Showing posts with label Mandelbrot. Show all posts
Showing posts with label Mandelbrot. Show all posts

7 Jul 2009

Posthumanism and Pixels, Condensed Version

by Corry Shores
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[The following is a condensed version of a previous text. There you will find proper citation. I thank Professor Ulrich Melle, Professor Aziz Zambak, and my Father Ebbie V. Shores for their contributions to this research. And may I thank kvond at Frames/ sing for opening new directions for the philosophy of Analog & Digital. Later we will examine his developments and ways to integrate these ideas with Spinoza's thinking.]



Corry Shores



Do Posthumanists Dream of Pixilated Sheep?

Bostrom and Sandberg's Brain Emulation,

Examined and Critiqued



Enhancement technologies may someday give us capacities far beyond what we dream humanly possible. We could become post-human. Nick Bostrom & Anders Sandberg suggest that we might survive our body's death by living as a computer simulation. They issued a report from a conference where experts in all relevant fields collaborated to determine the path to "whole brain emulation." This technology will in the very least be an effective research tool for the neurosciences. It could even aid philosophical research too. Their "roadmap" defends certain philosophical assumptions required for this technology's success. So by determining the reasons why it succeeds or fails, we can obtain empirical data for philosophical debates regarding our mind and selfhood. I have chosen four issues to discuss: emergentism, analog vs. digital, chance, and personal identity.

Brain emulation succeeds if a computer program replicates human neural functioning. Yet for the authors, its success increases when it perfectly replicates one specific person’s brain. She might then survive her body’s death by living as the simulation.

This prospect has posthumanist proponents. Their view presupposes certain traits of human consciousness and selfhood. Hans Moravec for example thinks our personal identities exist independently to our bodies. According to his pattern-identity theory of selfhood, we are no more than the patterns and the processes found in our brains and bodies. William Bainbridge explains that we are neither man nor machine. We are just the dynamic patterns of information that can be realized in a wide variety of materials. Hence our personal patterns might be found in this body or in that computer. Either way we are the same person.

To emulate someone’s neural patterns, we first scan a particular brain to obtain precise detail of its structures and their interactions. Using this data, we program a simulation that will behave essentially the same as the original brain. Now first consider a gnat’s wild flight pattern. It seems irrational and random. But the motion of a whole swarm is smooth, controlled, and intelligent, as though the whole group of gnats has a mind of its own. To simulate the swarm, perhaps we will not need to understand how the whole swarm thinks. We instead just learn the way one gnat behaves and interacts with other ones. When we combine thousands of these simulated gnats, the swarm’s collective intelligence should thereby appear. Whole brain emulation presupposes this principle. The simulation will mimic the human brain’s functioning on the cellular level. Then automatically, higher-and-higher orders of organization should spontaneously arise. Finally human consciousness might emerge at the highest level of organization.

Early in this technology's development, we should only expect simpler brain states, like wakefulness and sleep. But in its ultimate form, whole brain emulation would enable us to make back-up copies of our minds. Then we might somehow survive our body’s death.

According to Bostrom & Sandberg, whole brain emulation should replicate all the original brain’s relevant properties so to produce a 1-to-1 model of the brain’s functioning. In this sense, the brain and its emulation are black boxes. We feed each one on its own the same sequence of stimuli. If they both respond with the same sequence of reactions, then the two separate machines are functionally equivalent. In this way, the same mind could be realized in two physically different systems. Hilary Putnam claims that electronic computers can be functionally equivalent to mechanical ones and even to humans using pencil and paper. Their insides may differ drastically, but their outward behaviors are identical.

There are various levels of emulation success. The highest ones are the most philosophically interesting.

When the technology achieves individual brain emulation, Bostrom & Sandberg write, it produces emergent activity characteristic of one particular brain. With further success, we would emulate someone’s personal identity. Perhaps somehow it would be numerically the same person. But at least it would continue-on as that person even after her body dies. We achieve such a simulation when it becomes rationally self-concerned for the brain it emulates.

Minds emerge from the brain’s pattern of physical dynamics. If you replicate this pattern-dynamic in some other physical medium, the same mental phenomena should likewise emerge. One mind would then be realized in a multiplicity of different physical embodiments. So whole brain emulation’s success would provide evidence for the theory of multiple realizability.

According to emergentist theories, all reality is made-up of a single kind of stuff. But its parts aggregate and assemble into dynamic organizational patterns. The higher levels exhibit properties not found in the lower ones. But, there can be no higher order without lower ones underlying it.

