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Oct
29th
Tue
permalink

Beauty of Mathematics

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without the gorgeous trappings of painting or music."

Betrand Russell, British philosopher, logician, mathematician, historian, and social critic (1872-1970)

By Yann Pineill & Nicolas Lefaucheux, parachutes.tv

Oct
28th
Mon
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Douglas Hofstadter: The Man Who Would Teach Machines to Think

"All the limitative theorems of metamathematics and the theory of computation suggest that once the ability to represent your own structure has reached a certain critical point, that is the kiss of death: it guarantees that you can never represent yourself totally. Gödel’s Incompleteness Theorem, Church’s Undecidability Theorem, Turing’s Halting Theorem, Tarski’s Truth Theorem — all have the flavour of some ancient fairy tale which warns you that “To seek self-knowledge is to embark on a journey which … will always be incomplete, cannot be charted on any map, will never halt, cannot be described.”

Douglas Hofstadter, 1979, cited in Vinod K. Wadhawan, Complexity Explained: 17. Epilogue, Nirmukta, 04 April 2010.

image M. C. Escher, Print Gallery. Hofstadter calls this Escher work a “pictorial parable for Godel’s Incompleteness Theorem.” Why? Look to the center of the painting, is there any way logical way to complete it? — source (Click picture to enlarge)

"On [Douglas] Hofstadter's office wall is a somewhat tattered reproduction of one of his favorite mind-twisting M. C. Escher prints, The Print Gallery.” In it, a boy stands looking at a print depicting a town where a woman looks down from her window at the roof of a gallery in which - yes - the boy stands looking at the print. We appreciate the paradox without being thrown by it, because we are outside looking in. Something like that creates our own unfathomable feelings of self. The self, subject and object, perceiver and perceived, is everywhere in the paradox.

It is a ”circling back,” the Tortoise tells Achilles, ”of a complex representation of the system together with its representations of all the rest of the world.”

”It is just so hard, emotionally,” Achilles tells the Tortoise, ”to acknowledge that a ‘soul’ emerges from so physical a system.” (…)

But philosophers like  [Daniel] Dennett believe, with Hofstadter, that scientific answers can be found without cheating by reducing the question to a manageable scale. (…) [T]he danger of looking only at the lowest biological level is in losing sight of the essential humanity that, in Hofstadter’s view, exists in the pattern and in the paradox. ”There seems to be no alternative to accepting some sort of incomprehensible quality to existence,” as Hofstadter puts it. ”Take your pick.” 

James Gleick, on Douglas R. Hofstadter in Exploring the Labyrinth of the Mind, The New York Times, August 21, 1983.

"In 1931, the Austrian-born logician Kurt Gödel had famously shown how a mathematical system could make statements not just about numbers but about the system itself. Consciousness, Hofstadter wanted to say, emerged via just the same kind of “level-crossing feedback loop.” (…)

“Cognition is recognition,” he likes to say. He describes “seeing as” as the essential cognitive act: you see some lines as “an A,” you see a hunk of wood as “a table,” you see a meeting as “an emperor-has-no-clothes situation” and a friend’s pouting as “sour grapes” and a young man’s style as “hipsterish” and on and on ceaselessly throughout your day. That’s what it means to understand. But how does understanding work? For three decades, Hofstadter and his students have been trying to find out, trying to build “computer models of the fundamental mechanisms of thought.”

“At every moment,” Hofstadter writes in Surfaces and Essences, his latest book (written with Emmanuel Sander), “we are simultaneously faced with an indefinite number of overlapping and intermingling situations.” It is our job, as organisms that want to live, to make sense of that chaos. We do it by having the right concepts come to mind. This happens automatically, all the time. Analogy is Hofstadter’s go-to word. The thesis of his new book, which features a mélange of A’s on its cover, is that analogy is “the fuel and fire of thinking,” the bread and butter of our daily mental lives.

“Look at your conversations,” he says. “You’ll see over and over again, to your surprise, that this is the process of analogy-making.” Someone says something, which reminds you of something else; you say something, which reminds the other person of something else—that’s a conversation. It couldn’t be more straightforward. But at each step, Hofstadter argues, there’s an analogy, a mental leap so stunningly complex that it’s a computational miracle: somehow your brain is able to strip any remark of the irrelevant surface details and extract its gist, its “skeletal essence,” and retrieve, from your own repertoire of ideas and experiences, the story or remark that best relates.

“Beware,” he writes, “of innocent phrases like ‘Oh, yeah, that’s exactly what happened to me!’ … behind whose nonchalance is hidden the entire mystery of the human mind.” (…)

[Hofstadter] spends most of his time in his study, two rooms on the top floor of his house, carpeted, a bit stuffy, and messier than he would like. His study is the center of his world. He reads there, listens to music there, studies there, draws there, writes his books there, writes his e‑mails there. (Hofstadter spends four hours a day writing e‑mail. “To me,” he has said, “an e‑mail is identical to a letter, every bit as formal, as refined, as carefully written … I rewrite, rewrite, rewrite, rewrite all of my e‑mails, always.”) He lives his mental life there, and it shows. Wall-to-wall there are books and drawings and notebooks and files, thoughts fossilized and splayed all over the room. It’s like a museum for his binges, a scene out of a brainy episode of Hoarders.

“Anything that I think about becomes part of my professional life,” he says. Daniel Dennett, who co-edited The Mind’s I with him, has explained that “what Douglas Hofstadter is, quite simply, is a phenomenologist, a practicing phenomenologist, and he does it better than anybody else. Ever.” He studies the phenomena—the feelings, the inside actions—of his own mind. “And the reason he’s good at it,” Dennett told me, “the reason he’s better than anybody else, is that he is very actively trying to have a theory of what’s going on backstage, of how thinking actually happens in the brain.” (…)

He makes photocopies of his notebook pages, cuts them up with scissors, and stores the errors in filing cabinets and labeled boxes around his study.

For Hofstadter, they’re clues. “Nobody is a very reliable guide concerning activities in their mind that are, by definition, subconscious,” he once wrote. “This is what makes vast collections of errors so important. In an isolated error, the mechanisms involved yield only slight traces of themselves; however, in a large collection, vast numbers of such slight traces exist, collectively adding up to strong evidence for (and against) particular mechanisms.” Correct speech isn’t very interesting; it’s like a well-executed magic trick—effective because it obscures how it works. What Hofstadter is looking for is “a tip of the rabbit’s ear … a hint of a trap door.

As the wind tunnel was to the Wright brothers, so the computer is to FARG. The quick unconscious chaos of a mind can be slowed down on the computer, or rewound, paused, even edited. In Hofstadter’s view, this is the great opportunity of artificial intelligence. Parts of a program can be selectively isolated to see how it functions without them; parameters can be changed to see how performance improves or degrades. When the computer surprises you—whether by being especially creative or especially dim-witted—you can see exactly why. “I have always felt that the only hope of humans ever coming to fully understand the complexity of their minds,” Hofstadter has written, “is by modeling mental processes on computers and learning from the models’ inevitable failures.” (…)

But very few people, even admirers of GEB, know about the book or the programs it describes. And maybe that’s because FARG’s programs are almost ostentatiously impractical. Because they operate in tiny, seemingly childish “microdomains.” Because there is no task they perform better than a human.

The modern era of mainstream AI—an era of steady progress and commercial success that began, roughly, in the early 1990s and continues to this day—is the long unlikely springtime after a period, known as the AI Winter, that nearly killed off the field.

It came down to a basic dilemma. On the one hand, the software we know how to write is very orderly; most computer programs are organized like a well-run army, with layers of commanders, each layer passing instructions down to the next, and routines that call subroutines that call subroutines. On the other hand, the software we want to write would be adaptable—and for that, a hierarchy of rules seems like just the wrong idea. Hofstadter once summarized the situation by writing, “The entire effort of artificial intelligence is essentially a fight against computers’ rigidity.”

Machine learning

The “expert systems” that had once been the field’s meal ticket were foundering because of their brittleness. Their approach was fundamentally broken. Take machine translation from one language to another, long a holy grail of AI. The standard attack involved corralling linguists and translators into a room and trying to convert their expertise into rules for a program to follow. The standard attack failed for reasons you might expect: no set of rules can ever wrangle a human language; language is too big and too protean; for every rule obeyed, there’s a rule broken.

If machine translation was to survive as a commercial enterprise—if AI was to survive—it would have to find another way. Or better yet, a shortcut.

The technique is called “machine learning.” The goal is to make a device that takes an English sentence as input and spits out a French sentence. One such device, of course, is the human brain—but the whole point is to avoid grappling with the brain’s complexity. So what you do instead is start with a machine so simple, it almost doesn’t work: a machine, say, that randomly spits out French words for the English words it’s given.

Imagine a box with thousands of knobs on it. Some of these knobs control general settings: given one English word, how many French words, on average, should come out? And some control specific settings: given jump, what is the probability that shot comes next? The question is, just by tuning these knobs, can you get your machine to convert sensible English into sensible French?

It turns out that you can. What you do is feed the machine English sentences whose French translations you already know. (Candide, for example, used 2.2 million pairs of sentences, mostly from the bilingual proceedings of Canadian parliamentary debates.) You proceed one pair at a time. After you’ve entered a pair, take the English half and feed it into your machine to see what comes out in French. If that sentence is different from what you were expecting—different from the known correct translation—your machine isn’t quite right. So jiggle the knobs and try again. After enough feeding and trying and jiggling, feeding and trying and jiggling again, you’ll get a feel for the knobs, and you’ll be able to produce the correct French equivalent of your English sentence.

By repeating this process with millions of pairs of sentences, you will gradually calibrate your machine, to the point where you’ll be able to enter a sentence whose translation you don’t know and get a reasonable result. And the beauty is that you never needed to program the machine explicitly; you never needed to know why the knobs should be twisted this way or that. (…)

Google has projects that gesture toward deeper understanding: extensions of machine learning inspired by brain biology; a “knowledge graph” that tries to map words, like Obama, to people or places or things. But the need to serve 1 billion customers has a way of forcing the company to trade understanding for expediency. You don’t have to push Google Translate very far to see the compromises its developers have made for coverage, and speed, and ease of engineering. Although Google Translate captures, in its way, the products of human intelligence, it isn’t intelligent itself. It’s like an enormous Rosetta Stone, the calcified hieroglyphics of minds once at work. (…)

Ever since he was about 15, Hofstadter has read The Catcher in the Rye once every 10 years. In the fall of 2011, he taught an undergraduate seminar called “Why Is J. D. Salinger’s The Catcher in the Rye a Great Novel?” He feels a deep kinship with Holden Caulfield. When I mentioned that a lot of the kids in my high-school class didn’t like Holden—they thought he was a whiner—Hofstadter explained that “they may not recognize his vulnerability.” You imagine him standing like Holden stood at the beginning of the novel, alone on the top of a hill, watching his classmates romp around at the football game below. “I have too many ideas already,” Hofstadter tells me. “I don’t need the stimulation of the outside world.” (…)

“Ars longa, vita brevis,” Hofstadter likes to say. “I just figure that life is short. I work, I don’t try to publicize. I don’t try to fight.”

There’s an analogy he made for me once. Einstein, he said, had come up with the light-quantum hypothesis in 1905. But nobody accepted it until 1923. “Not a soul,” Hofstadter says. “Einstein was completely alone in his belief in the existence of light as particles—for 18 years.

“That must have been very lonely.” “

— James Somers, to read the full article click The Man Who Would Teach Machines to Think, The Atlantic, Oct 23 2013

Douglas Hofstadter, is an American professor of cognitive science whose research focuses on the sense of “I”, consciousness, analogy-making, artistic creation, literary translation, and discovery in mathematics and physics. He is best known for his book Gödel, Escher, Bach: an Eternal Golden Braid, first published in 1979. It won both the Pulitzer Prize for general non-fiction.

See also:

The Mathematical Art of M.C. Escher, Lapidarium notes

Feb
10th
Sun
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Universality: In Mysterious Pattern, Math and Nature Converge

image

"In 1999, while sitting at a bus stop in Cuernavaca, Mexico, a Czech physicist named Petr Šeba noticed young men handing slips of paper to the bus drivers in exchange for cash. It wasn’t organized crime, he learned, but another shadow trade: Each driver paid a “spy” to record when the bus ahead of his had departed the stop. If it had left recently, he would slow down, letting passengers accumulate at the next stop. If it had departed long ago, he sped up to keep other buses from passing him. This system maximized profits for the drivers. And it gave Šeba an idea. (…)

The interaction between drivers caused the spacing between departures to exhibit a distinctive pattern previously observed in quantum physics experiments. (…) “We felt here some kind of similarity with quantum chaotic systems.” (…) A “spy” network makes the decentralized bus system more efficient. As a consequence, the departure times of buses exhibit a ubiquitous pattern known as “universality.” (…)

Subatomic particles have little to do with decentralized bus systems. But in the years since the odd coupling was discovered, the same pattern has turned up in other unrelated settings. Scientists now believe the widespread phenomenon, known as “universality,” stems from an underlying connection to mathematics, and it is helping them to model complex systems from the internet to Earth’s climate. (…)

                image

The red pattern exhibits a precise balance of randomness and regularity known as “universality,” which has been observed in the spectra of many complex, correlated systems. In this spectrum, a mathematical formula called the “correlation function” gives the exact probability of finding two lines spaced a given distance apart. (…)

The pattern was first discovered in nature in the 1950s in the energy spectrum of the uranium nucleus, a behemoth with hundreds of moving parts that quivers and stretches in infinitely many ways, producing an endless sequence of energy levels. In 1972, the number theorist Hugh Montgomery observed it in the zeros of the Riemann zeta function, a mathematical object closely related to the distribution of prime numbers. In 2000, Krbálek and Šeba reported it in the Cuernavaca bus system. And in recent years it has shown up in spectral measurements of composite materials, such as sea ice and human bones, and in signal dynamics of the Erdös–Rényi model, a simplified version of the internet named for Paul Erdös and Alfréd Rényi. (…)

Each of these systems has a spectrum — a sequence like a bar code representing data such as energy levels, zeta zeros, bus departure times or signal speeds. In all the spectra, the same distinctive pattern appears: The data seem haphazardly distributed, and yet neighboring lines repel one another, lending a degree of regularity to their spacing. This fine balance between chaos and order, which is defined by a precise formula, also appears in a purely mathematical setting: It defines the spacing between the eigenvalues, or solutions, of a vast matrix filled with random numbers. (…)

It seems to be a law of nature,” said Van Vu, a mathematician at Yale University who, with Terence Tao of the University of California, Los Angeles, has proven universality for a broad class of random matrices.

Universality is thought to arise when a system is very complex, consisting of many parts that strongly interact with each other to generate a spectrum. The pattern emerges in the spectrum of a random matrix, for example, because the matrix elements all enter into the calculation of that spectrum. But random matrices are merely “toy systems” that are of interest because they can be rigorously studied, while also being rich enough to model real-world systems, Vu said. Universality is much more widespread. Wigner’s hypothesis (named after Eugene Wigner, the physicist who discovered universality in atomic spectra) asserts that all complex, correlated systems exhibit universality, from a crystal lattice to the internet.

     

Mathematicians are using random matrix models to study and predict some of the internet’s properties, such as the size of typical computer clusters. (Illustration: Matt Britt)

The more complex a system is, the more robust its universality should be, said László Erdös of the University of Munich, one of Yau’s collaborators. “This is because we believe that universality is the typical behavior.”

