This Explains Everything

by Mr. John Brockman

Computer scientist, University of California–Berkeley, School of Information; author, Search User Interfaces

  From the earliest days of computer programming up through the present, we have been faced with the unfortunate reality that the field does not know how to design error-free programs.

  Why can’t we tame the writing of computer programs to emulate the successes of other areas of engineering? Perhaps the most lyrical thinker to address this question is Frederick Brooks, author of The Mythical Man-Month. (If one notes that this unfortunately titled book was first published in 1975, it’s easier to ignore the sexist language littering it; the points Brooks made thirty-seven years ago are almost all accurate today, except the assumption that all programmers are “he.”)

  Espousing the joys of programming, Brooks writes:

  The programmer, like the poet, works only slightly removed from pure thought-stuff. He builds his castles in the air, from air, creating by exertion of the imagination. Few media of creation are so flexible, so easy to polish and rework, so readily capable of realizing grand conceptual structures. . . . Yet the program construct, unlike the poet’s words, is real in the sense that it moves and works, producing visible outputs separate from the construct itself. It prints results, draws pictures, produces sounds, moves arms. The magic of myth and legend has come true in our time.

  But this magic comes with the bite of its flip side:

  In many creative activities the medium of execution is intractable. Lumber splits; paints smear; electrical circuits ring. These physical limitations of the medium constrain the ideas that may be expressed, and they also create unexpected difficulties in the implementation.

  . . . Computer programming, however, creates with an exceedingly tractable medium. The programmer builds from pure thought-stuff: concepts and very flexible representations thereof. Because the medium is tractable, we expect few difficulties in implementation; hence our pervasive optimism. Because our ideas are faulty, we have bugs; hence our optimism is unjustified.

  Just as there is an arbitrarily large number of ways to arrange the words in an essay, a staggering variety of different programs can be written to perform the same function. The universe of possibility is too wide open, too unconstrained, to permit elimination of errors.

  There are additional compelling causes of programming errors, most important the complexiting of autonomously interacting independent systems with unpredictable inputs, often driven by even more unpredictable human actions interconnected on a worldwide network. But in my view, the beautiful explanation is the one about unfettered thought-stuff.

  CAGEPATTERNS

  HANS-ULRICH OBRIST

  Curator, Serpentine Gallery, London; author, Ai Weiwei Speaks; coauthor (with Rem Koolhaas), Project Japan: Metabolism Talks; editor, A Brief History of Curating

  In art, the title of a work can often be its first explanation. And in this context I am thinking especially of the titles of Gerhard Richter. In 2006, when I visited Richter in his studio in Cologne, he had just finished a group of six corresponding abstract paintings to which he gave the title Cage.

  There are many correlations between Richter’s paintings and the compositions of John Cage. In a book about the Cage series, Robert Storr has traced them from Richter’s attendance of a Cage performance at the Festum Fluxorum Fluxus in Düsseldorf in 1963 to analogies in their artistic processes. Cage has often applied chance procedures in his compositions, notably with the use of the I Ching. Richter, in his abstract paintings, also intentionally allows effects of chance. In these paintings, he applies the oil paint on the canvas with a large squeegee. He selects the colors on the squeegee, but the actual trace the paint leaves on the canvas is to a large extent the outcome of chance. The result then forms the basis for Richter’s decisions on how to continue with the next layer. In such an inclusion of “controlled chance,” an artistic similarity between Cage and Richter can be found. In addition to the reference to John Cage, Richter’s title Cage also has a visual association, as the six paintings have a hermetic, almost impermeable appearance. The title points to different layers of meaning.

