Archive for April, 2011

April 30, 2011

Markets can indeed organize very complex production processes, to an extent that might well seem miraculous to anyone who tried to reason about it in the abstract. But that doesn’t mean that markets are the only, or invariably the best, way to organize production.

John Quiggin

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Superposition of Feelings

April 29, 2011

The act of writing is like the collapse of a quantum waveform. So many things are in your mind, interacting with each other, unsaid. Many truths — some at odds with others — could be spun into a thread. But whatever you write crystallizes as the story. The other ephemera die.

Since speech is serial, it’s hard to portray the composite quality of real-time motivations, perceptions, emotions, impulses, sentiments, choices, …. I’ve heard that Chinese poetry can multi-track — and perhaps many great writers can — but not me. 

Quantum Quacks

Even Roger Penrose was roundly mocked for suggesting that quantum interactions in the brain relate to free will. Going the other direction, What the bleep do we know? draws several intellectually limp conclusions from quantum mechanics, e.g. that QM implies the possibility of telekinesis. (I would say that the authors must have leapt to conclusions from blurbs & pamphlets, except that Niels Bohr and Wolfgang Pauli also took spiritualistic and parapsychological views on QM.)

So it would seem that connections between QM and psychology are limited to quackery.

Metaphor

However, QM is just an abstract mathematical theory. You don’t have to plug physical parameters into the formalism. In that sense you can abscond the superposition-and-collapse metaphor out of the subatomic realm where it was invented and apply it to other things — like thought.

In other words, you don’t have to talk about quantum superposition. You can talk about emotional superposition, opinion superposition, mood superposition, colour superposition (like, are these images green? red? blue? 1 2 3 4 5), personality superposition, guilt superposition, … and more.

If I say: “I had a superposition of thoughts during the bear attack which collapsed into a 1-D sequence when I told the story,” that is valid.

It’s neither what-the-bleep nor relying on quantum effects on my brain. I just appropriated the mathematical metaphor of superposition and the mathematical metaphor of collapse, to express how I can’t really tell you the whole story of the bear attack, and how the telling perverts the story itself.

April 28, 2011

is a mysteryiiiiiiiii by Cristian Elizalde via firefluff

How to multiply matrices

April 27, 2011

This is for my homies in math class.

Mathematical matrices are blocks of numbers, arrayed in 2-D. (Higher-dimensional arrays are called tensors.)


  1. Left “times” right equals target. Each entry in the target is the result of a series of +’s and ×’s along the red and blue. A long sum of pairwise products.
  2. Your left hand goes across and your right hand goes up/down.
    where
    .
  3. There need to be as many abcdefg’s as there are 1234567’s or else the operation can’t be done.
  4. Also you can tell how big the output matrix will be. There can be three blue rows so the output has three rows. There can be four red columns so the output has four columns.
  5. This is the “inner product” because multiplying vector-shaped blocks (tall blocks) like Aᵀ•B results in an equal or smaller sized output.

    (There is also an “outer product” which is a different way of combining the info from the two matrices. That gives you an equal or larger shaped result when you multiply vector/list-shaped tall blocks A∧B.)

  6. Try playing around with this one or that one.

Matrix multiplication is the simplest example of a linear operator, the broad class of which explains quantum mechanics and ODE’s. You can also apply different matrices at different points as in a vector field — on a flat surface or a curvy, holey surface.

April 26, 2011

Today we’re going to ask—-and hopefully answer—-this question of whether there’s free will or not. If you want to know where I stand, I’ll tell you: I believe in free will. Why? Well, the neurons in my brain just fire in such a way that my mouth opens and I say I have free will. What choice do I have?

Scott Aaronson

James Ferraro – James Ferraro-∞∞

April 25, 2011

by James Ferraro

April 24, 2011

April 23, 2011

A lot of people think of “geometric” art as being math-y, in the same sense that the band Maps & Atlases is math-y.

But I don’t think lines, circles, squares, tessellations, grids, and polygons are more mathematical than globsleaves, aleatorics, coloursnets, or scribbles. In fact, I can link to a math post about each: lines, circles, hypersquarespolytopes, aleatoricstessellations, blobs, gridsleaves, nets, scribbles, colours.

The mathematical thought that occurs to me when looking at this painting is how, in composition, every spot on the canvas influences every other spot. Holger Lippmann couldn’t have swapped a few of these circles because it would have ruined the effect.

Similarly in painting like this, if you added a splotch of yellow in the bottom right, that would affect the look of several other parts of the canvas.

Algebraically, the pieces of the composition are like a highly connected graph (in “how good it looks” space).

If you regressed compositional outcome against the content of each point in the painting (or just against the style of each circle), the relevant explanatory variables would be highly interactive terms. All the monomial, binomial, trinomial, … terms would be irrelevant.

 

The image is: 29417FlowerCircles_13_grid3 by holger lippmann, via wowgreat

Logic and Emotion are Not Opposites

April 22, 2011

Despite what the MBTI claims, logical and emotional are not opposites. 

You mean to tell me that,

  • because I cried when I watched Brokeback Mountain and Never Let Me Go;
  • because I empathise with a character in a book;
  • because I’m moved by the majesty of nature;
  • because I relate to people, because I listen, because I think other perspectives are valid;
  • because I like art,

that I can’t follow a deductive argument? That I can’t understand modus ponens or learn to program a computer? Or conversely, that because a kid scored 100% on the math test, that s/he has no feelings?

That doesn’t make sense.

Just like the artist-versus-businessman-versus-engineer divide, logical-versus-emotional is an artificial, invalid, made-up limitation on people.

Determinant

April 21, 2011

A matrix ℳ represents a sequence of + and × operations. At the end you’ve linearly transformed a space (sheared it, expanded it, rotated it — but kept the origin where it is.)

sheared Mona Lisa

Did the amount of stuff in the picture change when you did that? If you kept everything in proportion then det |ℳ| = 1. If not, then det |ℳ| ≠ 1.

If the amount of stuff increased by 10% then det |ℳ|=1.1. If you effectively shrank the picture in half, then det |ℳ|=.5. And so on.

The determinant |ℳ| is the change in volume after the linear transformation.

This metaphor extends to 3-D and beyond.

  • If water is flowing linearly in a stream, then |ℳ| needs to be 1, or else water (matter) would be being created.
  • If money is flowing linearly in a billion-dimensional economic system, then |ℳ| is hopefully just a little bit above 1, if value is being created. (Central banks need to print |ℳ| times more money to prevent deflation.) 
  • And a hundred-dimensional linear dynamical system’s phase space grows by |ℳ| at every step.