— Natalie Wolchover, In Mysterious Pattern, Math and Nature Converge, Wired, Feb 6, 2013. (Photo: Marco de Leija)

See also:

Mathematics of Disordered Quantum Systems and Matrices, IST Austria.

Jan
22nd
Tue
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Kevin Slavin: How algorithms shape our world

“But the Turing test cuts both ways. You can’t tell if a machine has gotten smarter or if you’ve just lowered your own standards of intelligence to such a degree that the machine seems smart. If you can have a conversation with a simulated person presented by an AI program, can you tell how far you’ve let your sense of personhood degrade in order to make the illusion work for you?

People degrade themselves in order to make machines seem smart all the time. Before the crash, bankers believed in supposedly intelligent algorithms that could calculate credit risks before making bad loans. We ask teachers to teach to standardized tests so a student will look good to an algorithm. We have repeatedly demonstrated our species’ bottomless ability to lower our standards to make information technology look good. Every instance of intelligence in a machine is ambiguous.

The same ambiguity that motivated dubious academic AI projects in the past has been repackaged as mass culture today. Did that search engine really know what you want, or are you playing along, lowering your standards to make it seem clever? While it’s to be expected that the human perspective will be changed by encounters with profound new technologies, the exercise of treating machine intelligence as real requires people to reduce their mooring to reality.”

Jaron Lanier, You are Not a Gadget (2010)

Kevin Slavin argues that we’re living in a world designed for — and increasingly controlled by — algorithms. In this riveting talk from TEDGlobal, he shows how these complex computer programs determine: espionage tactics, stock prices, movie scripts, and architecture.

"We’re writing things (…) that we can no longer read. And we’ve rendered something illegible, and we’ve lost the sense of what’s actually happening in this world that we’ve made. (…)

“We’re running through the United States with dynamite and rock saws so that an algorithm can close the deal three microseconds faster, all for a communications framework that no human will ever know; that’s a kind of manifest destiny.”

Kevin Slavin, Entrepreneur, Raconteur Assistant Professor of Media Arts and Sciences, MIT Media Lab, Kevin Slavin: How algorithms shape our world, TED, July 2011.

See also:

☞ Jane Wakefield, When algorithms control the world, BBC, Aug 23, 2011.

Dec
11th
Tue
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Researchers discover surprising complexities in the way the brain makes mental maps

                     image
Spatial location is closely connected to the formation of new memories. Until now, grid cells were thought to be part of a single unified map system. New findings from the Norwegian University of Science and Technology demonstrate that the grid system is in fact composed of a number of independent grid maps, each with unique properties. Each map displays a particular resolution (mesh size), and responds independently to changes in the environment. A system of several distinct grid maps (illustrated on left) can support a large number of unique combinatorial codes used to associate new memories formed with specific spatial information (illustrated on right).

Your brain has at least four different senses of location – and perhaps as many as 10. And each is different, according to new research from the Kavli Institute for Systems Neuroscience, at the Norwegian University of Science and Technology. (…)

The findings, published in the 6 December 2012 issue of Nature, show that rather than just a single sense of location, the brain has a number of “modules” dedicated to self-location. Each module contains its own internal GPS-like mapping system that keeps track of movement, and has other characteristics that also distinguishes one from another.

"We have at least four senses of location," says Edvard Moser, director of the Kavli Institute. "Each has its own scale for representing the external environment, ranging from very fine to very coarse. The different modules react differently to changes in the environment. Some may scale the brain’s inner map to the surroundings, others do not. And they operate independently of each other in several ways."

This is also the first time that researchers have been able to show that a part of the brain that does not directly respond to sensory input, called the association cortex, is organized into modules. The research was conducted using rats. (…)

Technical breakthroughs

A rat’s brain is the size of a grape, while the area that keeps track of the sense of location and memory is comparable in size to a small grape seed. This tiny area holds millions of nerve cells.

A research team of six people worked for more than four years to acquire extensive electrophysiological measurements in this seed-sized region of the brain. New measurement techniques and a technical breakthrough made it possible for Hanne Stensola and her colleagues to measure the activity in as many as 186 grid cells of the same rat brain. A grid cell is a specialized cell named for its characteristic of creating hexagonal grids in the brain’s mental map of its surroundings.

"We knew that the ‘grid maps’ in this area of the brain had resolutions covering different scales, but we did not know how independent the scales were of each other," Stensola said. "We then discovered that the maps were organized in four to five modules with different scales, and that each of these modules reacted slightly differently to changes in their environment. This independence can be used by the brain to create new combinations - many combinations - which is a very useful tool for memory formation.

After analysing the activity of nearly 1000 grid cells, researchers were able to conclude that the brain has not just one way of making an internal map of its location, but several. Perhaps 10 different senses of location.

Perhaps 10 different senses of location

image
The entorhinal cortex is a part of the neocortex that represents space by way of brain cells that have GPS-like properties. Each cell describes the environment as a hexagonal grid mesh, earning them the name ‘grid cells’. The panels show a bird’s-eye view of a rat’s recorded movements (grey trace) in a 2.2x2.2 m box. Each panel shows the activity of one grid cell (blue dots) with a particular map resolution as the animal moved through the environment. Credit: Kavli Institute for Systems Neuroscience, NTNU

Institute director Moser says that while researchers are able to state with confidence that there are at least four different location modules, and have seen clear evidence of a fifth, there may be as many as 10 different modules.

He says, however, that researchers need to conduct more measurements before they will have covered the entire grid-cell area. “At this point we have measured less than half of the area,” he says.

Aside from the time and challenges involved in making these kinds of measurements, there is another good reason why researchers have not yet completed this task. The lower region of the sense of location area, the entorhinal cortex, has a resolution that is so coarse or large that it is virtually impossible to measure it.

"The thinking is that the coordinate points for some of these maps are as much as ten metres apart," explains Moser. "To measure this we would need to have a lab that is quite a lot larger and we would need time to test activity over the entire area. We work with rats, which run around while we make measurements from their brain. Just think how long it would take to record the activity in a rat if it was running back and forth exploring every nook and cranny of a football field. So you can see that we have some challenges here in scaling up our experiments."

New way to organize

Part of what makes the discovery of the grid modules so special is that it completely changes our understanding of how the brain physically organizes abstract functions. Previously, researchers have shown that brain cells in sensory systems that are directly adjacent to each other tend to have the same response pattern. This is how they have been able to create detailed maps of which parts of the sensory brain do what.

The new research shows that a modular organization is also found in the highest parts of the cortex, far away from areas devoted to senses or motor outputs. But these maps are different in the sense that they overlap or infiltrate other. It is thus not possible to locate the different modules with a microscope, because the cells that work together are intermingled with other modules in the same area.

“The various components of the grid map are not organized side by side,” explains Moser. “The various components overlap. This is the first time a brain function has been shown to be organized in this way at separate scales. We have uncovered a new way for neural network function to be distributed.”

A map and a constant

The researchers were surprised, however, when they started calculating the difference between the scales. They may have discovered an ingenious mathematical coding system, along with a number, a constant. (Anyone who has read or seen “The Hitchhiker’s Guide to the Galaxy” may enjoy this.) The scale for each sense of location is actually 42% larger than the previous one. “

We may not be able to say with certainty that we have found a mathematical constant for the way the brain calculates the scales for each sense of location, but it’s very funny that we have to multiply each measurement by 1.42 to get the next one. That is approximately equal to the square root of the number two,” says Moser.

Maps are genetically encoded

Moser thinks it is striking that the relationship between the various functional modules is so orderly. He believes this orderliness shows that the way the grid map is organized is genetically built in, and not primarily the result of experience and interaction with the environment.

So why has evolution equipped us with four or more senses of location?

Moser believes the ability to make a mental map of the environment arose very early in evolution. He explains that all species need to navigate, and that some types of memory may have arisen from brain systems that were actually developed for the brain’s sense of location.

“We see that the grid cells that are in each of the modules send signals to the same cells in the hippocampus, which is a very important component of memory,” explains Moser. “This is, in a way, the next step in the line of signals in the brain. In practice this means that the location cells send a different code into the hippocampus at the slightest change in the environment in the form of a new pattern of activity. So every tiny change results in a new combination of activity that can be used to encode a new memory, and, with input from the environment, becomes what we call memories.”

Researchers discover surprising complexities in the way the brain makes mental maps, Medical press, Dec 5, 2012.

The article is a part of doctoral research conducted by Hanne and Tor Stensola, and has been funded through an Advanced Investigator Grant that Edvard Moser was awarded by the European Research Council (ERC).

See also:

☞ Hanne Stensola, Tor Stensola, Trygve Solstad, Kristian Frøland, May-Britt Moser & Edvard I. Moser, The entorhinal grid map is discretized, Nature, 5 Dec 2012.
Mind & brain tag on Lapidarium notes

Jun
3rd
Sun
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Self as Symbol. The loopy nature of consciousness trips up scientists studying themselves

              
                                                          M.C. Escher’s “Drawing Hands”

"The consciousness problem remains popular on lists of problems that might never be solved.

Perhaps that’s because the consciousness problem is inherently similar to another famous problem that actually has been proved unsolvable: finding a self-consistent set of axioms for deducing all of mathematics. As the Austrian logician Kurt Gödel proved eight decades ago, no such axiomatic system is possible; any system as complicated as arithmetic contains true statements that cannot be proved within the system.

Gödel’s proof emerged from deep insights into the self-referential nature of mathematical statements. He showed how a system referring to itself creates paradoxes that cannot be logically resolved — and so certain questions cannot in principle be answered. Consciousness, in a way, is in the same logical boat. At its core, consciousness is self-referential awareness, the self’s sense of its own existence. It is consciousness itself that is trying to explain consciousness.

Self-reference, feedback loops, paradoxes and Gödel’s proof all play central roles in the view of consciousness articulated by Douglas Hofstadter in his 2007 book I Am a Strange Loop. Hofstadter is (among other things) a computer scientist, and he views consciousness through lenses unfamiliar to most neuroscientists. In his eyes, it’s not so bizarre to compare math and numbers to the mind and consciousness. Math is, after all, deeply concerned with logic and reason — the stuff of thought. Mathematical paradoxes, Hofstadter points out, open up “profound questions concerning the nature of reasoning — and thus concerning the elusive nature of thinking — and thus concerning the mysterious nature of the human mind itself.”

Enter the loop

In particular, Hofstadter seizes on Gödel’s insight that a mathematical formula — a statement about a number — can itself be represented by a number. So you can take the number describing a formula and insert that number into the formula, which then becomes a statement about itself. Such a self-referential capability introduces a certain “loopiness” into mathematics, Hofstadter notes, something like the famous Escher print of a right hand drawing a left hand, which in turn is drawing the right hand. This “strange loopiness” in math suggested to Hofstadter that something similar is going on in human thought.

So when he titled his book “I Am a Strange Loop,” Hofstadter didn’t mean that he was personally loopy, but that the concept of an individual — a persistent identity, an “I,” that accompanies what people refer to as consciousness — is a loop of a certain sort. It’s a feedback loop, like the circuit that turns a whisper into an ear-piercing screech when the microphone whispered into is too close to the loudspeaker emitting the sound.

But consciousness is more than just an ordinary feedback loop. It’s a strange loop, which Hofstadter describes as a loop capable of perceiving patterns in its environment and assigning common symbolic meanings to sufficiently similar patterns. An acoustic feedback loop generates no symbols, just noise. A human brain, though, can assign symbols to patterns. While patterns of dots on a TV screen are just dots to a mosquito, to a person, the same dots evoke symbols, such as football players, talk show hosts or NCIS agents. Floods of raw sensory data trigger perceptions that fall into categories designated by “symbols that stand for abstract regularities in the world,” Hofstadter asserts. Human brains create vast repertoires of these symbols, conferring the “power to represent phenomena of unlimited complexity and thus to twist back and to engulf themselves via a strange loop.”

Consciousness itself occurs when a system with such ability creates a higher-level symbol, a symbol for the ability to create symbols. That symbol is the self. The I. Consciousness. “You and I are mirages that perceive themselves,” Hofstadter writes.

This self-generated symbol of the self operates only on the level of symbols. It has no access to the workings of nerve cells and neurotransmitters, the microscopic electrochemical machinery of neurobiological life. The symbols that consciousness contemplates don’t look much like the real thing, the way a map of Texas conveys nothing of the grass and dirt and asphalt and bricks that cover the physical territory.

And just like a map of Texas remains remarkably stable over many decades — it doesn’t change with each new pothole in a Dallas street — human self-identity remains stable over a lifetime, despite constant changes on the micro level of proteins and cells. As an individual grows, matures, changes in many minute ways, the conscious self’s identity remains intact, just as Texas remains Texas even as new skyscrapers rise in the cities, farms grow different crops and the Red River sometimes shifts the boundary with Oklahoma a bit.

If consciousness were merely a map, a convenient shortcut symbol for a complex mess of neurobiological signaling, perhaps it wouldn’t be so hard to figure out. But its mysteries multiply because the symbol is generated by the thing doing the symbolizing. It’s like Gödel’s numbers that refer to formulas that represent truths about numbers; this self-referentialism creates unanswerable questions, unsolvable problems.

A typical example of such a Gödelian paradox is the following sentence: This sentence cannot be true.

Is that sentence true? Obviously not, because it says it isn’t true. But wait — then it is true. Except that it can’t be. Self-referential sentences seem to have it both ways — or neither way.

And so perceptual systems able to symbolize themselves — self-referential minds — can’t be explained just by understanding the parts that compose them. Simply describing how electric charges travel along nerve cells, how small molecules jump from one cell to another, how such signaling sends messages from one part of the brain to another — none of that explains consciousness any more than knowing the English alphabet letter by letter (and even the rules of grammar) will tell you the meaning of Shakespeare’s poetry.

Hofstadter does not contend, of course, that all the biochemistry and cellular communication is irrelevant. It provides the machinery for perceiving and symbolizing that makes the strange loop of consciousness possible. It’s just that consciousness does not itself deal with molecules and cells; it copes with thoughts and emotions, hopes and fears, ideas and desires. Just as numbers can represent the complexities of all of mathematics (including numbers), a brain can represent the complexities of experience (including the brain itself). Gödel’s proof showed that math is “incomplete”; it contains truths that can’t be proven. And consciousness is a truth of a sort that can’t be comprehended within a system of molecules and cells alone.

That doesn’t mean that consciousness can never be understood — Gödel’s work did not undermine human understanding of mathematics, it enriched it. And so the realization that consciousness is self-referential could also usher in a deeper understanding of what the word means — what it symbolizes.

Information handler

Viewed as a symbol, consciousness is very much like many of the other grand ideas of science. An atom is not so much a thing as an idea, a symbol for matter’s ultimate constituents, and the modern physical understanding of atoms bears virtually no resemblance to the original conception in the minds of the ancient Greeks who named them. Even Francis Crick’s gene made from DNA turned out to be much more elusive than the “unit of heredity” imagined by Gregor Mendel in the 19th century. The later coinage of the word gene to describe such units long remained a symbol; early 20th century experiments allowed geneticists to deduce a lot about genes, but nobody really had a clue what a gene was.

“In a sense people were just as vague about what genes were in the 1920s as they are now about consciousness,” Crick said in 1998. “It was exactly the same. The more professional people in the field, which was biochemistry at that time, thought that it was a problem that was too early to tackle.”