  Beyond Richter’s abstract paintings, analogies to Cage can be found in other of his works. His book Patterns is my favorite book of 2011. It shows Richter’s experiment of taking an image of his Abstract Painting [CR: 724-4] and dividing it vertically into strips: first 2, then 4, 8, 16, 32, 64, 128, 256, 512, 1,024, 2,048, up to 4,096 strips. This methodology leads to 8,190 strips. Throughout the process, the strips become thinner and thinner. The experiment then leads to the strips being mirrored and repeated, which leads to a diversity of patterns. The outcomes are 221 patterns published on 246 double-page images. In Patterns, Richter set the precise rules, but he didn’t manipulate the outcome, so the pictures are again an interaction of a defined system and chance.

  Patterns is one of many outstanding art books Richter has done over the last couple of years, such as Wald (2008), or Ice (1981), which includes a special layout of the artist with his stunning photos of a trip to the Antarctic. The layout of those books is composed of intervals with different arrangements of the photos but also blank spaces—like pauses. Richter told me that his layout has to do with music, Cage, and silence.

  In 2007, Richter designed a 20-meter-high arched stained-glass window to fill the south transept of Cologne Cathedral. The Cologne Cathedral Window comprises 11,000 hand-blown squares of glass in 72 colors derived from the palette of the original Medieval glazing that was destroyed during the Second World War. Half of the squares were allotted by a random generator, the other half were like a mirror image to them. Control is once more ceded here to some extent, suggesting his interest in Cage’s ideas to do with chance and the submission of the individual will to forces beyond one’s control. “Coincidences are only useful,” Richter has told me, “because they’ve been worked out—that means either eliminated or allowed or emphasized.”

  In Halberstadt, a performance of Cage’s piece “ORGAN2/ASLSP” (1987) recently took place. “ASLSP” stands for “as slow as possible.” Cage has not further specified this instruction, so that each performance of the score will be different. The actual performance will take 639 years to be completed. The slowness in Cage’s piece is an essential aspect for our time. With globalization and the Internet, all processes have been accelerated to a speed in which no time for critical reflection remains. The present “Slow movement” thus advises us to take time for well-chosen decisions together with a more locally oriented approach. The idea of slowness is one of the many aspects that continue to make Cage most relevant for the 21st century.

  Richter’s concise title, Cage, can be unfolded into an extensive interpretation of these abstract paintings (and of other works)—but, one can say, the short form already contains everything. The title, like an explanation of a phenomenon, unlocks the works, describing their relation to one of the most important cultural figures of the 20th century, John Cage, who shares with Richter the great themes of chance and uncertainty.

  THE TRUE ROTATIONAL SYMMETRY OF SPACE

  SETH LLOYD

  Professor of quantum mechanical engineering, MIT; author, Programming the Universe

  The following deep, elegant, and beautiful explanation of the true rotational symmetry of space comes from the late Sidney Coleman, as presented to his graduate physics class at Harvard. This explanation takes the form of a physical act that you will perform yourself. Although elegant, the explanation is verbally awkward to explain and physically awkward to perform. It may need to be practiced a few times. So limber up and get ready: You are about to experience in a deep and personal way the true rotational symmetry of space!

  At bottom, the laws of physics are based on symmetries, and the rotational symmetry of space is one of the most profound of these symmetries. The most rotationally symmetric object is a sphere. So take a sphere such as a soccer ball or basketball that has on it a mark, logo, or unique lettering at some spot. Rotate the sphere about any axis: The rotat
ional symmetry of space implies that the shape of the sphere is invariant under rotation. In addition, if there is a mark on the sphere, then when you rotate the sphere by 360˚, the mark returns to its initial position. Go ahead; try it. Hold the ball in both hands and rotate it by 360˚ until the mark returns.

  That’s not so awkward, you may say. But that’s because you have not yet demonstrated the true rotational symmetry of space. To demonstrate this symmetry requires fancier moves. Now hold the ball cupped in one hand, palm facing up. Your goal is to rotate the sphere while keeping your palm up. This is trickier, but if Michael Jordan can do it, so can you.

  The steps are as follows:

  Keeping your palm up, rotate the ball inward toward your body. At 90˚, one quarter of a full rotation, the ball is comfortably tucked under your arm.

