**Hilbert with Ladders**, 2004 by Don Relyea

## Archive for March, 2011

### Noncommutative distances between industries

March 30, 2011The distance from your house to the grocery must be the same as the distance back, but 20th-century mathematicians speculated about circumstances where this might not be the case.

**Very small-scale physics** is non-commutative in some ways and so is distance in finance.

But non-commutative logic isn’t really that exotic or abstract.

**Imagine you’re hiring.**You could hire someone from the pri**v**ate sector,**c**harity sector, or pu**b**lic sector. It’s easier for**v**managers to cross over into**b**|**c**than for**c**|**b**managers to cross over into**v**.So pri

**v**ate is close to pu**b**lic, but not the other way around. Or rather,**v**is clos*er*to**b**than**b**is to**v**.**δ**|**v**,**b**| <**δ**|**b**,**v**| . (same for**δ| v**,**c |**)- Perhaps something similar is true of
**management consulting, or i-banking**? Such is the belief, at least, of recent Ivy grads who don’t know what to do but want to “keep their options open”.This might be more of a statement about average distance to other industries

**∑ᵢ δ**| consulting,**xᵢ**| being low, rather than a comparison between**δ**| consulting,**x**| and**δ**|**x**, consulting | . Can you cross over from energy consulting to actual energy companies just as easily as the reverse?

- Imagine you’re want a
**marketing consultant**. Maybe some “verticals” are more respected than others? So that a firm from vertical**1**could cross over into vertical**2**but not vice versa. - Is it easier for
**sprinters**to cross over into**distance running**, or vice versa? I think distance runners have a more difficult time getting fast. If it’s easier for one type to cross over, then**δ|**sprinter**,**longdist**|****≠****δ|**longdist**,**sprinter**|**. - It’s easier to roll things downhill than uphill. So the energy distance
**δ |**top**,**bottom**| < δ |**bottom**,**top**|**. - It’s usually cheaper to ship one direction than the other.
**Protip:**if you’re shipping*PACA*(donated clothes) from the USA to Central America, crate your donation on a Chiquita vessel returning to point of export.

**Noncommutative distance, homies.** (quasimetric) And I didn’t invoke quantum field theory or Alain Connes. Just business as usual.

### March 29, 2011

You know what’s surprising?

### Rotations are linear transformations.

I guess lo conocí but no entendí. Like, I could write you the matrix formula for a rotation by **θ** degrees:

But **why** is that linear? **Lines are straight and circles bend.** When you rotate something you are moving it along a circle. So how can that be linear?

I guess 2-D linear mappings ℝ²→ℝ² surprise our natural 1-D way of thinking about “straightness”.

### March 27, 2011

The Ascent of Mandarin (via sandy dreams in my backpack, man)

### When economists use maths to describe people, it’s somehow clearer when they’re less accurate. People tried to model the entire world and failed — but the logic of simple metaphors means … more.
Below is a good example.
The main question motivating the paper below is: would **school vouchers** improve the educational system, and would they **increase inequality**?
*A Complicated Endogenous Jargony Title* by G. Glomm and B. Ravikumar
**PAPER SUMMARY**
The paper studies the difference between private and public systems of education.
The public system means that everyone pays taxes to finance schools and the quality of schools is the same for everyone.
The private educational system means that people choose how much to pay for education and the quality of schooling depends on the amount of money a student’s parent puts into it.
The model used on the paper is an **overlapping generations model**. In the first period agents allocate a unit of time between studies (i.e. acquiring human capital) and leisure. In the second period agents get income equal to their human capital.
The main findings of the paper are the following:
First,** inequality declines** faster with the **public** system of **education**.
Second, **private** education results in **higher** per capita **income** unless the initial inequality is not very high.
Third, if income distribution is skewed to the left, then majority voting will result in the choice of the public education system.
*end of excerpt*
**METAPHORS AND FAIRY TALES**
Alright, so what does that mean? We took a mathematical model which resembles reality. It is wildly inaccurate; much more inaccurate than assuming the sun is a point mass for instance. But … the conclusion still does seem to say something. But what? And what, precisely, can you do with the knowledge gained from reading the above? Does it prove anything about real school vouchers? Does it even make a valid suggestion about real school voucher policy?
It’s not exactly an if-the-world-were-like-this story and it’s certainly not a here’s-how-it-is-and-here’s-what-it-means story. More like a mathematical fairy tale? Or something.
This ambiguity and vagueness are actually what made me return to mathematics after years away from it.
It’s exciting to think that logic and maths might apply to relevant, contentious questions like political disputes. It’s also cool how economists rely on judgment to construct assumptions, “pure logic” to reach conclusions, and then judgment again to apply the conclusions.