It turned out that with genes, their physical implementation didn’t really matter as much as the information storage and processing that genes engaged in. DNA is in essence a map, containing codes allowing one set of molecules to be transcribed into others necessary for life. It’s a lot easier to make a million copies of a map of Texas than to make a million Texases; DNA’s genetic mapping power is the secret that made the proliferation of life on Earth possible. Similarly, consciousness is deeply involved in representing information (with symbols) and putting that information together to make sense of the world. It’s the brain’s information processing powers that allow the mind to symbolize itself.

Koch believes that focusing on information could sharpen science’s understanding of consciousness. A brain’s ability to find patterns in influxes of sensory data, to send signals back and forth to integrate all that data into a coherent picture of reality and to trigger appropriate responses all seem to be processes that could be quantified and perhaps even explained with the math that describes how information works.

“Ultimately I think the key thing that matters is information,” Koch says. “You have these causal interactions and they can be quantified using information theory. Somehow out of that consciousness has to arrive.” An inevitable consequence of this point of view is that consciousness doesn’t care what kind of information processors are doing all its jobs — whether nerve cells or transistors.

“It’s not the stuff out of which your brain is made,” Koch says. “It’s what that stuff represents that’s conscious, and that tells us that lots of other systems could be conscious too.”

Perhaps, in the end, it will be the ability to create unmistakable features of consciousness in some stuff other than a biological brain that will signal success in the quest for an explanation. But it’s doubtful that experimentally exposing consciousness as not exclusively human will displace humankind’s belief in its own primacy. People will probably always believe that it can only be the strange loop of human consciousness that makes the world go ’round.

“We … draw conceptual boundaries around entities that we easily perceive, and in so doing we carve out what seems to us to be reality,” Hofstadter wrote. “The ‘I’ we create for each of us is a quintessential example of such a perceived or invented reality, and it does such a good job of explaining our behavior that it becomes the hub around which the rest of the world seems to rotate.”

Tom Siegfried, American journalist, author, Self as Symbol, Science News, Feb 11, 2012.

See also:

☞ Laura Sanders, Ph.D. in Molecular Biology from the University of Southern California in Los Angeles, Emblems of Awareness, Science News, Feb 11, 2012.

                                            Degress of thought

                                          (Credit: Stanford University)

"Awareness typically tracks with wakefulness — especially in normal states of consciousness (bold). People in coma or under general anesthesia score low on both measures, appearing asleep with no signs of awareness. Sometimes, wakefulness and awareness become uncoupled, such as among people in a persistent vegetative state. In this case, a person seems awake and is sometimes able to move but is unaware of the surroundings."  (…)

“Messages constantly zing around the brain in complex patterns, as if trillions of tiny balls were simultaneously dropped into a pinball machine, each with a prescribed, mission-critical path. This constant flow of information might be what creates consciousness — and interruptions might destroy it. (…)

“If you knock on a wooden table or a bucket full of nothing, you get different noises,” Massimini says. “If you knock on the brain that is healthy and conscious, you get a very complex noise.” (…)

In the same way that “life” evades a single, clear definition (growth, reproduction or a healthy metabolism could all apply), consciousness might turn out to be a collection of remarkable phenomena, Seth says. “If we can explain different aspects of consciousness, then my hope is that it will start to seem slightly less mysterious that there is consciousness at all in the universe.” (…)

Recipe for consciousness

Somehow a sense of self emerges from the many interactions of nerve cells and neurotransmitters in the brain — but a single source behind the phenomenon remains elusive.

            

                                                      Illustration: Nicolle Rager Fuller

1. Parietal cortex Brain activity in the parietal cortex is diminished by anesthetics, when people fall into a deep sleep and in people in a vegetative state or coma. There is some evidence suggesting that the parietal cortex is where first-person perspective is generated.

2. Frontal cortex Some researchers argue that parts of the frontal cortex (along with connections to the parietal cortex) are required for consciousness. But other scientists point to a few studies in which people with damaged frontal areas retain consciousness.

3. Claustrum An enigmatic, thin sheet of neural tissue called the claustrum has connections with many other regions. Though the structure has been largely ignored by modern scientists, Francis Crick became keenly interested in the claustrum’s role in consciousness just before his death in 2004.

4. Thalamus As one of the brain’s busiest hubs of activity, the thalamus is believed by many to have an important role in consciousness. Damage to even a small spot in the thalamus can lead to consciousness disorders.

5. Reticular activating system Damage to a particular group of nerve cell clusters, called the reticular activating system and found in the brain stem, can render a person comatose.”

☞ Bruce Hood, The Self Illusion: How the Brain Creates Identity
Theories of consciousness. Make Up Your Own Mind (visualization)
Malcolm MacIver on why did consciousness evolve, and how can we modify it?
Consciousness tag on Lapidarium

Mar
26th
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Science historian George Dyson: Unravelling the digital code
      
                                                     George Dyson (Photo: Wired)

"It was not made for those who sell oil or sardines."

— G. W. Leibniz, ca. 1674, on his calculating machine

A universe of self-replicating code

Digital organisms, while not necessarily any more alive than a phone book, are strings of code that replicate and evolve over time. Digital codes are strings of binary digits — bits. Google is a fantastically large number, so large it is almost beyond comprehension, distributed and replicated across all kinds of hosts. When you click on a link, you are replicating the string of code that it links to. Replication of code sequences isn’t life, any more than replication of nucleotide sequences is, but we know that it sometimes leads to life.

Q [Kevin Kelly]: Are we in that digital universe right now, as we talk on the phone?

George Dyson: Sure. You’re recording this conversation using a digital recorder — into an empty matrix of addresses on a microchip that is being filled up at 44 kilobytes per second. That address space full of numbers is the digital universe.

Q: How fast is this universe expanding?

G.D.: Like our own universe at the beginning, it’s more exploding than expanding. We’re all so immersed in it that it’s hard to perceive. Last time I checked, the digital universe was expanding at the rate of five trillion bits per second in storage and two trillion transistors per second on the processing side. (…)

Q: Where is this digital universe heading?

G.D.: This universe is open to the evolution of all kinds of things. It’s cycling faster and faster. Even with Google and YouTube and Facebook, we can’t consume it all. And we aren’t aware what this space is filling up with. From a human perspective, computers are idle 99 per cent of the time. While they’re waiting for us to come up with instructions, computation is happening without us, as computers write instructions for each other. As Turing showed, this space can’t be supervised. As the digital universe expands, so does this wild, undomesticated side.”

— George Dyson interviewed by Kevin Kelly in Science historian George Dyson: Unravelling the digital code, Wired, Mar 5, 2012.

"Just as we later worried about recombinant DNA, what if these things escaped? What would they do to the world? Could this be the end of the world as we know it if these self-replicating numerical creatures got loose?

But, we now live in a world where they did get loose—a world increasingly run by self-replicating strings of code. Everything we love and use today is, in a lot of ways, self-reproducing exactly as Turing, von Neumann, and Barricelli prescribed. It’s a very symbiotic relationship: the same way life found a way to use the self-replicating qualities of these polynucleotide molecules to the great benefit of life as a whole, there’s no reason life won’t use the self-replicating abilities of digital code, and that’s what’s happening. If you look at what people like Craig Venter and the thousand less-known companies are doing, we’re doing exactly that, from the bottom up. (…)

What’s, in a way, missing in today’s world is more biology of the Internet. More people like Nils Barricelli to go out and look at what’s going on, not from a business or what’s legal point of view, but just to observe what’s going on.

Many of these things we read about in the front page of the newspaper every day, about what’s proper or improper, or ethical or unethical, really concern this issue of autonomous self-replicating codes. What happens if you subscribe to a service and then as part of that service, unbeknownst to you, a piece of self-replicating code inhabits your machine, and it goes out and does something else? Who is responsible for that? And we’re in an increasingly gray zone as to where that’s going. (…)

Why is Apple one of the world’s most valuable companies? It’s not only because their machines are so beautifully designed, which is great and wonderful, but because those machines represent a closed numerical system. And they’re making great strides in expanding that system. It’s no longer at all odd to have a Mac laptop. It’s almost the normal thing.

But I’d like to take this to a different level, if I can change the subject… Ten or 20 years ago I was preaching that we should look at digital code as biologists: the Darwin Among the Machines stuff. People thought that was crazy, and now it’s firmly the accepted metaphor for what’s going on. And Kevin Kelly quoted me in Wired, he asked me for my last word on what companies should do about this. And I said, “Well, they should hire more biologists.”

But what we’re missing now, on another level, is not just biology, but cosmology. People treat the digital universe as some sort of metaphor, just a cute word for all these products. The universe of Apple, the universe of Google, the universe of Facebook, that these collectively constitute the digital universe, and we can only see it in human terms and what does this do for us?

We’re missing a tremendous opportunity. We’re asleep at the switch because it’s not a metaphor. In 1945 we actually did create a new universe. This is a universe of numbers with a life of their own, that we only see in terms of what those numbers can do for us. Can they record this interview? Can they play our music? Can they order our books on Amazon? If you cross the mirror in the other direction, there really is a universe of self-reproducing digital code. When I last checked, it was growing by five trillion bits per second. And that’s not just a metaphor for something else. It actually is. It’s a physical reality.

We’re still here at the big bang of this thing, and we’re not studying it enough. Who’s the cosmologist really looking at this in terms of what it might become in 10,000 years? What’s it going to be in 100 years? Here we are at the very beginning and we just may simply not be asking the right questions about what’s going on. Try looking at it from the other side, not from our side as human beings. Scientists are the people who can do that kind of thing. You can look at viruses from the point of view of a virus, not from the point of view of someone getting sick.

Very few people are looking at this digital universe in an objective way. Danny Hillis is one of the few people who is. His comment, made exactly 30 years ago in 1982, was that "memory locations are just wires turned sideways in time". That’s just so profound. That should be engraved on the wall. Because we don’t realize that there is this very different universe that does not have the same physics as our universe. It’s completely different physics. Yet, from the perspective of that universe, there is physics, and we have almost no physicists looking at it, as to what it’s like. And if we want to understand the sort of organisms that would evolve in that totally different universe, you have to understand the physics of the world in which they are in.  It’s like looking for life on another planet. Danny has that perspective. Most people say just, “well, a wire is a wire. It’s not a memory location turned sideways in time.” You have to have that sort of relativistic view of things.

We are still so close to the beginning of this explosion that we are still immersed in the initial fireball. Yet, in that short period of time, for instance, it was not long ago that to transfer money electronically you had to fill out paper forms on both ends and then wait a day for your money to be transferred. And, in a very few years, it’s a dozen years or so, most of the money in the world is moving electronically all the time.

The best example of this is what we call the flash crash of May 6th, two years ago, when suddenly, the whole system started behaving unpredictably. Large amounts of money were lost in milliseconds, and then the money came back, and we quietly (although the SEC held an investigation) swept it under the rug and just said, “well, it recovered. Things are okay.” But nobody knows what happened, or most of us don’t know.

There was a great Dutch documentary—Money and Speed: Inside the Black Box—where they spoke to someone named Eric Scott Hunsader who actually had captured the data on a much finer time scale, and there was all sorts of very interesting stuff going on. But it’s happening so quickly that it’s below what our normal trading programs are able to observe, they just aren’t accounting for those very fast things. And this could be happening all around us—not just in the world of finance. We would not necessarily even perceive it, that there’s a whole world of communication that’s not human communication. It’s machines communicating with machines. And they may be communicating money, or information that has other meaning—but if it is money, we eventually notice it. It’s just the small warm pond sitting there waiting for the spark.

It’s an unbelievably interesting time to be a digital biologist or a digital physicist, or a digital chemist. A good metaphor is chemistry. We’re starting to address code by template, rather than by numerical location—the way biological molecules do.

We’re living in a completely different world. The flash crash was an example: you could have gone out for a cup of coffee and missed the whole thing, and come back and your company lost a billion dollars and got back 999 million, while you were taking your lunch break. It just happened so fast, and it spread so quickly.

So, yes, the fear scenario is there, that some malevolent digital virus could bring down the financial system. But on the other hand, the miracle of this flash crash was not that it happened, but that it recovered so quickly. Yet, in those milliseconds, somebody made off with a lot of money. We still don’t know who that was, and maybe we don’t want to know.

The reason we’re here today (surrounded by this expanding digital universe) is because in 1936, or 1935, this oddball 23-year-old undergraduate student, Alan Turing, developed this theoretical framework to understand a problem in mathematical logic, and the way he solved that problem turned out to establish the model for all this computation. And I believe we wold have arrived here, sooner or later, without Alan Turing or John von Neumann, but it was Turing who developed the one-dimensional model, and von Neumann who developed the two-dimensional implementation, for this increasingly three-dimensional digital universe in which everything we do is immersed. And so, the next breakthrough in understanding will also I think come from some oddball. It won’t be one of our great, known scientists. It’ll be some 22-year-old kid somewhere who makes more sense of this.

But, we’re going back to biology, and of course, it’s impossible not to talk about money, and all these other ways that this impacts our life as human beings. What I was trying to say is that this digital universe really is so different that the physics itself is different. If you want to understand what types of life-like or self-reproducing forms would develop in a universe like that, you actually want to look at the sort of physics and chemistry of how that universe is completely different from ours. An example is how not only its time scale but how time operates is completely different, so that things can be going on in that world in microseconds that suddenly have a real effect on ours.

Again, money is a very good example, because money really is a sort of a gentlemen’s agreement to agree on where the money is at a given time. Banks decide, well, this money is here today and it’s there tomorrow. And when it’s being moved around in microseconds, you can have a collapse, where suddenly you hit the bell and you don’t know where the money is. And then everybody’s saying, “Where’s the money? What happened to it?” And I think that’s what happened. And there are other recent cases where it looks like a huge amount of money just suddenly disappeared, because we lost the common agreement on where it is at an exact point in time. We can’t account for those time periods as accurately as the computers can.

One number that’s interesting, and easy to remember, was in the year 1953, there were 53 kilobytes of high-speed memory on planet earth. This is random access high-speed memory. Now you can buy those 53 kilobytes for an immeasurably small, thousandth of one cent or something. If you draw the graph, it’s a very nice, clean graph. That’s sort of Moore’s Law; that it’s doubling. It has a doubling time that’s surprisingly short, and no end in sight, no matter what the technology does. We’re doubling the number of bits in a extraordinarily short time.

And we have never seen that. Or I mean, we have seen numbers like that, in epidemics or chain reactions, and there’s no question it’s a very interesting phenomenon. But still, it’s very hard not to just look at it from our point of view. What does it mean to us? What does it mean to my investments? What does it mean to my ability to have all the music I want on my iPhone? That kind of thing. But there’s something else going on. We’re seeing a fraction of one percent of it, and there’s this other 99.99 percent that people just aren’t looking at.

The beginning of this was driven by two problems. The problem of nuclear weapons design, and the problem of code breaking were the two drivers of the dawn of this computational universe. There were others, but those were the main ones.

What’s the driver today? You want one word? It’s advertising. And, you may think advertising is very trivial, and of no real importance, but I think it’s the driver. If you look at what most of these codes are doing, they’re trying to get the audience, trying to deliver the audience. The money is flowing as advertising.

And it is interesting that Samuel Butler imagined all this in 1863, and then in his book Erewhon. And then 1901, before he died, he wrote a draft for “Erewhon Revisited.” In there, he called out advertising, saying that advertising would be the driving force of these machines evolving and taking over the world. Even then at the close of 19th century England, he saw advertising as the way we would grant power to the machines.

If you had to say what’s the most powerful algorithm set loose on planet earth right now? Originally, yes, it was the Monte Carlo code for doing neutron calculations. Now it’s probably the AdWords algorithm. And the two are related: if you look at the way AdWords works, it is a Monte Carlo process. It’s a sort of statistical sampling of the entire search space, and a monetizing of it, which as we know, is a brilliant piece of work. And that’s not to diminish all the other great codes out there.