  Keep on rotating in the same direction, palm up. At 180˚, half a rotation, your arm sticks out in back of your body to keep the ball cupped in your palm.

  As you keep rotating to 270˚, three-quarters of a rotation, in order to maintain your palm facing up, your arm sticks awkwardly out to the side, ball precariously perched on top.

  At this point, you may feel that it is impossible to rotate the last 90˚ to complete one full rotation. If you try, however, you will find that you can continue rotating the ball keeping your palm up by raising your upper arm and bending your elbow so that your forearm sticks straight forward. The ball has now rotated by 360˚—one full rotation. If you’ve done everything right, however, your arm should be crooked in a maximally painful and awkward position.

  To relieve the pain, continue rotating by an additional 90˚ to one-and-a-quarter turns, palm up all the time. The ball should now be hovering over your head, and the painful tension in your shoulder should be somewhat lessened.

  Finally, like a waiter presenting a tray containing the pièce de resistance, continue the motion for the final three-quarters of a turn, ending with the ball and your arm (what a relief!) back in its original position.

  If you have managed to perform these steps correctly and without personal damage, you will find that the trajectory of the ball has traced out a kind of twisty figure eight, or infinity sign (∞), in space and has rotated around not once but twice. The true symmetry of space is not rotation by 360˚ but by 720˚.

  Although this exercise might seem no more than a fancy and painful basketball move, the fact that the true symmetry of space is rotation not once but twice has profound consequences for the nature of the physical world at its most microscopic level. It implies that “balls,” such as electrons, attached to a distant point by flexible and deformable “strings,” such as magnetic field lines, must be rotated around twice to return to their original configuration. Digging deeper, the twofold rotational nature of spherical symmetry implies that two electrons, both spinning in the same direction, cannot be placed in the same place at the same time. This exclusion principle in turn underlies the stability of matter. If the true symmetry of space were rotating around only once, then all the atoms of your body would collapse into nothingness in a tiny fraction of a second. Fortunately, however, the true symmetry of space consists of rotating around twice, and your atoms are stable, a fact that should console you as you ice your shoulder.

  THE PIGEONHOLE PRINCIPLE REVISITED

  CHARLES SEIFE

  Professor of journalism, New York University; former writer, Science; author, Proofiness: The Dark Arts of Mathematical Deception

  Sometimes even the simple act of counting can tell you something profound. One day, back in the late 1990s, when I was a correspondent for New Scientist magazine, I got an e-mail from a flack waxing rhapsodic about an extraordinary piece of software. It was a revolutionary data-compression program so efficient it would squash every digital file by 95 percent or more without losing a single bit of data. Wouldn’t my magazine jump at the chance to tell the world about the computer program that would make hard drives hold twenty times more information than before?

  No, my magazine wouldn’t.

  No such compression algorithm could possibly exist; it was the algorithmic equivalent of a perpetual-motion machine. The software was a fraud. The reason: the pigeonhole principle.

  The pigeonhole principle is a simple counting argument. It says that if you’ve got n pigeons and manage to stuff them into fewer than n boxes, then at least one box must have more than one pigeon in it. As blindingly obvious as this is, it’s a powerful tool. For example, imagine that the compression software really worked as advertised and every file shrank by a factor of 20 with no loss of fidelity. Every single file 2,000 bits long would be squashed down to a mere 100 bits, and then, when the algorithm was reversed, it would expand back into its original form, unscathed.

  When compressing files, you bump up against the pigeonhole principle. There are many more 2,000-bit pigeons (22000, to be exact) than 100-bit boxes (2100). If an algorithm stuffs the former into the latter, at least one box must contain multiple pigeons. Take that box—that 100-bit file—and reverse the algorithm, expanding the file into its original 2,000-bit form. You can’t! Since there are multiple 2,000-bit files that all wind up being squashed into the same 100-bit file, the algorithm has no way of knowing which one was the true original—it can’t reverse the compression.

