Posts tagged with why

I’ve never understood this and the more I learn about computers the less sense it makes to me. Why would a computer software company purchase from 37Signals a piece of software that does the same thing as IRC, which is free?

[T]he firms that leaned most heavily on lobbyists have outperformed the S&P 500 by a whopping 11 percent per year since 2002.

—Brad Plumer
report by Strategas; chart appears both in and

[T]he firms that leaned most heavily on lobbyists have outperformed the S&P 500 by a whopping 11 percent per year since 2002.

Brad Plumer

report by Strategas; chart appears both in and

This has confused me ever since I learned about evolution.

As Mr. Shepherd explained it to me in the second grade, zebras are hard to kill because, when the zeal runs, their collective stripes make it hard for a predator to visually pick out an individual zebra to tackle.


So the benefit of stripes comes from

  • high contrast
  • highly structured shapes
  • in a group

In economic terms, increasing returns to scale — and increasing returns to coördinated group action (“Let’s all be stripey”).

coördination games

COÖPERATION by Selfish Genes?

So how can that process get started, if the returns to stripes don’t pile up until you get to a really high level of contrast, a very certain structure, and group coördination? Even with a constant scaling factor, how could genes that promote themselves, inside an individual zebra, coördinate to get many zebras to all adopt stripes at once?


It’s not like genes can talk to each other. How could they have coöperated to achieve a better group outcome? Humans do so with language. If there’s a fire in the building and everyone’s crowding the exit, someone can yell: “Stop! We all need to back up and wait in a line. Then we’ll get through this exit faster and all live. None of us will get through if we push each other.” But the genes don’t have a way to communicate for better group outcomes — right?

Maybe stripes could evolve if it only took a few mutations to turn on high-contrast stripes (then the possibility of coöperation arising randomly would be greater … but still small). But I also wonder if there isn’t a population dynamics answer. Or a game theory answer. Is sexual selection involved? Is the sexual transfer of genes involved?



I think about evolution in an abstract way. Even though I know there’s meiosis and specific proteins and RNA’s change the expression and so on, … I just think about genes-as-strings. They mutate and cross-pollinate, with the sexier strings pollinating more. Nature selects (in a Brownian manner) from the pool for the next generation.


Does anybody know a good book or paper about the mathematics of sexual selection, like in a dynamical systems model? Or some other explanation for how the zebra got its stripes?



Readers answered with a lot of mathematical biology links (great!). As I read through what they’ve sent, I’ll add to this list:

  • The pattern of zebra stripes is fully determined by day 21-35 of embryonic development (out of a year-long gestation). Melanoblasts mark out the patterns on the zembryo. Turing Patterns in Animal Coats, via Artemy Kolchinsky

Children are used to not understanding everything that’s said around them.
Ed Catmull, founder of Pixar

Cy Twombly

Cy Twombly


Contrary to common folklore, causal relationships can be distinguished from spurious covariations using inductive reasoning.

Judea Pearl, Causality

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:

R(\theta) = \begin{bmatrix} \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}

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”.


New answer, June 2013:

New answer, July 2014: Really it’s better to think of rotations (and other linear operators like derivative, certain kernels, Laplace or Fourier transform) as homomorphisms or isomorphisms rather than “linear” like a line.
linear maps as multiplication
Algebra works the same after a homomorphism. That’s what “homomorphism” means, and “linear” is sometimes used as a synonym for “homomorphism”.

plot(eXp, xlab="exponent in the power series", ylab="value of constant", main="Spectrum of exp", log="y", cex.lab=1.1, cex.axis=.9, type="h", lwd=8, lend="butt", col="#333333")    eXp <- c(1, 1/2, 1/6, 1/2/3/4, 1/2/3/4/5, 1/2/3/4/5/6, 1/2/3/4/5/6/7, 1/2/3/4/5/6/7/8, 1/2/3/4/5/6/7/8/9, 1/2/3/4/5/6/7/8/9/10, 1/2/3/4/5/6/7/8/9/10/11),    eXp <- c(1, 1/2, 1/6, 1/2/3/4, 1/2/3/4/5, 1/2/3/4/5/6, 1/2/3/4/5/6/7, 1/2/3/4/5/6/7/8, 1/2/3/4/5/6/7/8/9, 1/2/3/4/5/6/7/8/9/10, 1/2/3/4/5/6/7/8/9/10/11)

So really saying a transform is linear is just saying that things work the same after the transform. The concept of linear transforms, then, is useful in the following way:

  1. I start with a situation I don’t understand
  2. I come up with a transformation which simplifies things for my wee brain but leaves the essentials unchanged
  3. I resolve whatever was the original issue in the easier-yet-essentially-the-same space
  4. I transform back to where Is tarted.

A canonical example in physics is rotating the coordinate system (basis isomorphism) to make calculations easier. Like making one dimension disappear by lining things up better.


Quantum mechanics and string theory use different isomorphisms that are more complicated than rotating the coordinate system.

Bonus: Sean Carroll shows us (for free) how Minkowski spacetime uses the concept of rotation—across space-and-time (as if these were orthogonal dimensions as much as the typical xy plane) to explain time dilation and length contraction, the salient features of special relativity.

  1. The world is rational.
  2. Human reason can, in principle, be developed more highly (through certain techniques).
  3. There are systematic methods for the solution of all problems (also art, etc.).
  4. There are other worlds and rational beings of a different and higher kind.
  5. The world in which we live is not the only one in which we shall live or have lived.
  6. There is incomparably more knowable a priori than is currently known.
  7. The development of human thought since the Renaissance is thoroughly intelligible (durchaus einsichtige).
  8. Reason in mankind will be developed in every direction.
  9. Formal rights comprise a real science.
  10. Materialism is false.
  11. The higher beings are connected to the others by analogy, not by composition.
  12. Concepts have an objective existence.
  13. There is a scientific (exact) philosophy and theology, which deals with concepts of the highest abstractness; and this is also most highly fruitful for science.
  14. Religions are, for the most part, bad— but religion is not.

Question: what is the opposite of belief?

Not disbelief. Too final, certain, closed. Itself a kind of belief.

Salman Rushdie, The Satanic Verses

When I was a maths teacher some curious students (Fez and Andrew) asked, “Does i, √−1, exist? Does infinity ∞ exist?” I told this story.

You explain to me what 4 is by pointing to four rocks on the ground, or dropping them in succession — Peano map, Peano map, Peano map, Peano map. Sure. But that’s an example of the number 4, not the number 4 itself.

So is it even possible to say what a number is? No, let’s ask something easier. What a counting number is. No rationals, reals, complexes, or other logically coherent corpuses of numbers.

Willard van Orman Quine had an interesting answer. He said that the number seventeen “is” the equivalence class of all sets of with 17 elements.

Accept that or not, it’s at least a good try. Whether or not numbers actually exist, we can use math to figure things out. The concepts of √−1 and serve a practical purpose just like the concept of (you know, the obvious moral cap on income tax). For instance

  • if power on the power line is traveling in the direction +1 then the wire is efficient; if it travels in the direction √−1 then the wire heats up but does no useful work. (Er, I guess alternating current alternates between −1 and −1.)
  • allows for limits and therefore derivatives and calculus. Just one example apiece.

Do 6-dimensional spheres exist? Do matrices exist? Do power series exist? Do vector fields exist? Do eigenfunctions exist? Do 400-dimensional spaces exist? Do dynamical systems exist? Yes and no, in the same way.