We live in a world where we measure numbers of computers in billions, and numbers of what we call servers, which are the equivalent of in the old days, of what would be called mainframes. Those are in the millions, hundreds of millions.

Two of the pioneers of this—to single out only two pioneers—were John Von Neumann and Alan Turing. If they were here today Turing would be 100. Von Neumann would be 109. I think they would understand what’s going on immediately—it would take them a few minutes, if not a day, to figure out, to understand what was going on. And, they both died working on biology, and I think they would be immediately fascinated by the way biological code and digital code are now intertwined. Von Neumann’s consuming passion at the end was self-reproducing automata. And Alan Turing was interested in the question of how molecules could self-organize to produce organisms.

They would be, on the other hand, astonished that we’re still running their machines, that we don’t have different computers. We’re still just running your straight Von Neumann/Turing machine with no real modification. So they might not find our computers all that interesting, but they would be diving into the architecture of the Internet, and looking at it.

In both cases, they would be amazed by the direct connection between the code running on computers and the code running in biology—that all these biotech companies are directly reading and writing nucleotide sequences in and out of electronic memory, with almost no human intervention. That’s more or less completely mechanized now, so there’s direct translation, and once you translate to nucleotides, it’s a small step, a difficult step, but, an inevitable step to translate directly to proteins. And that’s Craig Venter’s world, and it’s a very, very different world when we get there.

The question of how and when humans are going to expand into the universe, the space travel question, is, in my view, almost rendered obsolete by this growth of a digitally-coded biology, because those digital organisms—maybe they don’t exist now, but as long as the system keeps going, they’re inevitable—can travel at the speed of light. They can propagate. They’re going to be so immeasurably far ahead that maybe humans will be dragged along with it.

But while our digital footprint is propagating at the speed of light, we’re having very big trouble even getting to the eleven kilometers per second it takes to get into lower earth orbit. The digital world is clearly winning on that front. And that’s for the distant future. But it changes the game of launching things, if you no longer have to launch physical objects, in order to transmit life.”

George Dyson, author and historian of technology whose publications broadly cover the evolution of technology in relation to the physical environment and the direction of society, A universe of self-replicating code, Edge, Mar 26, 2012.

See also:

Jameson Dungan on information and synthetic biology
Vlatko Vedral: Decoding Reality: the universe as quantum information
Rethinking “Out of Africa: A Conversation with Christopher Stringer (2011)
A Short Course In Synthetic Genomics, The Edge Master Class with George Church & Craig Venter (2009)
Eat Me Before I Eat You! A New Foe For Bad Bugs: A Conversation with Kary Mullis (2010)
Mapping The Neanderthal Genome. A Conversation with Svante Pääbo (2009)
Engineering Biology”: A Conversation with Drew Endy (2008)
☞ “Life: A Gene-Centric View A Conversation in Munich with Craig Venter & Raichard Dawkins (2008)
Ants Have Algorithms: A Talk with Ian Couzin (2008)
Life: What A Concept, The Edge Seminar, Freeman Dyson, J. Craig Venter, George Church, Dimitar Sasselov, Seth Lloyd, Robert Shapiro (2007)
Code II J. Doyne Farmer v. Charles Simonyi (1998)
Jason Silva on singularity, synthetic biology and a desire to transcend human boundaries

Mar
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Is The World An Idea?

                              
                                                  Plato, Hulton Archive/Getty Images

Plato was one that made the divide between the world of ideas and the world of the senses explicit. In his famous Allegory of the Cave, he imagined a group of prisoners who had been chained to a cave all their lives; all they could see were shadows projected on a wall, which they conceived as their reality. Unbeknownst to them, a fire behind them illuminated objects and created the shadows they saw, which could be manipulated to deceive them. In contrast, the philosopher could see reality as it truly is, a manifestation of ideas freed from the deception of the senses. In other words, if we want to understand the true nature of reality, we shouldn’t rely on our senses; only ideas are truly pure, freed from the distortions caused by our limited perception of reality.

Plato thus elevated the human mind to a god-like status, given that it can find truth through reason, in particular through the rational construction of ideal “Forms,” which are the essence of all objects we see in reality. For example, all tables share the Form of “tableness,” even if every table is different. The Form is an ideal and, thus, a blueprint of perfection. If I ask you to imagine a circle, the image of a circle you hold in your head is the only perfect circle: any representation of that circle, on paper or on a blackboard, will be imperfect. To Plato, intelligence was the ability to grasp the world of Forms and thus come closer to truth.

Due to its connection with the search for truth, it’s no surprise that Plato’s ideas influenced both scientists and theologians. If the world is made out of Forms, say geometrical forms, reality may be described mathematically, combining the essential forms and their relations to describe the change we see in the world. Thus, by focusing on the essential elements of reality as mathematical objects and their relations we could, perhaps, grasp the ultimate nature of reality and so come closer to timeless truths.

The notion that mathematics is a portal to final truths holds tremendous intellectual appeal and has influenced some of the greatest names in the history of science, from Copernicus, Kepler, Newton, and Einstein to many present-day physicists searching for a final theory of nature based upon a geometrical scaffolding, such as superstring theories. (…)

Taken in context, we can see where modern scientific ideas that relate the ultimate nature of reality to geometry come from. If it’s not God the Geometer anymore, Man the Geometer persists. That this vision offers a major drive to human creativity is undeniable.

We do imagine the universe in our minds, with our minds, and many scientific successes are a byproduct of this vision. Perhaps we should take Nicholas of Cusa,’s advice to heart and remember that whatever we achieve with our minds will be an expression of our own creativity, having little or nothing to do with ultimate truths.”

Marcelo Gleiser, Brazilian Professor of Natural Philosophy, Physics and Astronomy at Dartmouth College, USA, Is The World An Idea?, NPR, March 7, 2012.

See also:

Cognition, perception, relativity tag on Lapidarium notes

Jan
22nd
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What Happened Before the Big Bang? The New Philosophy of Cosmology

    

Tim Maudlin: “There are problems that are fairly specific to cosmology. Standard cosmology, or what was considered standard cosmology twenty years ago, led people to the conclude that the universe that we see around us began in a big bang, or put another way, in some very hot, very dense state. And if you think about the characteristics of that state, in order to explain the evolution of the universe, that state had to be a very low entropy state, and there’s a line of thought that says that anything that is very low entropy is in some sense very improbable or unlikely. And if you carry that line of thought forward, you then say “Well gee, you’re telling me the universe began in some extremely unlikely or improbable state” and you wonder is there any explanation for that. Is there any principle that you can use to account for the big bang state?

This question of accounting for what we call the “big bang state” — the search for a physical explanation of it — is probably the most important question within the philosophy of cosmology, and there are a couple different lines of thought about it. One that’s becoming more and more prevalent in the physics community is the idea that the big bang state itself arose out of some previous condition, and that therefore there might be an explanation of it in terms of the previously existing dynamics by which it came about. There are other ideas, for instance that maybe there might be special sorts of laws, or special sorts of explanatory principles, that would apply uniquely to the initial state of the universe.

One common strategy for thinking about this is to suggest that what we used to call the whole universe is just a small part of everything there is, and that we live in a kind of bubble universe, a small region of something much larger. And the beginning of this region, what we call the big bang, came about by some physical process, from something before it, and that we happen to find ourselves in this region because this is a region that can support life. The idea being that there are lots of these bubble universes, maybe an infinite number of bubble universes, all very different from one another. Part of the explanation of what’s called the anthropic principle says, “Well now, if that’s the case, we as living beings will certainly find ourselves in one of those bubbles that happens to support living beings.” That gives you a kind of account for why the universe we see around us has certain properties. (…)

Newton would call what he was doing natural philosophy, that’s actually the name of his book: Mathematical Principles of Natural Philosophy." Philosophy, traditionally, is what everybody thought they were doing. It’s what Aristotle thought he was doing when he wrote his book called Physics. So it’s not as if there’s this big gap between physical inquiry and philosophical inquiry. They’re both interested in the world on a very general scale, and people who work in the foundations of physics, that is, the group that works on the foundations of physics, is about equally divided between people who live in philosophy departments, people who live in physics departments, and people who live in mathematics departments.

Q: In May of last year Stephen Hawking gave a talk for Google in which he said that philosophy was dead, and that it was dead because it had failed to keep up with science, and in particular physics. Is he wrong or is he describing a failure of philosophy that your project hopes to address?

Maudlin: Hawking is a brilliant man, but he’s not an expert in what’s going on in philosophy, evidently. Over the past thirty years the philosophy of physics has become seamlessly integrated with the foundations of physics work done by actual physicists, so the situation is actually the exact opposite of what he describes. I think he just doesn’t know what he’s talking about. I mean there’s no reason why he should. Why should he spend a lot of time reading the philosophy of physics? I’m sure it’s very difficult for him to do. But I think he’s just … uninformed. (…)

Q: Do you think that physics has neglected some of these foundational questions as it has become, increasingly, a kind of engine for the applied sciences, focusing on the manipulation, rather than say, the explanation, of the physical world? 

Maudlin: Look, physics has definitely avoided what were traditionally considered to be foundational physical questions, but the reason for that goes back to the foundation of quantum mechanics. The problem is that quantum mechanics was developed as a mathematical tool. Physicists understood how to use it as a tool for making predictions, but without an agreement or understanding about what it was telling us about the physical world. And that’s very clear when you look at any of the foundational discussions. This is what Einstein was upset about; this is what Schrodinger was upset about.

Quantum mechanics was merely a calculational technique that was not well understood as a physical theory. Bohr and Heisenberg tried to argue that asking for a clear physical theory was something you shouldn’t do anymore. That it was something outmoded. And they were wrong, Bohr and Heisenberg were wrong about that. But the effect of it was to shut down perfectly legitimate physics questions within the physics community for about half a century. And now we’re coming out of that, fortunately.

Q And what’s driving the renaissance?

Maudlin: Well, the questions never went away. There were always people who were willing to ask them. Probably the greatest physicist in the last half of the twentieth century, who pressed very hard on these questions, was John Stewart Bell. So you can’t suppress it forever, it will always bubble up. It came back because people became less and less willing to simply say, “Well, Bohr told us not to ask those questions,” which is sort of a ridiculous thing to say.

Q: Are the topics that have scientists completely flustered especially fertile ground for philosophers? For example I’ve been doing a ton of research for a piece about the James Webb Space Telescope, the successor to the Hubble Space Telescope, and none of the astronomers I’ve talked to seem to have a clue as to how to use it to solve the mystery of dark energy. Is there, or will there be, a philosophy of dark energy in the same way that a body of philosophy seems to have flowered around the mysteries of quantum mechanics?

Maudlin: There will be. There can be a philosophy of anything really, but it’s perhaps not as fancy as you’re making it out. The basic philosophical question, going back to Plato, is “What is x?” What is virtue? What is justice? What is matter? What is time? You can ask that about dark energy - what is it? And it’s a perfectly good question.

There are different ways of thinking about the phenomena which we attribute to dark energy. Some ways of thinking about it say that what you’re really doing is adjusting the laws of nature themselves. Some other ways of thinking about it suggest that you’ve discovered a component or constituent of nature that we need to understand better, and seek the source of. So, the question — What is this thing fundamentally? — is a philosophical question, and is a fundamental physical question, and will lead to interesting avenues of

Q: One example of philosophy of cosmology that seems to have trickled out to the layman is the idea of fine tuning - the notion that in the set of all possible physics, the subset that permits the evolution of life is very small, and that from this it is possible to conclude that the universe is either one of a large number of universes, a multiverse, or that perhaps some agent has fine tuned the universe with the expectation that it generate life. Do you expect that idea to have staying power, and if not what are some of the compelling arguments against it?

Maudlin: A lot of attention has been given to the fine tuning argument. Let me just say first of all, that the fine tuning argument as you state it, which is a perfectly correct statement of it, depends upon making judgments about the likelihood, or probability of something. Like, “how likely is it that the mass of the electron would be related to the mass of the proton in a certain way?” Now, one can first be a little puzzled by what you mean by “how likely” or “probable” something like that is. You can ask how likely it is that I’ll roll double sixes when I throw dice, but we understand the way you get a handle on the use of probabilities in that instance. It’s not as clear how you even make judgments like that about the likelihood of the various constants of nature (an so on) that are usually referred to in the fine tuning argument.

Now let me say one more thing about fine tuning. I talk to physicists a lot, and none of the physicists I talk to want to rely on the fine tuning argument to argue for a cosmology that has lots of bubble universes, or lots of worlds. What they want to argue is that this arises naturally from an analysis of the fundamental physics, that the fundamental physics, quite apart from any cosmological considerations, will give you a mechanism by which these worlds will be produced, and a mechanism by which different worlds will have different constants, or different laws, and so on.  If that’s true, then if there are enough of these worlds, it will be likely that some of them have the right combination of constants to permit life. But their arguments tend not to be “we have to believe in these many worlds to solve the fine tuning problem,” they tend to be “these many worlds are generated by physics we have other reasons for believing in.”

If we give up on that, and it turns out there aren’t these many worlds, that physics is unable to generate them, then it’s not that the only option is that there was some intelligent designer. It would be a terrible mistake to think that those are the only two ways things could go. You would have to again think hard about what you mean by probability, and about what sorts of explanations there might be. Part of the problem is that right now there are just way too many freely adjustable parameters in physics. Everybody agrees about that. There seem to be many things we call constants of nature that you could imagine setting at different values, and most physicists think there shouldn’t be that many, that many of them are related to one another.

Physicists think that at the end of the day there should be one complete equation to describe all physics, because any two physical systems interact and physics has to tell them what to do. And physicists generally like to have only a few constants, or parameters of nature. This is what Einstein meant when he famously said he wanted to understand what kind of choices God had —using his metaphor— how free his choices were in creating the universe, which is just asking how many freely adjustable parameters there are. Physicists tend to prefer theories that reduce that number, and as you reduce it, the problem of fine tuning tends to go away. But, again, this is just stuff we don’t understand well enough yet.

Q: I know that the nature of time is considered to be an especially tricky problem for physics, one that physicists seem prepared, or even eager, to hand over to philosophers. Why is that?

Maudlin: That’s a very interesting question, and we could have a long conversation about that. I’m not sure it’s accurate to say that physicists want to hand time over to philosophers. Some physicists are very adamant about wanting to say things about it; Sean Carroll for example is very adamant about saying that time is real. You have others saying that time is just an illusion, that there isn’t really a direction of time, and so forth. I myself think that all of the reasons that lead people to say things like that have very little merit, and that people have just been misled, largely by mistaking the mathematics they use to describe reality for reality itself. If you think that mathematical objects are not in time, and mathematical objects don’t change — which is perfectly true — and then you’re always using mathematical objects to describe the world, you could easily fall into the idea that the world itself doesn’t change, because your representations of it don’t.

There are other, technical reasons that people have thought that you don’t need a direction of time, or that physics doesn’t postulate a direction of time. My own view is that none of those arguments are very good. To the question as to why a physicist would want to hand time over to philosophers, the answer would be that physicists for almost a hundred years have been dissuaded from trying to think about fundamental questions. I think most physicists would quite rightly say “I don’t have the tools to answer a question like ‘what is time?’ - I have the tools to solve a differential equation.” The asking of fundamental physical questions is just not part of the training of a physicist anymore.