  The pigeonhole principle puts an absolute limit on what a compression algorithm can do. It can compress some files, often dramatically, but it can’t compress them all—at least, not if you insist on perfect fidelity.

  Counting arguments similar to this one have opened up entire new realms for us to explore. The German mathematician Georg Cantor used a kind of reverse pigeonhole-principle technique to show that it was impossible to fit the real numbers into boxes labeled by the integers—even though there is an infinite number of integers. The almost unthinkable consequence was that there are different levels of infinity. The infinity of the integers is dwarfed by the infinity of the reals, which, in turn, is dwarfed by yet another infinity and another infinity on top of that . . . an infinity of infinities, all unexplored until we learned to count them.

  Taking the pigeonhole principle into deep space has an even stranger consequence. A principle in physics, the holographic bound, implies that in any finite volume of space there is only a finite number of possible configurations of matter and energy. If, as cosmologists tend to believe, the universe is infinite, there is an infinite number of visible-universe-size volumes out there—enormous cosmos-size bubbles containing matter and energy. And if space is more or less homogeneous, there’s nothing particularly special about the cosmos-size bubble we live in. These assumptions, taken together, lead to a stunning conclusion. Infinite universe-size bubbles, with only a finite number of configurations of the matter and energy in each, mean that there’s not just an exact copy of our universe—and our Earth—out there; the transfinite version of the pigeonhole principle states that there’s an infinite number of copies of every (technically, “almost every,” which has a precise mathematical definition) possible universe. Not only are there infinite copies of you on infinite alternate Earths, there are infinite copies of countless variations on the theme: versions of you with a prehensile tail, versions of you with multiple heads, versions of you that have made a career juggling carnivorous rabbitlike animals in exchange for costume jewelry. Even something as simple as counting one, two, three can lead you to bizarre and unexpected realms.

  MOORE’S LAW

  RODNEY A. BROOKS

  Roboticist; Panasonic Professor of Robotics, emeritus, MIT; founder, chairman & CTO, Heartland Robotics, Inc.; author, Flesh and Machines: How Robots Will Change Us

  Moore’s Law originated in a four-page 1965 magazine article written by Gordon Moore, then at Fairchild Semiconductor and later one of the founders of Intel. In it, he predicted that the number of components on a single integrated circuit would rise from the then-current number of roughly 26 to roughly 216 in the following ten years—that is, the number of components would double every y
ear. He based this prediction on four empirical data points and one null data point, fitting a straight line on a graph plotting the log of the number of components on a single chip against a linear scale of calendar years. Intel later amended Moore’s Law to say that “the number of transistors on a chip roughly doubles every two years.”

  Moore’s Law is rightly seen as the fundamental driver of the information technology revolution in our world over the last fifty years. Doubling the number of transistors every so often has made our computers twice as powerful for the same price, doubled the amount of data they can store or display, made them twice as fast, made them smaller, made them cheaper, and in general improved them in every possible way by a factor of 2 on a clockwork schedule.

  But why does it happen? Automobiles have not obeyed Moore’s Law; neither have batteries, nor clothing, nor food production, nor the level of political discourse. All but the last have demonstrably improved due to the influence of Moore’s Law, but none has had the same relentless exponential improvements.

  The most elegant explanation for what makes Moore’s Law possible is that digital logic is all about an abstraction—and, in fact, a one-bit abstraction, a yes/no answer to a question—and that abstraction is independent of physical bulk.

  In a world that consists entirely of piles of red sand and piles of green sand, the size of the piles is irrelevant. A pile is either red or green, and you can take away half the pile, and it’s still either a pile of red sand or a pile of green sand. And you can take away another half, and another half, and so on, and still the abstraction is maintained. And repeated halving at a constant rate makes an exponential.

  That’s why Moore’s Law works for digital technology and doesn’t work for technologies that require physical strength, or physical bulk, or must deliver certain amounts of energy. Digital technology uses physics to maintain an abstraction and nothing more.