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Pituophis catenifer annectens ♂ On September 1, 1983, the Soviet Union shot down Korean Air Flight 007 which had violated its airspace.

The USA was quick to insist that the Soviets had fired on the plane with full knowledge that it was a civilian airliner. But we didn't know that. From the introduction to Seymour Hersh's book on KE007, The Target is Destroyed:
Those [intelligence agents] who chose to talk to me did so out of a conviction that political abuse of communications intelligence has become a reality in the Reagan administration, and a belief that to protest to their superiors about it would be futile and damaging to their careers. Some of those interviewed did retire from intelligence service shortly after the events described in this book. In a few cases, the mishandling of Flight 007 played a role in their decision to get out.
The USSR in turn claimed the flight had been deliberately sent into Soviet airspace at the request of the USA. That assertion was also ahead of any available evidence.

Five years later, the USS Vincennes shot down Iran Air Flight 655. Statements by US officials featured numerous untruths, e.g.:
  • the Vincennes was in international waters at the time (it was in Iranian waters)
  • Flight 655 was outside of a commercial air corridor (it wasn't)
  • Flight 655 was descending (it was climbing)
So yeah I'm skeptical about everything I hear in these early days after Malaysia Airlines Flight 17 went down. Today, math (but nothing scarier than matrix multiplication and a whiff of complex arithmetic).

When I first did photography, scaling an image was done by moving an enlarger head up and down and mirroring an image was a matter of turning the negative over. In computer graphics, scaling and mirroring are mathematical transformations: (x,y) coördinates of image elements are replaced by linear combinations of the original x and y.

Matrix multiplication is a concise way to express linear combinations:
the values of  a b c d  control the type of transformation Sample transformation matrices and their effects on text:

original scale mirror rotate shear
identity matrix both x and y scaled by 150% reflected about the x axis counterclockwise rotation a.k.a. oblique or skew
dog 150%qogtheta = 106°x = x + 0.3y


A key feature of using matrices is that multiple transformations in sequence can be folded together by matrix multiplication.

Sometimes a sequence of transformations turns out to be equivalent to a single familiar transformation. A carefully-chosen sequence of three shears is equivalent to rotation, good to know if for some reason you must do arbitrary rotations with a feeble program like Micros‑‑t Paint (which only rotates by multiples of 90° but which does know how to shear).

But matrices aren't just for graphics.

The Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, ...) has the recurrence relation Fn = Fn‑1 + Fn‑2, concisely expressible by powers of a 2×2 matrix:
0 1 1 1 (That this is correct for whole values of n is easily shown by induction.)

Raising a matrix to a power by repeated matrix multiplication is a slow process, but in this case there's a trick loosely akin to the rotation-by-three-shears maneuver. There's an advantageous way to decompose this 0 1 1 1 matrix into three factors. In the language of graphical transformations, the three components are a rotation (by about 211.7°), then a scale-with-reflection, then the inverse of the original rotation. Denoting the appropriate rotation matrix by R:
eigendecomposition Pairs of inverses R‑1 and R cancel out when raising the 0 1 1 1 matrix to powers, and raising a diagonal matrix to a power is as simple as raising each element to the desired power. (See here if that wasn't clear.) That is,
diagonalizable matrix to the nth power which gives the basis for a closed‑form expression for the nth Fibonacci number.

And because raising a matrix to non‑integral powers is even more fun than doing arbitrary rotations in Paint, why not calculate Fn for continuous real values of n? The negative number ‑0.618... in the diagonal matrix means the results are complex, but that just adds to the flavor.

Here's Fn plotted on the complex plane as n varies from [0,7]. The curve intersects the real axis at Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, 13. complex axis scale exaggerated (kind of like elevation on 3D topo maps)

Thanks to everyone whose pages are linked to above and to Ron Knott, on whose page I first saw Fn evaluated for continuous values of n. Caesalpinia gilliesi flower + bee
Philosophy is to science as pornography is to sex: it is cheaper, easier, and some people prefer it.
raven (Corvus corax) Since I was a teenager, lightbulbs sold in the USA have listed wattage (power consumed) and lumens (light produced) on the package. Even though most consumers don't know what a lumen is, the numbers can be compared and you can buy the more efficient bulb if you're so inclined.

A lightbulb's efficiency can also be expressed as a percentage: a unitless number, immediately comprehensible—but unheard of on lightbulb packaging or in most discussions of the relative merits of, say, incandescents and fluorescents. A cynic might say that bulb packages don't quote efficiency percentages because they're embarrasingly low (wanna guess how efficient an incandescent bulb is?). But to be fair, lumens take the color-dependent sensitivity of the eye into account and thus tell you something that an efficiency percentage doesn't.

In a trivial sense, every electrical device is 100% efficient. Any energy consumed comes out in one form or another. Question is, how much of the energy consumed goes toward the desired purpose and how much comes out in an undesired form (usually heat).

To gauge efficiency as a percentage, the desired output must be expressible in terms of energy. That's straightforward for a lightbulb (photons embody energy) or a motor (mechanical energy is easily calculated). But what about, say, a computer? Computers throw off a lot of heat—some more than others, making it tempting to deem one computer more efficient than another. But the desired work of a computer is calculation; how to measure that in terms of energy? Is there a theoretical minimum amount of energy needed to add two numbers?