Q: I recently came across a paper about Fermi’s Paradox and Self-Replicating Probes, and while it had kind of a science fiction tone to it, it occurred to me as I was reading it that philosophers might be uniquely suited to speculating about, or at least evaluating the probabilistic arguments for the existence of life elsewhere in the universe. Do you expect philosophers of cosmology to enter into those debates, or will the discipline confine itself to issues that emerge directly from physics?

Maudlin: This is really a physical question. If you think of life, of intelligent life, it is, among other things, a physical phenomenon — it occurs when the physical conditions are right. And so the question of how likely it is that life will emerge, and how frequently it will emerge, does connect up to physics, and does connect up to cosmology, because when you’re asking how likely it is that somewhere there’s life, you’re talking about the broad scope of the physical universe. And philosophers do tend to be pretty well schooled in certain kinds of probabilistic analysis, and so it may come up. I wouldn’t rule it in or rule it out.

I will make one comment about these kinds of arguments which seems to me to somehow have eluded everyone. When people make these probabilistic equations, like the Drake Equation, which you’re familiar with — they introduce variables for the frequency of earth-like planets, for the evolution of life on those planets, and so on. The question remains as to how often, after life evolves, you’ll have intelligent life capable of making technology.

What people haven’t seemed to notice is that on earth, of all the billions of species that have evolved, only one has developed intelligence to the level of producing technology. Which means that kind of intelligence is really not very useful. It’s not actually, in the general case, of much evolutionary value. We tend to think, because we love to think of ourselves, human beings, as the top of the evolutionary ladder, that the intelligence we have, that makes us human beings, is the thing that all of evolution is striving toward. But what we know is that that’s not true.

Obviously it doesn’t matter that much if you’re a beetle, that you be really smart. If it were, evolution would have produced much more intelligent beetles. We have no empirical data to suggest that there’s a high probability that evolution on another planet would lead to technological intelligence. There is just too much we don’t know.”

Tim Maudlin, (B.A. Yale, Physics and Philosophy; Ph.D. Pittsburgh, History and Philosophy of Science), interviewed by Ross Andersen, What Happened Before the Big Bang? The New Philosophy of Cosmology, The Atlantic, Jan 2012.

Illustrations: 1 - Cambridge Digital Gallery Newton Collection, 2 - Aristotle, Ptolemy, and Copernicus discussing astronomy, Published in 1632, Library of Congress.

See also:

The Concept of Laws. The special status of the laws of mathematics and physics
Raphael Bousso: Thinking About the Universe on the Larger Scales
Stephen Hawking on the univers’s origin
Universe tag on Lapidarium notes
Universe tag on Lapidarium

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Symmetry: A ‘Key to Nature’s Secrets’


The five regular polyhedra. Plato argued in Timaeus that these were the shapes of the bodies making up the elements: earth consists of little cubes, while fire, air, and water are made of polyhedra with four, eight, and twenty identical faces, respectively. The fifth regular polyhedron, with twelve identical faces, was supposed by Plato to symbolize the cosmos.’ (Illustration Mike King)

“The Oxford English Dictionary tells us that symmetry is “the quality of being made up of exactly similar parts.” (…) A symmetry is a principle of invariance. That is, it tells us that something does not change its appearance when we make certain changes in our point of view—for instance, by rotating it or moving it. (…)

Laws of nature, in the modern sense of mathematical equations that tell us precisely what will happen in various circumstances, first appeared as the laws of motion and gravitation that Newton developed as a basis for understanding Kepler’s description of the solar system. From the beginning, Newton’s laws incorporated symmetry: the laws that we observe to govern motion and gravitation do not change their form if we reset our clocks, or if we change the point from which distances are measured, or if we rotate our entire laboratory so it faces in a different direction. (…)

It was already clear in the 1950s that the laws of nature, whatever they are, also respect symmetries of other kinds, having nothing directly to do with space and time. (…) Electromagnetic phenomena did not respect these symmetries: protons and some hyperons are electrically charged; neutrons and other hyperons are not. (…)

The recognition of accidental symmetry not only resolved the old puzzle about approximate symmetries; it also opened up exciting new possibilities. It turned out that there are certain symmetries that could not be violated in any theory that has the same particles and the same exact local symmetries as the Standard Model and that is simple enough to be renormalizable. (…)

There is an attractive theory, called chaotic inflation, according to which the universe began without any special spatial symmetries, in a completely chaotic state. Here and there by accident the fields pervading the universe were more or less uniform, and according to the gravitational field equations it is these patches of space that then underwent an exponentially rapid expansion, known as inflation, leading to something like our present universe, with all nonuniformities in these patches smoothed out by the expansion. In different patches of space the symmetries of the laws of nature would be broken in different ways. Much of the universe is still chaotic, and it is only in the patches that inflated sufficiently (and in which symmetries were broken in the right ways) that life could arise, so any beings who study the universe will find themselves in such patches. (…)

If it turns out that chaotic inflation is correct, then much of what we observe in nature will be due to the accident of our particular location, an accident that can never be explained, except by the fact that it is only in such locations that anyone could live.”

Steven Weinberg, American theoretical physicist and Nobel laureate in Physics, Symmetry: A ‘Key to Nature’s Secrets’, The New York Review of Books, Oct 27, 2011

Marcus du Sautoy on Symmetry in Mathematics, and in Life | University of Oxford



In this fascinating lecture, Professor Marcus du Sautoy talks about his quest to discover Symmetry in Mathematics, and in Life. OUMNH 150th Anniversary.

Marcus du Sautoy is the Simonyi Professor for the Public Understanding of Science and a Professor of Mathematics at the University of Oxford.

See also:

☞ Marcus du Sauto, "Finding Moonshine" lecture at Google’s Brussels office, 2010 (video)
The Concept of Laws. The special status of the laws of mathematics and physics

Sep
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Supercomputer predicts revolution: Forecasting large-scale human behavior using global news media tone in time and space


Figure 1. Global geocoded tone of all Summary of World Broadcasts content, 2005. Note: Click on image to see animation.

"Feeding a supercomputer with news stories could help predict major world events, according to US research.

While the analysis was carried out retrospectively, scientists say the same processes could be used to anticipate upcoming conflict. (…)

The study’s information was taken from a range of sources including the US government-run Open Source Centre and BBC Monitoring, both of which monitor local media output around the world.

News outlets which published online versions were also analysed, as was the New York Times' archive, going back to 1945.

In total, Mr Leetaru gathered more than 100 million articles.

Reports were analysed for two main types of information: mood - whether the article represented good news or bad news, and location - where events were happening and the location of other participants in the story.

Mood detection, or “automated sentiment mining” searched for words such as “terrible”, “horrific” or “nice”.

Location, or “geocoding” took mentions of specific places, such as “Cairo” and converted them in to coordinates that could be plotted on a map.

Analysis of story elements was used to create an interconnected web of 100 trillion relationships. (…)

The computer event analysis model appears to give forewarning of major events, based on deteriorating sentiment.

However, in the case of this study, its analysis is applied to things that have already happened.

According to Kalev Leetaru, such a system could easily be adapted to work in real time, giving an element of foresight. (…)

"It looks like a stock ticker in many regards and you know what direction it has been heading the last few minutes and you want to know where it is heading in the next few.

"It is very similar to what economic forecasting algorithms do.” (…)

"The next iteration is going to city level and beyond and looking at individual groups and how they interact.

"I liken it to weather forecasting. It’s never perfect, but we do better than random guessing."

Supercomputer predicts revolution, BBC News, 9 September 2011

Culturomics 2.0: Forecasting large-scale human behavior using global news media tone in time and space

"News is increasingly being produced and consumed online, supplanting print and broadcast to represent nearly half of the news monitored across the world today by Western intelligence agencies. Recent literature has suggested that computational analysis of large text archives can yield novel insights to the functioning of society, including predicting future economic events. Applying tone and geographic analysis to a 30–year worldwide news archive, global news tone is found to have forecasted the revolutions in Tunisia, Egypt, and Libya, including the removal of Egyptian President Mubarak, predicted the stability of Saudi Arabia (at least through May 2011), estimated Osama Bin Laden’s likely hiding place as a 200–kilometer radius in Northern Pakistan that includes Abbotabad, and offered a new look at the world’s cultural affiliations. Along the way, common assertions about the news, such as “news is becoming more negative” and “American news portrays a U.S.–centric view of the world” are found to have merit.”

The emerging field of Culturomics” seeks to explore broad cultural trends through the computerized analysis of vast digital book archives, offering novel insights into the functioning of human society (Michel, et al., 2011). Yet, books represent the “digested history” of humanity, written with the benefit of hindsight. People take action based on the imperfect information available to them at the time, and the news media captures a snapshot of the real–time public information environment (Stierholz, 2008). News contains far more than just factual details: an array of cultural and contextual influences strongly impact how events are framed for an outlet’s audience, offering a window into national consciousness (Gerbner and Marvanyi, 1977). A growing body of work has shown that measuring the “tone” of this real–time consciousness can accurately forecast many broad social behaviors, ranging from box office sales (Mishne and Glance, 2006) to the stock market itself (Bollen, et al., 2011). (…)


Figure 2. Global geocoded tone of all Summary of World Broadcasts content, January 1979–April 2011 mentioning “bin Laden”

Most theories of civilizations feature some approximation of the degree of conflict or cooperation between each group. Figure 3 displays the average tone of all links between cities in each civilization, visualizing the overall “tone” of the relationship between each. Group 1, which roughly encompasses the Asiatic and Australian regions, has largely positive links to the rest of the world and is the only group with a positive connection to Group 4 (Middle East). Group 3 (Africa) has no positive links to any other civilization, while Group 2  (North and South America excluding Canada) has negative links to all but Group 1. As opposed to explicit measures of conflict or cooperation based on armed conflict or trade ties, this approach captures the latent view of conflict and cooperation as portrayed by the world’s news media.

            
Figure 3. Average tone of links between world “civilizations” according to SWB, 1979–2009.

Figure 4 shows the world civilizations according to the New York Times 1945–2005. It divides the world into five civilizations, but paints a very different picture of the world, with a far greater portion of the global landmass arrayed around the United States. Geographic affinity appears to play a far lesser role in this grouping, and the majority of the world is located in a single cluster with the United States. It is clear from comparing the SWB and NYT civilization maps that even within the news media there is no one “universal” set of civilizations, but that each country’s media system may portray the world very differently to its audience. By pooling all of these varied viewpoints together, SWB’s view of the world’s civilizations offers a “crowdsourced” aggregate view of civilization, but it too is likely subject to some innate Western bias.


Figure 4. World “civilizations” according to NYT, 1945–2005. A full–resolution version of this figure is available here

Monitoring first broadcast then print media over the last 70 years, nearly half of the annual output of Western intelligence global news monitoring is now derived from Internet–based news, standing testament to the Web’s disruptive power as a distribution medium. Pooling together the global tone of all news mentions of a country over time appears to accurately forecast its near–term stability, including predicting the revolutions in Egypt, Tunisia, and Libya, conflict in Serbia, and the stability of Saudi Arabia.

Location plays a critical role in news reporting, and “passively crowdsourcing” the media to find the locations most closely associated with Bin Laden prior to his capture finds a 200km – wide swath of northern Pakistan as his most likely hiding place, an area which contains Abbottabad, the city he was ultimately captured in. Finally, the geographic clustering of the news, the way in which it frames localities together, offers new insights into how the world views itself and the “natural civilizations” of the news media.

While heavily biased and far from complete, the news media captures the only cross–national real–time record of human society available to researchers. The findings of this study suggest that Culturomics, which has thus far focused on the digested history of books, can yield intriguing new understandings of human society when applied to the real–time data of news. From forecasting impending conflict to offering insights on the locations of wanted fugitives, applying data mining approaches to the vast historical archive of the news media offers promise of new approaches to measuring and understanding human society on a global scale.”

Kalev Leetaru is Senior Research Scientist for Content Analysis at the Institute for Computing in the Humanities, Arts, and Social Science at the University of Illinois, Center Affiliate of the National Center for Supercomputing Applications, and Research Coordinator at the University of Illinois Cline Center for Democracy. His award-winning work centers on the application of high performance computing to grand challenge problems using news and open sources intelligence. He holds three US patents and more than 50 University Invention Disclosures.

To see full research click University of Illinois at Chicago - UI, Volume 16, Number 9, 5 September 2011

See also:

Culturomics: Quantitative Analysis of Culture Using Millions of Digitized Books

"Construct a corpus of digitized texts containing about 4% of all books ever printed, and then analyze that corpus using advanced software and the investigatory curiosity of thousands, and you get something called "Culturomics," a field in which cultural trends are represented quantitatively.

In this talk Erez Lieberman Aiden and Jean-Baptiste Michel — co-founders of the Cultural Observatory at Harvard and Visiting Faculty at Google — show how culturomics can provide insights about fields as diverse as lexicography, the evolution of grammar, collective memory, the adoption of technology, the pursuit of fame, censorship, and historical epidemiology.”

— E. Lieberman Aiden, Harvard Society of Fellows & Jean-Baptiste Michel, FQEB Fellow at Harvard, Culturomics: Quantitative Analysis of Culture Using Millions of Digitized Books, May 10, 2011

See also:

What we learned from 5 million books, TED.com, 2011 (video)

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Universal Semantic Communication. Is it possible for two intelligent beings to communicate meaningfully, without any common language or background?

                              

"This question has interest on its own, but is especially relevant in the context of modern computational infrastructures where an increase in the diversity of computers is making the task of inter-computer interaction increasingly burdensome. Computers spend a substantial amount of time updating their software to increase their knowledge of other computing devices. In turn, for any pair of communicating devices, one has to design software that enables the two to talk to each other. Is it possible instead to let the two computing entities use their intelligence (universality as computers) to learn each others’ behavior and attain a common understanding? What is “common understanding?” We explore this question in this paper.

To formalize this problem, we suggest that one should study the “goal of communication:” why are the two entities interacting with each other, and what do they hope to gain by it? We propose that by considering this question explicitly, one can make progress on the question of universal communication.

We start by considering a computational setting for the problem where the goal of one of the interacting players is to gain some computational wisdom from the other player. We show that if the second player is “sufficiently” helpful and powerful, then the first player can gain significant computational power (deciding PSPACE complete languages).

Our work highlights some of the definitional issues underlying the task of formalizing universal communication, but also suggests some interesting phenomena and highlights potential tools that may be used for such communication. (…)

Consider the following scenario: Alice, an extraterrestrial, decides to initiate contact with a terrestrial named Bob by means of a radio wave transmission. How should he respond to her? Will he ever be able to understand her message? In this paper we explore such scenarios by framing the underlying questions computationally.

We believe that the above questions have intrinsic interest, as they raise some further fundamental questions. How does one formalize the concept of understanding? Does communication between intelligent beings require a “hardwired” common sense of meaning or language? Or, can intelligence substitute for such requirements? What role, if any, does computational complexity play in all this? (…)

Marvin Minsky suggested that communication should be possible from a philosophical standpoint, but did not provide any formal definitions or constructions.

LINCOS [an abbreviation of the Latin phrase lingua cosmica]: The most notable and extensive prior approach to this problem is due to Hans Freudenthal, who claims that it is possible to code messages describing mathematics, physics, or even simple stories in such a radio transmission which can be understood by any sufficiently humanlike recipient. Ideally, we would like to have such a rich language at our disposal; it should be clear that the “catch” lies in Freudenthal’s assumption of a “humanlike” recipient, which serves as a catch-all for the various assumptions that serve as the foundations for Freudenthal’s scheme.