Consider the H2O molecule. It does not itself bear the properties of liquidity, wetness, and transparency. However, a large enough aggregate of water molecules will exhibit these properties.

In our brains, no one single neuron is conscious. Yet, your minds emerge from the complex dynamic pattern of all our neurons communicating and computing in parallel. Roger Sperry offers compelling evidence. There are "split brain" patients whose right and left brain hemispheres are disconnected from one another. Nonetheless, they maintain unified consciousness.

William Hasker offers the analogy of magnetic fields, which are distinct from the magnets producing them. The iron atoms themselves need to be organized in alignment in order for a magnetic field to emerge on a higher scale. In a similar way, the particular organization of the brain’s neurons generates a field of ‘consciousness.’ This emergent consciousness-field permeates and haloes our brain-matter, occupying its space and traveling along with it.

Not everyone agrees that the mind emerges from the brain. Todd Feinberg is one example. Now in fact, he does think that consciousness results from the complex interaction of many layers of neural organization. However, he argues that consciousness does not get squirted-out from neural activity and thereby obtain a life of its own. Instead, the layers of neural activity are all mutually interdependent and simultaneously cooperative. Consider for example when we recognize our grandmother. One layer of neurons transmits information about the whole visual field. Another layer picks-out lines. Another one, shapes. Finally the information arrives at the grandmother cell, which only fires when it is she that we see. But this does not make the grandmother cell emergently higher. Rather, all the neural layers of organization must work together simultaneously to achieve this recognition. The brain is a vast network of interconnected circuits. So we cannot say that any layer of organization emerges over and above the others.

Feinberg’s objection may prove problematic for whole brain emulation. Bostrom & Sandberg explicitly state that we only need to simulate the lower levels of activity.

But if Feinberg’s holistic theory is correct, we cannot only emulate the lower levels and expect the rest to spontaneously emerge. For, we need already to understand the higher-levels in order to program the lower ones. So, whole brain emulation’s emergentist assumptions might not express the actual way that consciousness appears.

Notice how in the recent past, many digital technologies have replaced analog ones. It would seem that these two sorts of quantity-representation reside in contrary worlds: the continuous versus the discrete. An abacus is digital. It computes one discrete value or another, but it is blind to the values between its lowest digit-places. When we count to two on our fingers, meaningless empty space spans between our digits. So spreading our fingers further apart does not change their numerical value. Slide-rules, however, are analog. One ruler slides against another continuously. So it may calculate any possible real number along the continuum. It could potentially compute and display irrational numbers like pi or the golden ratio. However, a digital computer would never cease calculating into lower digit places. For, there can be no final figure when rendering irrational numbers into digits. But if you move the slide rule from three to four, you will for one instant display pi. Analog is dense. Between any two values is already a third one, and lying between those are yet even more, and so on infinitely. On account of this infinite divisibility, analog can compute and display an infinity of different values found within a finite range. But like the gaps between our fingers, digital at some point will be blind to a middle value, no matter how precise it is. So, because analog computers can deal with an infinity of values, they have more computing potential for certain applications.

Our emulated brain will receive simulated sense-signals. Does it matter if they are digital signals rather than analog? Many audiophiles swear by the unsurpassable superiority of analog. It might be less precise, but it always flows like natural sound waves. Digital, even as it becomes more accurate, still sounds to them artificial or cartoon-like. In other words, there might be a qualitative difference to how we experience analog and digital stimuli, even though it might take a person with extra sensitivities to bring this difference to our explicit awareness.

And if the continuous and discrete are so fundamentally different, then maybe a brain computing in analog would experience a qualitatively different feel of consciousness than if the brain were instead computing in digital. Perhaps digital emulations might even produce a mental awareness quite foreign to what humans normally experience.

Bostrom’s & Sandberg’s brain emulation exclusively uses digital computation. But, they acknowledge the argument that analog and digital are qualitatively different. And, they admit that implementing analog in brain emulation could present profound difficulties. Yet, there is no need to worry, they say.

They pose what is called “the argument from noise.” Analog devices always take some physical form. It is unavoidable that interferences and irregularities, called noise, will make the analog device imprecise. So analog might be capable of taking-on an infinite range of variations. However, it will never be absolutely accurate, because noise always causes it to veer-off slightly from where it should be. Yet digital has its own inaccuracies. It is always missing variables between its discrete values. Nonetheless, digital is improving. Little-by-little it is coming to handle more variables. It is filling in the gaps. Digital will never be completely dense like analog. Values will always slip through its fingers. And analog will always miss its mark. But soon the distance between digital’s smallest fingers will equal the distance that analog veers-away from its proper course. Digital’s blindness would then match analog’s sloppiness. So, we only need to wait for digital technology to improve enough that it can compute the same values with equivalent precision. Both will be equally inaccurate, but for fundamentally different reasons.

