola).

ℝ→ℝ drawings’ “slope” *feels* more like a pulse, a β (observed magnitude), a force, a pay rise, a spike in the price of petrol, a nasty vega wave that chokes out a hedge fund, cruising down the highway (speedometer not odometer), a basic not a derived parameter, a linear operator in the space of all functionals, a blip, a pushforward, an impression, a straight–line projection from data, a deep dive into a function’s infinite profundity, a “bite” in the words of Jan Koenderink.

**A derivative “is really” a pulse. And an integral “is really” an accumulation.**

This story, “Bird’s Eye View” by Radiolab (minute 12:00), nicely illustrates a differential-geometry-consistent view of derivative & integral in the pleasantly-unexpected space of rare languages.

English : Derivative :: Pormpuraaw : Integral

In the Pormpuraaw language of Cape York, Australia, people say things like “You have an ant on your south-west leg” and “Move your cup to the north-north-west a bit”. “*How ya goin’?”* one asks the other. *“Headed east-north-east in the middle distance.”*

- Little kids always know, even indoors, which cardinal direction they’re facing.
- This is very useful when you live in the outback without a GPS.
- American linguistics professor who was exploring there: “After about a week I developed a bird’s-eye view of myself on a map, like a video game, in the upper right corner of my mind’s eye.”

The mental map is like a running integral ∮ xᵗθᵗ dt of moves they make. (Or we could think of it decomposed into two integrals, one that tracks changes in orientation ∮ θᵗ and one that tracks accumulating changes in place ∮xᵗ.) In other words, a bird’s-eye view.

**left right forward back : derivative :: NSEW : integral**

Our English way of thinking is like a differential-geometry-consistent derivative. The time derivative “takes a bite” out of space and so is always relative to the particular moment in time. “Left” and “right” are concepts like this — relative, immediate, and having no length of their own. Just like the differential forms in Élie Cartan’s exterior algebra — tangent to our bodies.

There is a way to make this more precise and I *think* it would make sense to do it on ℂ || with a twistor || spinor. (Help, anyone? David?)

Our English conception of time & space is like a (time-)derivative of our movements. The Pormpuraawans’ conception of time & space is like an integral of their movements, orientation, and location. When we think of direction it’s an immediate slice of time. When they think of direction they’ve been *tracking* those relative-direction derivatives and they answer with the sum.