It is possible to state more precise assumptions which form the basis of Freudenthal’s scheme, but among these will be some fairly strong assumptions about how the recipient interprets the message. In particular, one of these is the assumption that all semantic concepts of interest can be characterized by lists of syntactic examples. (…)

Information Theory

The classical theory of communication does not investigate the meaning associated with information and simply studies the process of communicating the information, in its exact syntactic form. It is the success of this theory that motivates our work: computers are so successful in communicating a sequence of bits, that the most likely source of “miscommunication” is a misinterpretation of what these bits mean. (…)

Interactive Proofs and Knowledge

Finally, the theory of interactive proofs and knowledge [pdf] (and also the related M. Blum and S. Kannan. Designing programs that check their work) gets further into the gap between Alice and Bob, by ascribing to them different, conflicting intents, though they still share common semantics. It turns out this gap already starts to get to the heart of the issues that we consider, and this theory is very useful to us at a technical level. In particular, in this work we consider a setting where Bob wishes to gain knowledge from Alice. Of course, in our setting Bob is not mistrustful of Alice, he simply does not understand her. (…)

Modeling issues

Our goal is to cast the problem of “meaningful” communication between Alice and Bob in a purely mathematical setting. We start by considering how to formulate the problem where the presence of a “trusted third party” would easily solve the problem.

Consider the informal setting in which Alice and Bob speak different natural languages and wish to have a discussion via some binary channel. We would expect that a third party who knows both languages could give finite encoding rules to Alice and Bob to facilitate this discussion, and we might be tempted to require that Alice’s statements translate into the same statements in Bob’s language that the third party would have selected and vice-versa.

In the absence of the third party, this is unreasonable to expect, though: suppose that Alice and Bob were given encoding rules that were identical to those that a third party would have given them, except that some symmetric sets of words have been exchanged—say, Alice thinks “left” means “right,” “clockwise” means “counter-clockwise,” etc. Unless they have some way to tell that these basic concepts have been switched, observe that they would still have a conversation that is entirely sensible to each of them. [See also] Thus, if we are to have any hope at all, we must be prepared to accept interactions that are indistinguishable from successes as “successes” as well. We do not wish to take this to an extreme, though: Bob cannot distinguish among Alices who say nothing, and yet we would not classify their interactions as “successes.”

At the heart of the issues raised by the discussion above is the question: what does Bob hope to get out of this conversation with Alice? In general, why do computers, or humans communicate? Only by pinning down this issue can we ask the question, “can they do it without a common language?”

We believe that there are actually many possible motivations for communication. Some communication is motivated by physical needs, and others are motivated purely by intellectual needs or even curiosity. However these diverse settings still share some common themes: communication is being used by the players to achieve some effects that would be hard to achieve without communication. In this paper, we focus on one natural motivation for communication: Bob wishes to communicate with Alice to solve some computa- tional problems. (…)

In order to establish communication between Alice and Bob, Bob runs in time exponential in a parameter that could be described informally as the length of the dictionary that translates Bob’s language into Alice’s language. (Formally, the parameter is the description length of the protocol for interpreting Alice in his encoding of Turing machines.) (…) [p.3]

[To see proofs of theorems and more, click pdf]

Conclusions

In the previous sections we studied the question, “how can two intelligent interacting players attempt to achieve some meaningful communication in a universal setting, i.e., one in which the two players do not start with a common background?” We return now to the motivation for studying this question, and the challenges that need to be dealt with to address the motivations. (…)

We believe that this work has raised and addressed some fundamental questions of intrinsic interest. However this is not the sole motivation for studying this problem. We believe that these questions also go to the heart of “protocol issues” in modern computer networks. Modern computational infrastructures are built around the concept of communication and indeed a vast amount of effort is poured into the task of ensuring that the computers work properly as communication devices. Yet as computers and networks continue to evolve at this rapid pace, one problem is becoming increasingly burdensome: that of ensuring that every pair of computers is able to “understand” each other, so as to communicate meaningfully. (…)

Current infrastrusctures ensure this ability for pairs to talk to each other by explicitly going through a “setup” phase, where a third party who knows the specifications of both elements of a pair sets up a common language/protocol for the two to talk to each other, and then either or both players learn (download) this common language to establish communication. An everyday example of such an occurence is when we attempt to get our computer to print on a new printer. We download a device driver for our computer which is a common language written by someone who knows both our computer and the printer.

We remark that this issue is a fundamental one, and not merely an issue of improper design. Current protocols are designed with a fixed pair of types of devices in mind. However, we expect for our computers to be capable of communicating with all other communication devices, even ones that did not exist when our computer was built. While it would be convenient if all computers interacted with each other using a single fixed protocol that is static over time, this is no more reasonable to expect than asking humans to agree on a single language to converse in, and then to expect this language to stay fixed over time. Thus, to satisfy our expectations in the current setting, it is essential that computers are constantly updated so as to have universal connectivity over time. (…)

This work was motivated by a somewhat radical alternative scenario for communication. Perhaps we should not set computers up with common languages, but rather exploit the universality in our favor, by letting them evolve to a common language. But then this raises issues such as: how can the computers know when they have converged to a common understanding? Or, how does one of the computers realize that the computer it is communicating with is no longer in the same mode as they were previously, and so the protocol for communication needs to be adjusted? The problem described in the opening paragraph of the introduction is simply the extremal version of such issues, where the communicating players are modeled as having no common background. (…)

Perhaps the main contribution of this work is to suggest that communication is not an end in itself, but rather a means to achieving some general goal. Such a goal certainly exists in all the practical settings above, though it is no longer that of deciding membership in some set S. Our thesis is that one can broaden the applicability of this work to other settings by (1) precisely articulating the goal of communication in each setting and (2) constructing “universal protocols” that achieve these goals. (…)

One of the implicit suggestions in this work is that communicating players should periodically test to see if the assumption of common understanding still holds. When this assumption fails, presumably this happened due to a “mild” change in the behavior of one of the players. It may be possible to design communication protocols that use such a “mildness” assumption to search and re-synchronize the communicating players where the “exponential search” takes time exponential in the amount of change in the behavior of the players. Again, pinning down a precise measure of the change and designing protocols that function well against this measure are open issues.”

Brendan Juba, Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory, and Harvard University. School of Engineering and Applied Sciences - Theory of Computing group, Madhu Sudan, Indian computer scientist, professor of computer science at the Massachusetts Institute of Technology (MIT), Universal Semantic Communication I (pdf), MIT, 2010 (Illustration source)

See also:

☞ Brendan Juba, Madhu Sudan, Universal Semantic Communication II (pdf), MIT
☞ J. Bao, P. Basu, M. Dean, C. Partridge, A. Swami, W. Leland, J. A. Hendler, Towards a Theory of Semantic Communication (Extended Technical Report) (pdf)

Sep
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Quantum minds: Why we think like quarks - ‘To be human is to be quantum’

                     

"The quantum world defies the rules of ordinary logic. Particles routinely occupy two or more places at the same time and don’t even have well-defined properties until they are measured. It’s all strange, yet true - quantum theory is the most accurate scientific theory ever tested and its mathematics is perfectly suited to the weirdness of the atomic world. (…)

Human thinking, as many of us know, often fails to respect the principles of classical logic. We make systematic errors when reasoning with probabilities, for example. Physicist Diederik Aerts of the Free University of Brussels, Belgium, has shown that these errors actually make sense within a wider logic based on quantum mathematics. The same logic also seems to fit naturally with how people link concepts together, often on the basis of loose associations and blurred boundaries. That means search algorithms based on quantum logic could uncover meanings in masses of text more efficiently than classical algorithms.

It may sound preposterous to imagine that the mathematics of quantum theory has something to say about the nature of human thinking. This is not to say there is anything quantum going on in the brain, only that "quantum" mathematics really isn’t owned by physics at all, and turns out to be better than classical mathematics in capturing the fuzzy and flexible ways that humans use ideas. "People often follow a different way of thinking than the one dictated by classical logic," says Aerts. “The mathematics of quantum theory turns out to describe this quite well.”

It’s a finding that has kicked off a burgeoning field known as “quantum interaction”, which explores how quantum theory can be useful in areas having nothing to do with physics, ranging from human language and cognition to biology and economics. (…)

One thing that distinguishes quantum from classical physics is how probabilities work. Suppose, for example, that you spray some particles towards a screen with two slits in it, and study the results on the wall behind (see diagram below). Close slit B, and particles going through A will make a pattern behind it. Close A instead, and a similar pattern will form behind slit B. Keep both A and B open and the pattern you should get - ordinary physics and logic would suggest - should be the sum of these two component patterns.

But the quantum world doesn’t obey. When electrons or photons in a beam pass through the two slits, they act as waves and produce an interference pattern on the wall. The pattern with A and B open just isn’t the sum of the two patterns with either A or B open alone, but something entirely different - one that varies as light and dark stripes. (…)

The phenomenon may go well beyond physics, and one example of this is the violation of what logicians call the “sure thing” principle. This is the idea that if you prefer one action over another in one situation - coffee over tea in situation A, say, when it’s before noon - and you prefer the same thing in the opposite situation - coffee over tea in situation B, when it’s after noon - then you should have the same preference when you don’t know the situation: that is, coffee over tea when you don’t know what time it is.

Remarkably, people don’t respect this rule. (…)

Flexible logic

Suppose you ask people to put various objects, such as an ashtray, a painting and a sink, into one of two categories: “home furnishings” and “furniture”. Next, you ask if these objects belong to the combined category “home furnishings or furniture”. Obviously, if “ashtray” or “painting” belongs in home furnishings, then it certainly belongs in the bigger, more inclusive combined category too. But many experiments over the past two decades document what psychologists call the disjunction effect - that people often place things in the first category, but not in the broader one. Again, two possibilities listed simultaneously lead to strange results.

These experiments demonstrate that people aren’t logical, at least by classical standards. But quantum theory, Aerts argues, offers richer logical possibilities. For example, two quantum events, A and B, are described by so-called probability amplitudes, alpha and beta. To calculate the probability of A happening, you must square this amplitude alpha and likewise to work out the probability of B happening. For A or B to happen, the probability amplitude is alpha plus beta. When you square this to work out the probability, you get the probability of A (alpha squared) plus that of B (beta squared) plus an additional amount - an “interference term” which might be positive or negative.

This interference term makes quantum logic more flexible. In fact, Aerts has shown that many results demonstrating the disjunction effect fit naturally within a model in which quantum interference can play a role. The way we violate the sure thing principle can be similarly explained with quantum interference, according to economist Jerome Busemeyer of Indiana University in Bloomington and psychologist Emmanuel Pothos of the University of Wales in Swansea. "Quantum probabilities have the potential to provide a better framework for modelling human decision making," says Busemeyer.

The strange links go beyond probability, Aerts argues, to the realm of quantum uncertainty. One aspect of this is that the properties of particles such as electrons do not exist until they are measured. The experiment doing the measuring determines what properties an electron might have.

Hilbert's mathematics includes this effect by representing the quantum state of an electron by a so-called state vector - a kind of arrow existing in an abstract, high-dimensional space known as Hilbert space. An experiment can change the state vector arrow, projecting it in just one direction in the space. This is known as contextuality and it represents how the context of a specific experiment changes the possible properties of the electron being measured.

The meaning of words, too, changes according to their context, giving language a “quantum” feel. For instance, you would think that if a thing, X, is also a Y, then a “tall X” would also be a “tall Y” - a tall oak is a tall tree, for example. But that’s not always the case. A chihuahua is a dog, but a tall chihuahua is not a tall dog; “tall” changes meaning by virtue of the word next to it. Likewise, the way “red” is defined depends on whether you are talking about “red wine”, “red hair”, “red eyes” or “red soil”. “The structure of human conceptual knowledge is quantum-like because context plays a fundamental role,” says Aerts.

These peculiar similarities also apply to how search engines retrieve information. Around a decade ago, computer scientists Dominic Widdows, now at Google Research in Pittsburgh, Pennsylvania, and Keith van Rijsbergen of the University of Glasgow, UK, realised that the mathematics they had been building into search engines was essentially the same as that of quantum theory. (…)

Quantum leaps

An urgent challenge is to get computers to find meaning in data in much the same way people do, says Widdows. If you want to research a topic such as the “story of rock” with geophysics and rock formation in mind, you don’t want a search engine to give you millions of pages on rock music. One approach would be to include “-songs” in your search terms in order to remove any pages that mention “songs”. This is called negation and is based on classical logic. While it would be an improvement, you would still find lots of pages about rock music that just don’t happen to mention the word songs.

Widdows has found that a negation based on quantum logic works much better. Interpreting “not” in the quantum sense means taking “songs” as an arrow in a multidimensional Hilbert space called semantic space, where words with the same meaning are grouped together. Negation means removing from the search pages that shares any component in common with this vector, which would include pages with words like music, guitar, Hendrix and so on. As a result, the search becomes much more specific to what the user wants.

"It seems to work because it corresponds more closely to the vague reasoning people often use when searching for information," says Widdows. "We often rely on hunches, and traditionally, computers are very bad at hunches. This is just where the quantum-inspired models give fresh insights."

That work is now being used to create entirely new ways of retrieving information. Widdows, working with Trevor Cohen at the University of Texas in Houston, and others, has shown that quantum operations in semantic Hilbert spaces are a powerful means of finding previously unrecognised associations between concepts. This may even offer a route towards computers being truly able to discover things for themselves. (…)

Why should quantum logic fit human behaviour? Peter Bruza at Queensland University of Technology in Brisbane, Australia, suggests the reason is to do with our finite brain being overwhelmed by the complexity of the environment yet having to take action long before it can calculate its way to the certainty demanded by classical logic. Quantum logic may be more suitable to making decisions that work well enough, even if they’re not logically faultless. “The constraints we face are often the natural enemy of getting completely accurate and justified answers,” says Bruza.

This idea fits with the views of some psychologists, who argue that strict classical logic only plays a small part in the human mind. Cognitive psychologist Peter Gardenfors of Lund University in Sweden, for example, argues that much of our thinking operates on a largely unconscious level, where thought follows a less restrictive logic and forms loose associations between concepts.

Aerts agrees. “It seems that we’re really on to something deep we don’t yet fully understand.” This is not to say that the human brain or consciousness have anything to do with quantum physics, only that the mathematical language of quantum theory happens to match the description of human decision-making.

Perhaps only humans, with our seemingly illogical minds, are uniquely capable of discovering and understanding quantum theory. To be human is to be quantum.”

Mark Buchanan, American physicist and author, Quantum minds: Why we think like quarks, New Scientist, 05 September 2011 (Illustration source)

See also:

Vlatko Vedral: Decoding Reality: the universe as quantum information
Geoffrey West on Why Cities Keep Growing, Corporations and People Always Die, and Life Gets Faster
The Concept of Laws. The special status of the laws of mathematics and physics, Lapidarium

Jun
6th
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Geoffrey West on Why Cities Keep Growing, Corporations and People Always Die, and Life Gets Faster

         

What extent can biology and social organization (which are both quintessential complex adaptive systems) be put in a more quantitative, analytic, mathematizable, predictive framework so that we can understand them in the way that we understand “simple physical systems”?

It is very clear from the beginning that we will never have a theory of biological and social systems that is like physics — that is, something that’s precise that we can predict, like for example, the motion of the planets with great precision or the magnetic electron to 12 decimal places. Nothing approaching that can possibly be in these other sciences, because they are complex systems.