The answer is deliciously strange. Calculation inherently takes energy to the extent that the result throws away some of the information embodied in the input. If a circuit adds two numbers and outputs only the sum, the output contains less information that the input. If the sum is 19, you have lost the distinction between having been asked to add 17+2 or 0+19.

Writing a bit to a computer memory discards whatever value the memory previously contained and thus can't be done without expending energy.

The theoretical minimum cost of changing one bit is k T ln 2, where k is the Boltzmann constant (about 1.38×10-23 joules per degree Kelvin), T is the temperature of the circuit (Kelvin, natch) and ln 2 is the natural log of 2. Real-world computers use much much more than that.

Although the mininum energy inherent in calculation has some theoretical interest, there isn't much sense in expressing computer efficiency as a percentage.

But back to lighting. Y'all took a guess as to how efficient a lightbulb is, yes? A typical 100W incandescent generates about two watts of light, for an efficiency of about 2%. Sceloporus occidentalis Frogs across the street from my house this evening.

Nine days ago, I somewhat flippantly wrote
I haven't spelled out the reasons I deem free will to be illusory; the proof is left to the reader as an exercise.
Consider the arrow of time. Six years ago, I wrote (not flippantly)
With only a few exceptions (not believed to be relevant to the point of this discussion),1 interactions between subatomic particles are time-reversible. Macroscopic processes are anything but: water doesn't flow uphill, shards of exploded capacitors don't reassemble themselves, and the hair on my head doesn't get thicker.
If the apparent difference between past and future is not inherent in the processes underlying our actions, the familiar notion of the future as night-and-day different from the past starts to look suspect.

I realize this may come across as preposterous. Causality, agency, and volition are part of everyday life. We all try to influence the future, but hardly anyone tries to change past events.2

Everything is interrelated. Or—as 4 out of 5 mystics prefer to put it—all is one. Counterfactual conditionals like If I had left home earlier I wouldn't have missed the train are, strictly speaking, nonsense. My having left home when I did is an integral part of the past. Imagining that it were independent of everything else that happened is pure fiction: a practical fiction, but fiction nonetheless.

The future might be just as integrated with the past as parts of the past are with each other.

If the past is settled and the future is not, with events switching over from up‑in‑the‑air to cast‑in‑stone as the cutting edge of now passes by, note that the angle of that edge is relative. Einstein showed that whether A happens before B can depend on the observer. What is past and what is future depends not just on when you ask but also on who you ask.3

Copernicus helped disabuse humanity of the notion that the Earth held a central, privileged position. The idea that now has a privileged position in time might be just as arbitrary, not that I can imagine how it would feel to fully appreciate that.

Happy nineteenth, everyone.
1Whether or not CP violation has any bearing on conventional notions of the arrow of time is by no means settled. It doesn't seem to—but for all I know, it could totally demolish this argument.
2I can't take credit for "hardly anyone tries to change past events". It is one of several gems of phrasing I've come across in Wikipedia.
3Where the relativity of simultaneity applies, it doesn't violate conventional notions of causality. Even so, I find the subjectivity of the distinction between past and future curious.
li The Chinese character 理 (pinyin: lǐ) originally referred to markings in jade, but has come to refer to natural patterns that arise both in living organisms and inorganic material. It also has meanings associated with principle and reason.

Rules often control only the character of patterns, leaving the specifics of each instance to chance. Even identical twins don't have the same fingerprints.

All this is a roundabout way of coming to say that we had great cloud patterns yesterday morning. Rather than format a pic to the width of this column, please see a larger version. As best I can tell, free will is an illusion: something that isn't what it seems to be.

I doubt that quantum indeterminacy has a bearing on free will. We know what kind of systems are sensitive to quantum randomness and which are relatively immune, and there's no indication that neurons are the former.

The brain exhibits the flavor of unpredictability by dint of its scale. A network that large can operate 100% deterministically and yet be so chaotic that it bears no resemblance to what we normally think of as mere mechanisms.

I haven't spelled out the reasons I deem free will to be illusory; the proof is left to the reader as an exercise. I instead move on to other questions that follow. From Susan Blackmore's 2013 essay Living without free will:
But I want to leave aside the complexities of philosophical discussion and turn to a different question — a question that arises for anyone who, like me, rejects the notion of contra‑causal free will — if there is no free will, how should we live our lives?

There are two possible responses: One is to go on living 'as if' we have free will — in other words, to accept that free will is an illusion and yet choose to remain deluded (not a free choice of course, but one caused by prior events and circumstances). The other is to reject the illusion and aspire to live entirely without free will.

I have chosen the second option, but the first of these is by far the more common. Indeed, when I was lucky enough to be able to interview many leading philosophers and neuroscientists about consciousness, I was amazed by the number who chose to live 'as if'.
The whole essay is worth reading. It argues that the feeling of having free will is neither inevitable nor necessary for a full, rich, and just life.

We make use of 'as if' fictions all the time. Matter is largely empty space, yet I stand on the rung of a ladder as if it were solid. It doesn't feel dishonest to deem the ladder solid for practical purposes.

I give Susan Blackmore credit, though, for making the effort to see whether a sense of free will was perhaps optional. But by the end of her essay it's not clear that as if and without are the only two choices. I get more of a sense of transcending the question. Blackmore quotes Alan Watts:
We just decide without having the faintest understanding of how we do it. In fact it is neither voluntary nor involuntary.
ribbed for your pleasure
saguaro (Carnegia gigantea) flowers.