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.
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.