Nevertheless, that doesn’t mean that you couldn’t have a quantitative theory. It would simply mean that you would possibly have a theory that is cross-grained. Meaning that you would be able to ask questions, big questions, and answer them in an average idealized setting. (…)

Scaling phenomena

I started working some years ago on questions in biology. I started using the very powerful techniques developed in physics, and that have run through the history of physics, to think about scaling phenomena. The great thing about scaling is that if you observe scaling (that is, how the various characteristics of a system change when you change its size) and if you see regularity over several orders of magnitude, that typically means that there are underlying generic principles, that it is not an accident. If you see that in a system, it is opening a window onto some underlying, let’s use the word, “universal principle”.

The remarkable thing in biology that got me excited and has led to all of my present work (which has now gone beyond biology and into social organizations, cities, and companies) is that there was data, quite old and fundamental to all biological processes, about metabolism: Here is maybe the most complex physical chemical process possibly in the universe, and when you ask how it is scaled with size across mammals (as an example to keep it simple) you find that there is an extraordinary regularity.

This is surprising because we believe in natural selection, and natural selection has built into it this idea that history plays an important role. There’s the environmental niche for every organism, every component of an organism, every cell type is unique and has its own unique history. So if you plotted, for example the metabolic rate on the Y axis and size on the X axis, because of the extraordinary diversity and complexity of the system and the historical contingency, you would expect points all over the map representing, of course, history and geography and so on.

Well, you find quite the contrary. You find a very simple curve, and that curve has a very simple mathematical formula. It comes out to be a very simple power law. In fact, the power law not only is simple in itself mathematically, but here it has an exponent that is extraordinarily simple. The exponent was very close to the number three quarters.

First of all, that was amazing in itself, that you see scaling. But more importantly was that the scaling is manifested across all of life into eco-systems and down within cells. So this scaling law is truly remarkable. It goes from intracellular up to ecosystems almost 30 orders of magnitude. They’re the same phenomenon. (…)

That is, it scales as a simple power law. The extraordinary thing about it is that the power law has an exponent, which is always a simple multiple of one quarter. What you determine just from the data is that there’s this extraordinary simple number, four, which seems to dominate all biology and across all taxonomic groups from the microscopic to the macroscopic.

This can hardly be an accident. If you see scaling, it is manifesting something that transcends history, geography, and therefore the evolved engineered structure of the organism because it applies to me, all mammals, and the trees sitting out there, even though we’re completely different designs.

The big question is where in the hell does that number come from? And what is it telling us about the structure of the biology?  And what is it telling us about the constraints under which evolution occurred? That was the beginning of all this.

Are cities and companies just extensions of biology?

I’ll say a few words about what we propose as the solution. But to jump ahead, the idea was that once we had that body of work, understanding the origin of these scaling laws was to take it over into social organizations. And so the question that drove the extension of this work was, “are cities and companies just extensions of biology?”

They came out of biology. That’s where they came from. But is New York just actually, in some ways, a great big whale? And is Microsoft a great big elephant? Metaphorically we use biological terms, for example the DNA of the company or the ecology of the marketplace. But are those just metaphors or is there some serious substance that we can quantify with those?

There are two things that are very important that come out of the biology of the scaling —it’s theoretical and conceptual framework.

One: Since the metabolic rate scales non-linearly with size — all of these things scale non-linearly with size — and they scale with exponents that are less than one, what that means is that if the metabolic rate per cell is decreasing with size, the metabolic rate of our cells, my cells, are working harder than my horses. But my dogs are working even harder, in a systematic predictive way.

What does that say? That says there’s an extraordinary economy of scale.

Just to give you an example, if you increase the size of an organism by a factor of ten to the fourth, four is the magnitude, you would have expected naively to have ten to the fourth times as much energy. You would have the ten to the fourth times more cells. Ten thousand times more cells. Not true. You only need a thousand times. There’s an extraordinary savings in the energy use, and that cuts across all resources as well.

When we come to social organizations, there’s an interesting question. Do we have economies of scale or what? How do cities work, for example? How do companies work in this framework? That’s one thing.

The second thing is, (again, comes from the data and the conceptual framework explains it) the bigger you are, the slower everything is. The bigger you are, you live longer. Oxygen diffuses slower across your various membranes. You take longer to mature, you grow slower, but all in a systematic, mathematizable, predictable way. The pace of life systematically slows down following these quarter power scales. And again, we’ll ask those questions about life … social life and economies.

The work I got involved in was to try to understand these scaling laws. And to make it a very short story, what was proposed apart from the thinking was, look, this is universal. It cuts across the design of organisms. Whether you are insects, fish, mammals or birds, you get the same scaling laws. It is independent of design. Therefore, it must be something that is about the structure of the way things are distributed.

You recognize what the problem is. You have ten14cells. You have this problem. You’ve got to sustain them, roughly speaking, democratically and efficiently. And however natural selection solved it, it solved it by evolving hierarchical networks.

There is a very simple way of doing it. You take something macroscopic, you go through a hierarchy and you deliver them to very microscopic sites, like for example, your capillaries to your cells and so on. And so the idea was, this is true at all scales. It is true of an ecosystem; it is true within the cell. And what these scaling laws are manifesting are the generic, universal, mathematical, topological properties of networks.

The question is, what are the principles that are governing these networks that are independent of design? After a lot of work we postulated the following, just to give an idea.

First, they have to be space filling. They have to go everywhere. They have to feed every cell, every piece of the organism.

Secondly, they have things like invariant units. That is when you evolve from a human being to a whale (to make it a simple story) you do not change the basic units. The cells of the whale or the capillaries of whale, which are the kind of fundamental units, are pretty much indistinguishable from yours and mine. There is this invariance. When you evolve to a new species, you use the same units but you change the network. That’s the idea in this picture.

And the last one is of the infinitude of networks that have these properties - space filling and invariant total units. The ones that have actually evolved by the process of continuous feedback implicit in natural selection are those that have in some way optimized the system.

For example, the amount of work that your heart has to do to pump blood around your circulatory system to keep you alive is minimized with respect to the design of the system. You can put it into mathematics. You have a network theory, you mathematize the network, and then you make variations of the network and ask what is the one that minimizes the amount of energy your heart has to use to pump blood through it.

The principle is simple. Mathematically, it is quite complicated and challenging, but you can solve all of that. And you do that so that you can maximize the amount of energy you can put into fitness to make children. You want to minimize the amount of energy just to keep you alive, so that you can make more babies. That’s the simplest big picture.

All of those results about scaling are derived. A quarter, four, emerges. And what is the four?  It turns out the four isn’t a four. The four is actually a “three plus one”, meaning it’s the dimensionality of the space we live in plus one, which is actually to do, loosely speaking, with the fractal nature of these networks, the fact that there’s a sub-similar property.

In D dimensions, you read D plus one (that’s my physicist self speaking). Instead of being three quarters for metabolic rate, it would be D over D plus one.

Life in some funny way is actually five dimensional. It’s three space, one time, and one kind of fractal. That’s five. So we’re kind of five dimensional creatures in some curious way, mathematically.

This network theory was used to predict all kinds of things. You can answer questions like why is it we sleep eight hours. Why does a mouse have to sleep 15 hours?  And why does an elephant only have to sleep four and a whale two?  Well, we can answer that. Why do we evolve at the rate we do?  How does cancer work in terms of vasculature and its necrosis? And so on.

A whole bunch of questions can follow from this. One of the most important is growth. Understanding growth. How do we grow?  And why do we stop growing, for example?  Well, we can answer that. The theory answers that. And it’s quite powerful, and it explains why it is we have this so-called sigmoidal growth where you grow quickly and then you stop. And it explains why that is and it predicts when you stop, and it predicts the shape of that curve for an animal.

Here is this wonderful body of work that explains many things — some fundamental, some to do with very practical problems like understanding sleep, aging. The question is, can we take that over to other kinds of network systems. One of the obvious types of systems is a city. Another obvious one is a company. The first question you have to ask is, okay, this was based on the observation of scaling. Scaling was the window. It’s interesting of itself, but actually, it’s more interesting as a revelatory tool to open onto fundamental principles.

What did we learn from scaling in biology? We not only learned the network theory, but we learned that despite the fact that the whale lives in the ocean, the giraffe has a long neck, and the elephant a truck, and we walk on two feet and the mouse scurries around, at some 85, 90 percent level, we’re all scaled versions of one another.

There’s kind of one mammal, and every other mammal, no matter what size it is and where it existed, is actually some well-defined mathematically scaled version of that one master mammal, so to speak. And that is kind of amazing.

In other words, the size of a mammal, or any organism for that matter, can tell you how long it should live, how many children it should have, how oxygen diffuses across its lungs, what is the length of the ninth branch of its circulatory system, how its blood is flowing, how quickly it will grow, et cetera.

A provocative question is, is New York just a scaled up San Francisco, which is a scaled up Santa Fe?  Superficially, that seems unlikely because they look so different, especially Santa Fe. I live in Santa Fe and it’s a bunch of dopey buildings, and here I am in New York overwhelmed by huge skyscrapers. On the other hand, a whale doesn’t look much like a giraffe. But in fact, they’re scaled versions of one another, at this kind of cross-grained 85, 90 percent level.

Of course, you can’t answer this question just by sitting in an armchair. You have to go out and get the data and ask, “If I look at various metrics describing a city, do they scale in some simple way?”

Is there one line, so to speak, upon which all of them sit? Or when I look at all these metrics and I plot them, do I just see this random mess, which says that each city is unique and dominated by its geography and its history? In which case there’s not much you can do, and you’ve got to attack and think about cities as individual.

I got into this work, because first of all, I believe it’s a truly challenging, fundamental, science problem.

I think this is very much science of the 21st century, because it is the kind of problem that scientists have ignored. It is under the umbrella of a complex adaptive system and we need to come to terms with understanding the structure and dynamics and organization of such systems because they’re the ones that determine our lives and our extraordinary phenomenon that we have developed on this planet.

Can we understand them as scientists? The prevailing way of investigating them is social sciences and economics  — which have primarily less to do with generic principles and more to do with case studies and narrative (which is of course, very important). But the question is, can we complement them and make a science of cities, so to speak, and a science of corporations?

It is a very important question, certainly for scaling, because if it’s true that every city is unique, then of course, there’s no real science of cities. Every case would be special.

Another remarkable fact is that the planet has urbanizing at an exponential rate. Namely, 200 years ago, here sitting in Manhattan, almost everything around me would be a field. There would be a teeny settlement down at Wall Street somewhere of a small number of people. But most of the people would be living in these fields all the way up Manhattan into upstate New York. Indeed, at that time, less than four percent of the United States was urban. Primarily, it was agricultural. And now, only 200 years later, it’s almost the reverse. More like 82 percent is urban and less than 20 percent is agricultural. This has happened at an extraordinarily fast rate — and in fact, faster than exponential.

The point to recognize is that all of the tsunami of problems we’re facing, from global warming, the environment, to the questions of financial markets and risk, crime, pollution, disease and so forth, all of them are urban.

They all have their origin in cities. They have become dominant since the Industrial Revolution. Most importantly, they’ve been with us for the last two or 300 years, and somehow, we’ve only noticed them in the last ten or 15 years as if they’d never been here. Why? Because they’ve been increasing exponentially. We are on an exponential.

Cities are the cause of the problem, and they’re also the cause of the good life. They are the centers of wealth creation, creativity, innovation, and invention. They’re the exciting places. They are these magnets that suck people in. And that’s what’s been happening. And so they are the origin of the problems, but they are the origin of the solutions. And we need to come to terms with that, and we need to understand how cities work in a more scientific framework, meaning to what extent can we make it into a quantitative predictive, mathematizible kind of science.

Is that even possible?  And is it useful? That’s quest.

The first thing was to ask the question, do they scale? I put together a wonderful team of people, and I’d like to mention their names, because they play an extremely important and seminal role.

One is a man named Luis Bettencourt also a physicist who is at Los Alamos and the Santa Fe Institute. A man named José Lobo, who was at Cornell when I first got him involved, an urban economist and now he’s at Arizona State. Another is a student, Deborah Strumsky, who was at Harvard when she joined us, and is now at the University of North Carolina. And there are others, but these were the main characters. Most importantly, they were people that were part of a trans-disciplinary kind of group. And they brought together the data. They did the data mining, the statistics, analysis, et cetera. They have the expertise and the credentials.

The result of all of that was a long, tedious kind of process. To make a long story short, indeed, we found that cities scaled. Just amazing. Cities do scale. Not only do they scale, but also there’s universality to their scaling. Let me just tell you a little bit about of what we discovered from the data to begin with.

The first result that we actually got was with my German colleagues, Dirk Helbing, and his then student, Christian Kuhnert, who then worked with me. One of the first results was a very simple one —the number of gas stations as a function of city size in European cities.

What was discovered was that they behaved sort of like biology. You found that they are scaled beautifully, and it scaled as a power law, and the power law was less than one, indicating an economy of scale. Not surprisingly, the bigger the city, the less gas stations you need per capital. There is an economy of scale.

But it’s scaled! That is, it was systematic! You tell me the size of a city and I’ll tell you how many gas stations it has — that kind of idea. And not only that, it’s scaled at exactly the same way across all European cities. Kind of interesting!

But then, we discovered two things later that were quite remarkable. First, every infrastructural quantity you looked at from total length of roadways to the length of electrical lines to the length of gas lines, all the kinds of infrastructural things that are networked throughout a city, scaled in the same way as the number of gas stations. Namely, systematically, as you increase city size, I can tell you, roughly speaking, how many gas stations there are, what is the total length of roads, electrical lines, et cetera, et cetera. And it’s the same scaling in Europe, the United States, Japan and so on.

It is quite similar to biology. The exponent, instead of being three quarters was more like .85. So it’s a different exponent, but similar. But it’s an economy of scale.

The truly remarkable result was when we looked at quantities that I will call “socioeconomic”. That is, quantities that have no analog in biology. These are quantities, phenomena that did not exist until about 10,000 years ago when men and women started talking to one another and working together and forming serious communities leading to what we now call cities, i.e. things like wages, the number of educational institutions, the number of patents produced, et cetera. Things that have no analog in biology, things we invented.

And if you ask, first of all, do they scale? The answer is yes, in a regular way. Then, how do they scale?  And this was the surprise to me; I’m embarrassed to say. It should have been obvious prior, but they scaled in what we called a super linear fashion. Instead of being an exponent less than one, indicating economies of scale, the exponent was bigger than one, indicating what economists call increasing returns to scale.

What does that say? That says that systematically, the bigger the city, the more wages you can expect, the more educational institutions in principle, more cultural events, more patents are produced, it’s more innovative and so on. Remarkably, all to the same degree. There was a universal exponent which turned out to be approximately 1.15 which translated to English says something like the following:  If you double the size of a city from 50,000 to a hundred thousand, a million to two million, five million to ten million, it doesn’t matter what, systematically, you get a roughly 15 percent increase in productivity, patents, the number of research institutions, wages and so on, and you get systematically a 15 percent saving in length of roads and general infrastructure.

There are systematic benefits that come from increasing city size, both in terms of the individual getting something — which attracts people to the city, and in terms of the macroscopic economy. So the big cities are good in this sense.

However, some bad and ugly come with it. And the bad and ugly are things like a systematic increase in crime and various diseases, like AIDS, flu and so on. Interestingly enough, it scales all to the same 15 percent, if you double the size. Or put slightly differently, another way of saying it is, if you have a city of a million people and you broke it down into ten cities of a hundred thousand, you would require for that ten cities of a hundred thousand, 30 to 40 percent more roads, and 30 to 40 percent general infrastructure. And you would get a systematic decrease in wages and productivity and invention. Amazing. But you’d also get a decrease in crime, pollution and disease, systematically. So there are these trade-offs.

