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.
Espen Friberg & Magnus Voll Mathiassen, Rural Science
Published by NSEW, who sent us the zine.
very nice drawings
図書室への寄贈を頂きました / Recently Added : JESSICA WILLIAMS (NSEW)
オスローにて N S E W を運営する、Jessica Williams さんの Back and Forth と Lost and Found を追加いたしました。これらの作品は、形、言葉、そして写真のオーケストレーションと再利用から把握できるように、反復的で、分類された感傷性を特徴としています。またそれは、作者にとってとても個人的な秘密を、差し控えながらもあらわにしています。Back and Forth は下にお見せしている小さな手作りのポスターと一緒に送られてきました。
Back & Forth
Lost & Found
The crisp ooze of biting into the word scrupulous is akin to biting into a piece of hard candy with a surprise center. How I love the sound of words. Austere gives off the feeling just like it’s definition; somber.
Right now I’m listening to Nine Inch Nails, though I’m not really sure why exactly I picked it of all things to put on at a quarter past ten. Although I am enlightened as to why Jared Leto likes this particular song so much.
So I was in the elevator again just now (living in an apartment really confines you) and admiring my wonderfully ruffled hair in the mirror (of which there are four, in case you were wondering, and they are amazing when I have a reason to check myself out ie; when I’m all dressed up) and I asked myself why oh why I was so attractive. My subconscious answered with, “L’unno man. Why don’t you ask the legion of people who would want to date you?” And I proceeded to ponder on the number but again that nosy parker of my subconscious interrupted with, “The number of people who would actually want to date you are less than the number of people in this lift.” Sadly, I was the sole occupant. I had to agree anyway.
Napping in the afternoon is a thousand times easier than falling asleep at night. Walking through the carpark always provokes rhetorical hard to answer questions. Singing about the meaning of being lonely is prohibited after the lights are dimmed. Effectively communicating without an ounce of misunderstanding might be insurmountable.
I had the honor of witnessing the graceful descend of a mango from a tree today. Because falling fruits fetch friendly flies.
I can be happy, in fact, I was happy all day yesterday!