March 26, 2011
### March 25, 2011

http://www.bbc.co.uk/emp/9player.swf?revision=10344_10570

*The Taiping “Heavenly Kingdom”*

**historical mèmes:**

- appropriating convenient parts of a foreign, exotic religion for one’s own needs. Buddhism in USA over the last 25 years; Christianity in China 100 years ago.
- sex is bad
- smoking, drinking is bad
- simplistic thinking [shared property === equality]
- Animal Farm | idealistic rulers then abuse their power
- stable, sophisticated culture vs. battle-hardened barbarian invaders
- what makes a society rife for revolution: unmarried young men

### Hot Showers

March 24, 2011You know what’s really nice about the U.S. of A.? Being able to take **hot showers with lots of water**, for pretty much as long as you like.

Thanks, prosperity.

### March 23, 2011

“The anchovies were nowhere *near* the sardines and the tuna. That’s because they were near the pizza toppings.

But it was only a problem because this was a three-dimensional grocery store. If it had been a **thirty-dimensional grocery store** they could have been near the pizza *and* the sardines.”

I’ve been putting off a post about **phase space** for nearly a year now, and watching this talk made me remember that I’ll soon have to do it.

Geoff Hinton talks here about a number of different spaces that are *not* the physical space that math was initially developed on.

- The
**bag of words**model of a document takes each word in a text to be a dimension of the document, with like 80,000 possibilities each or however many words there are in English. (The 80,000 possibilities are the underlying corpus.) **Latent semantic analysis**of the bag-of-words type is how Google now does its search rankings. (PageRank only constitutes something like 30% of SERP ranking anymore, because [a] it’s too easy to game and [b] it’s inspecific to what’s being searched on. Domain authority and LSI comprise the rest. <—separate article)- Seeing a
**pixel-by-pixel representation**of a 2-D image as a list vector is problematic because the first pixel in a 200×300 Facebook profile image is next to the second pixel and also next to the 201st pixel. I.e. one needs a 2-array. - Hinton talks about abstract
**feature space**and energies — equivalently evolutionary fitness or economic utility — and ravines and mountains upon this manifold. - The number of dimensions here is like the number of parameters (same as free parameters or degrees of freedom or arbitrary parameters in stats class) and in a neural net each “synapse” or graph edge is a lever you can pull.
- The same metaphor — and this is a metaphor in a grand sense which I hope to cover before the year is up — applies to the equalizer on your uncle’s home stereo, i.e. the number of terms in the Fourier decomposition.

### March 22, 2011

Not saying I agree with this classification … but consider it a more topologically complex alternative to the **≅ℝ²** picture of **{**positive vs negative affect**} ⨯ {**high/low energy**}** used in the “Miller mood map”.

I’m not even sure if that’s the correct term for the **≅ℝ²** story — but that story is wrong. Google “valence” and “affect” or “mood” to get the skinny on it.

**Math questions:**

- where is
**0**? - Is there a
**0**? - How do you go from one state to another?
- Which states are easiest to reach (“closest to”) from which other states?
- Is distance non-commutative? Differing length to get from
**A→D→C**versus**A→E→C**? And different in reverse**C→E→A**?