What does this mean?  What is this coming from?  And what do they imply?  Let me just say one of the things that they imply.

If cities are dominated by wealth creation and innovation, i.e. the super linear scaling laws, there’s increasing returns to scale. How does that impact growth? What does that do for growth? Well, it turns out, of course, had it been biology and it had been dominated by economies of scale, you would have got a sigmoid curve, and you would have stopped growing. Bad for cities, we believe, and bad for economies.

Economies must be, in a capitalist system, ever expanding. It’s good that we have super linear scaling, because what that says is you have open-ended growth. And that’s very good. Indeed, if you can check it against data, it agrees very well. But there’s something very bad about open-ended growth.

One of the bad things about open-ended growth, growing faster than exponentially, is that open-ended growth eventually leads to collapse. It leads to collapse mathematically because of something called finite times singularity. You hit something that’s called a singularity, which is a technical term, and it turns out as you approach this singularity, the system, if it reaches it, will collapse. You have to avoid that singularity in order to stop collapsing. It’s great on the one hand that you have this open ended growth. But if you kept going, of course, it doesn’t make any sense. Eventually, you run out of resources anyway, but you would collapse. And that’s what the theory says.

How do you avoid that? Well, how have we avoided it? We’ve avoided it by innovation. By making a major innovation that so to speak, resets the clock and you can kind of start over again with new boundary conditions. We’ve done that by making major discoveries or inventions, like we discover iron, we discover coal. Or we invent computers, or we invent IT. But it has to be something that really changes the cultural and economic paradigm. It kind of resets the clock and we start over again.

There’s a theorem you can prove that says that if you demand continuous open growth, you have to have continuous cycles of innovation. Well, that’s what people believe, and it’s the way people have suggested that’s how you get out of the Malthusian paradox. This all agrees within itself but there is a huge catch.

I said earlier that in biology you have economies of scale, scaling that is sub linear, three quarters less than one, and that the pace of life gets slower the bigger you are. In cities and social systems, you have the opposite. You have the super linear scaling. You have increasing returns to scale. The bigger you are, the more you have rather than less.

It turns out when you go through the theoretical framework that leads to the opposite to biology the pace of life increases with size. So everything that’s going on in New York today is systematically going faster than it is in San Francisco, than it is in Santa Fe, even the speed of walking.

There’s data, and if you plot it, you will see that the speed of walking in cities, actually, I said the data is actually taken primarily in European cities, but you can see this systematic increase in some reasonable agreement with the theory.

The first thing is that we have this increasing pace of life. We have open-ended growth, increase in pace of life, and the threat of collapse because of the singularity. But there’s a big catch about this innovation. Theory says, sure, you can get out of collapse by innovating, but you have to innovate faster and faster.

Something that took 10,000 years 20,000 years ago to make a change, now takes 25 years. So this is not the clock that is governing social life. There’s a clock that’s getting faster and faster. And so you have to innovate faster and faster in order to avoid the collapse. And it all comes out of this exponential growth driven by super linear scaling.

The question then is, is this sustainable? The system will collapse, because eventually you would have to be making a major innovation, like you know, IT every six months. Well, that’s completely crazy. First of all, we’re human beings. We can’t adapt to that, even.  But we can’t do it, so this is very threatening.

This leads then to all kinds of questions about global sustainability and how can you construct a conceptual framework that gives rise to having wealth creation, innovation, this kind of quality and standard of life, wealth production, and yet, not grow in such a way that you are probing the singularity and collapsing. That’s the challenge. That’s certainly something that we have to face.

Let me just say a few words about ideas as to why it is there’s scaling in cities. What we’ve shown is that there’s universality that on the one hand, you have this sub linear scaling, economies of scale for infrastructure like biology. But the dominant part of the city, wealth creation, innovation and the socio economic kinds of quantities, that have no analog in biology, scale super linearly.

This is true for any metric you want to think of and across the world. If you look at Japanese data or Chinese data or data from Chile or Colombia or the Netherlands or Portugal or the United States, it all looks the same. Yet these cities have nothing to do with one another.

It says that geography and history played a subdominant role as it did in biology in a sense. And so if you tell me the size of a city in the United States, I can tell you with some 85 percent accuracy how many police it would have, how many AIDS cases, how long the length of the roads are, how many patents it’s producing and so on, on the average.

Of course, you can use that as a baseline for talking about actual individual cities, how they over and underperform relative to this idealized scaling number. But the question is, where in the hell does that come from?  What is it that’s universal that transcends countries and cultures?

Well obviously, it’s what cities are really about, not these buildings and the roads and things, but the people. It’s people. What we believe is that the scaling laws are a manifestation of social networks, of the universality of the way human beings interact, what we’re doing now, talking to one another, exchanging ideas, and doing tasks together, and so on.

It is the nature of those networks and the clustering — very importantly, the hierarchical clustering of those networks, the family structure, the way families interact, and then all the way out through businesses and so on, that there’s a kind of universality to that that is representative of the kind of scale at which humans interact.

For example, even though families in China and the United States traditionally may look different, most people cannot interact seriously, in a serious, dedicated way with more than five or six people. It doesn’t matter how big the family is actually. Despite Facebook, you cannot have a hundred best friends anywhere in the world.

These things are representative of the universal nature of the social networking. Our belief is that it is the nature of that and the hierarchy of it. For example, not only the hierarchy in size, but the hierarchy in the fact that you’re strongest interaction is with your family. You have a much weaker interaction with your colleagues in your job, and in your job situation, you have a much weaker interaction even with the CEO of the company, and all the way around the hierarchy. There is this presumed self-similar structure that goes up through the hierarchy in terms of the size of the hierarchy and in terms of the strength of interaction.

We believe it is that hierarchy which is transcending all of the aspects of the city and is being represented by these kinds of laws. So how is it that when we plot, we can plot GDP of the city, the number of AIDS cases and wages on one plot, and they overlap one another? They’re just the same line. Well, that’s because from this viewpoint, they’re all manifestations of people interacting with one another.

Predominately companies are dominated by economies of scale rather than innovation

The last piece of this is to take it to companies. Again, I must say that when I first started working on this, I just assumed companies were little cities so to speak. I also assumed they were dominated by creativity and so on.

It took us a long time to get data for companies, because unlike cities, you have to pay for that data. But we’ve just done it. (…)

In fact, we’ve done it primarily without paying attention to sector, although we’ve done some decomposition into sector. What I’m going to talk about here is regardless of sector. If you just take all companies equally for a moment, indeed, what you see if you plot the various metrics of a company from sales and profits, taxation, assets versus company size, using the metric of employees (you could use others — you could use sales itself but we used employees) you find scaling.

There’s much more variation, many more outliers among companies than there are among cities, and more among cities than there are among organisms. But nevertheless, you see very good evidence of scaling. And the thing that surprised us about this scaling was that it was like biology, not like cities. It was sub linear predominately.

That was surprising because sub linear in the kind of conceptual framework we developed was a reflection of economies of scale, and super linear as a reflection of wealth creation and innovation. It is said that predominately companies are dominated by economies of scale rather than innovation.

If it were dominated by economies of scale, sub linear scaling, unlike cities (which have open ended growth) companies would grow and then stop growing. And not only that, if you extrapolated from biology, they would indeed, die, ultimately.

We looked at the growth curves as the metrics of the company, like its assets or its profits, as a function of time, or its number of employees as a function of time. Indeed, the generic behavior is a sigmoid. They grow fast and they stop. All the big companies stop at roughly the same value, which is intriguing of it self. I think that number is about half a trillion dollars.

We have a wonderful graph that has about ten thousand companies plotted on one graph and they are these growth curves. You see this kind of spaghetti looking graph by just eyeballing it. Everything grows and stops growing. That’s what it looks like. We’re still in the middle of analyzing a lot of this.

The picture emerges. Companies are more like organisms. They grow and asymptote. Cities are open ended.

More importantly, what we discovered is that on the one hand, sales increased linearly with company size. On the other hand, profits increased sub linearly of an exponent of about one eighth. This data is all U.S. data on publicly traded companies.

Sales to profits are systematically decreasing so that eventually, the profit to sales margin is going to zero. If you just extrapolate this, indeed, if you look at the data, you see that the fluctuations in all these quantities are proportional to the size of the company. The fluctuation is getting bigger and bigger. The profits are decreasing relative to sales. Even though the profits are increasing the bigger you are, where you think, “we made several billion dollars” what you realize is that you’re in an environment where the fluctuation is eventually bigger than that. This is possibly the mechanism by which companies die.

Let me tell you the interpretation. Again, this is still speculative.

The great thing about cities, the thing that is amazing about cities is that as they grow, so to speak, their dimensionality increases. That is, the space of opportunity, the space of functions, the space of jobs just continually increases. And the data shows that. If you look at job categories, it continually increases. I’ll use the word “dimensionality.”  It opens up. And in fact, one of the great things about cities is that it supports crazy people. You walk down Fifth Avenue, you see crazy people, and there are always crazy people. Well, that’s good. It is tolerant of extraordinary diversity.

This is in complete contrast to companies, with the exception of companies maybe at the beginning (think of the image of the Google boys in the back garage, with ideas of the search engine no doubt promoting all kinds of crazy ideas and having maybe even crazy people around them).

Well, Google is a bit of an exception because it still tolerates some of that. But most companies start out probably with some of that buzz. But the data indicates that at about 50 employees to a hundred, that buzz starts to stop. And a company that was more multi dimensional, more evolved becomes one-dimensional. It closes down.

Indeed, if you go to General Motors or you go to American Airlines or you go to Goldman Sachs, you don’t see crazy people. Crazy people are fired. Well, to speak of crazy people is taking the extreme. But maverick people are often fired.

It’s not surprising to learn that when manufacturing companies are on a down turn, they decrease research and development, and in fact in some cases, do actually get rid of it, thinking “oh, we can get that back, in two years we’ll be back on track.”

Well, this kind of thinking kills them. This is part of the killing, and this is part of the change from super linear to sublinear, namely companies allow themselves to be dominated by bureaucracy and administration over creativity and innovation, and unfortunately, it’s necessary. You cannot run a company without administrative. Someone has got to take care of the taxes and the bills and the cleaning the floors and the maintenance of the building and all the rest of that stuff. You need it. And the question is, “can you do it without it dominating the company?” The data suggests that you can’t.

The question is, as a scientist, can we take these ideas and do what we did in biology, at least based on networks and other ideas, and put this into a quantitative, mathematizable, predictive theory, so that we can understand the birth and death of companies, how that stimulates the economy?  How it’s related to cities? How does it affect global sustainability and have a predictive framework for an idealized system, so that we can understand how to deal with it and avoid it? If you’re running a bigger company, you can recognize what the metrics are that are driving you to mortality, and possibly put it off, and hopefully even avoid it.

Otherwise we have a theory that tells you when Google and Microsoft will eventually die, and die might mean a merger with someone else.

That’s the idea and that’s the framework, and that’s what this is.”

Geoffrey West, British theoretical physicist, former president and distinguished professor of the Santa Fe Institute, Why Cities Keep Growing, Corporations and People Always Die, and Life Gets Faster, Edge, 23 May 2011. (Photo Illustration by Hubert Blanz)

Geoffrey West: The surprising math of cities and corporations



Physicist Geoffrey West has found that simple, mathematical laws govern the properties of cities — that wealth, crime rate, walking speed and many other aspects of a city can be deduced from a single number: the city’s population. In this mind-bending talk from TEDGlobal he shows how it works and how similar laws hold for organisms and corporations.

Geoffrey West: The surprising math of cities and corporations, TED.com, July 2011

Why Cities Keep on Growing, Corporations Always Die, and Life Gets Faster | Fora.tv



As organisms, cities, and companies scale up, they all gain in efficiency, but then they vary. The bigger an organism, the slower. Yet the bigger a city is, the faster it runs. And cities are structurally immortal, while corporations are structurally doomed. Scaling up always creates new problems; cities can innovate faster than the problems indefinitely, while corporations cannot.

These revolutionary findings come from Geoffrey West’s examination of vast quantities of data on the metabolic/economic behavior of organisms and organizations. A theoretical physicist, West was president of Santa Fe Institute from 2005 to 2009 and founded the high energy physics group at Los Alamos National Laboratory.

— Full program: Long Now Foundation, Fora.tv, Cowell Theatre, San Francisco, CA, Event Date: 07.25.2011

See also:

☞ Geoffrey West, Scaling Laws In Biology And Other Complex Systems, Google Tech talks August 1, 2007
☞ Geoffrey West, On the Scale and Unity of Life from Cells to Cities, Research channel lecture
The sameness of organisms, cities, and corporations: Q&A with Geoffrey West, TED, 26 July 2011.
☞ Jonah Lehrer, A Physicist Solves the City, The New York Times, Dec 17, 2010
Vlatko Vedral: Decoding Reality: the universe as quantum information
☞ Mark Buchanan, Quantum minds: Why we think like quarks - ‘To be human is to be quantum’, New Scientist, 05 Sep 2011
‘To understand is to perceive patterns’ - B. Fuller, Powell, Johnson, West, Kurzweil & video narration by J. Silva

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Universal Property of Musical Scales Discovered

"Researchers at the Institute for Logic, Language and Computation (ILLC) of the University of Amsterdam have discovered a universal property of musical scales. Until now it was assumed that the only thing scales throughout the world have in common is the octave.

                      
                           (Credit: Image courtesy of Universiteit van Amsterdam (UVA))

The many hundreds of scales in existence seem to possess a deeper commonality: if their tones are compared in a two or three-dimensional way by means of a coordinate system, they form convex or star-convex structures.

Convex structures are patterns without indentations or holes, such as a circle, square or oval. 

Almost all music in the world is based on an underlying scale from which compositions are built. In Western music, the major scale (do-re-mi-fa-sol-la-ti-do) is the best known scale. However, there are many other scales in use, such as the minor and the chromatic scale. Besides these ‘traditional’ scales there are also artificial scales created by modern composers. At a superficial level, scales consist of an ascending or descending sequence of tones where the initial and final tones are separated by an octave, which means the frequency of the final tone is twice that of the initial tone (the fundamental).

1000 scales

By placing scales in a coordinate system (an ‘Euler lattice’) they can be studied as multidimensional objects. Dr. Aline Honingh and Prof. Rens Bod from the ILLC did this for nearly 1,000 scales from all over the world, from Japan to Indonesia and from China to Greece. To their surprise, they discovered that all traditional scales produced star-convex patterns. This was also the case with almost 97% of non-traditional, scales conceived by contemporary composers, even though contemporary composers often state they have designed unconventional scales. This percentage is very high, because the probability that a random series of notes will produce a star-convex pattern is very small. Honingh and Bod try to explain this phenomenon by using the notion of consonance (harmony of sounds). They connect their research results with language and visual perception where convex patterns have also been detected, possibly indicating a cognitive universal (a general cognitive property).”

Universal Property of Music Discovered, ScienceDaily, Mar. 25, 2011.

"Finally it may be noteworthy that star-convexity is not unique for musical scales, but seems to be a prevalent property in many other areas of human perception, from language (Gardenfors and Williams 2001) to vision (Jaeger 2009). In this light, the star-convexity of scales may perhaps only be an instantiation of a more general cognitive property for the domain of music.

See also: Aline Honingh, Rens Bod, In search of universal properties of musical scales (pdf), Institute for Logic, Language and Computation University of Amsterdam