## Archive for June, 2011

### Radiolab – Bird’s-Eye View

June 26, 2011<![CDATA[// <![CDATA[

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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.

### June 24, 2011

Colourful Colorado

### June 23, 2011

scrambled (by Emma McNally1) via wowgreat

### Nonterminating decimals do not make sense.

June 22, 2011The **Banach-Tarski paradox** proves how f#cked up the real numbers are. Logical peculiarities confuse our intuitions about “length”, “density”, “volume”, etc. within the continuum (ℝ) of nonterminating decimals. Which is why Measure Theory is a graduate-level mathematics course. These peculiarities were noticed around the turn of the 20th century and perhaps never satisfactorily resolved. (Hence I disagree with the use of real numbers in economic theory: they aren’t what you think they are.)

**Axiom of Choice ****→ Garbage**

The paradox states that **if you assumed the axiom of choice** (or Zorn’s Lemma or the well-ordering of ℝ or the trichotomy law), **then you could take one ball and make two balls out of it**. It follows that you could make seven balls or thirty-seven out of just one. That doesn’t sound like real matter (it’s not; it’s the infinitely infinite mathematical continuum).

I can’t think of anything in real life that that *does* sound like. Conservation-of-mass-type constraints hold in economics (finite budget), probability (∑pᵢ=1), text mining, and in all the phase and state spaces I can think of as well. Generally you don’t make something out of nothing.

**If it’s broke, throw it out.**

The logical rule-of-inference *Modus Tollens* says that **if A→B and ¬B, then ¬A**. For example if leaving the fridge open overnight leads to rotten food, and the food is not rotten, I conclude that the fridge was not open overnight. Let A = Axiom of Choice and B = Banach-Tarski Paradox. Axiom of Choice leads to Banach Tarski paradox; said paradox is false; so why don’t we reject the Axiom of Choice? I have never gotten a satisfactory answer about that. ℝ is still used as a base corpus in dynamical systems, economics, fuzzy logic, finance, fluid dynamics, and as far as I can tell, everywhere.

**How does the proof of paradox work?**

The proof gives instructions of how to:

- Partition a solid ball into five unmeasurable disjoint subsets.
- Move them around (rigidly, without adding mass).
- Get a new solid ball, whilst leaving the first ball intact.

The internet has several readable, detailed explanations of the above. You’ll end up reading about Fuchsian groups, Henri Lebesgue’s measure, and hyperbolic geometry (& the Poincaré disk) along the way.

Stan Wagon has also written a Mathematica script to display the subsets in a hyperbolic geometry (whence these pictures come). Thanks, Stan!