But perhaps the argument from noise reduces the analog/digital distinction to a quantitative difference rather than a qualitative one. And analog is so prevalent in neural functioning that we should not so quickly brush it off.

Note first that our nervous system’s electrical signals are discrete pulses, like Morse code. In that sense they are digital. However, the frequency of the pulses can vary continuously. As well, there are many other neural quantities that are analog in this way.

Fred Dretske argues that our memories store information in analog. We might consider an event that occurred somewhere within a 15 minute time-span. But later, we also might search our memory for finer details to indicate more specifically when that event occurred. Because we can always further refine our remembered determinations, it could very well be that our brains record data in analog form.

But suppose anyway that the argument from noise is correct, and that we can dismiss analog’s computational superiority. Would there still be some reason to implement analog technology?

Recent research on neural-network learning supplies an answer. Analog noise interference is significantly more effective than digital at aiding adaptation. Being "wrong" allows neurons to explore new possibilities for computational values and connections. This enables us to learn and adapt to a chaotically changing environment. Using digitally-simulated neural noise might be inadequate. Analog is better. For, it affords our neurons an infinite array of alternate configurations. Hence, in response to Bostrom’s & Sandberg’s argument from noise, I propose this argument for noise. Analog’s inaccuracies take the form of continuous variation. In my view, this is precisely what makes it necessary for whole brain emulation.

Neural noise can result from external interferences like magnetic fields. Or internal random fluctuations might make the signals unpredictable. In both cases, chance & chaos reign our brains. And in fact, these random, indeterminate, and probabilistic events assist our brain’s computations. It implements noise to keep us adjusted to the world’s changes and uncertainties.

Some also theorize that noise is essential to the human brain’s creativity. Johnson-Laird claims that creative mental processes are never predictable. On this basis, he suggests a way to make computers think creatively. We make them alter their own functioning by submitting their programs to artificially-generated random variations. According to Daniel Dennett, such indeterminism is precisely what endows us with what we call free will. Likewise, Bostrom & Sandberg suggest we introduce random noise into our simulation by using pseudo-random number generators. They are not truly random, because eventually the pattern will repeat. But if it takes a very long time before the repetitions appear, then probably it would be sufficiently close to real randomness. It would be a major obstacle, Bostrom & Sandberg write, if artificial noise is not random enough for whole brain emulation.

Research suggests that we may characterize our neural irregularities as pink noise, or what is called 1/f noise. Benoit Mandelbrot classifies 1/f noise as what he terms “wild randomness.” This sort of random might not be so easily simulated. The stock market for example is wildly random. In such natural systems, astronomically improbable fluctuations occur frequently. There is no way to predict when they will appear or how drastic they will be.

For this reason, he considers wild variation to be a state of indeterminism that is qualitatively different than the usual mild variations we encounter at the casino. For, there is infinite variance in the distributions of wild randomness. Anything can happen at any time. He says, “the fluctuation from one value to the next is limitless and frightening.” And this is the wildness of our brains.

Paul Shepard considers our minds to be wild in an even more literal sense: we are wild animals. He distinguishes tameness from domestication. Cows are domesticated. They have been bred to suit our needs. And now their genes would probably not prepare them to live in the wild without human protections. But the human species has merely been tamed by culture and not domesticated like cows. Genetically, we are still the same wild creatures who hunted the Pleistocene savannas. So to emulate the human brain is to simulate the workings not of a rational machine, but of a wild animal. He writes, “The savage mind is ours! ... as a species we have in us the call of the wild.”


But let’s suppose that the brain’s wild randomness can be adequately simulated. Will brain emulation still attain its fullest success of perfectly replicating a specific person’s own identity? Bostrom & Sandberg recognize that neural noise will prevent precise one-to-one emulation. However, they think that the noise will not prevent the simulation from producing meaningful brain states. But to pursue further the personal identity question, let’s imagine that we want to emulate a certain casino slot machine. A relevant property is its unpredictability. So, do we want the emulation and the original to both give consistently the same outcomes? That would happen if we precisely duplicate of all the original’s relevant physical properties. But what about its essential unpredictability? The physically-accurate emulation could predict in advance all the original’s forthcoming read-outs. Or instead, would a more faithful copy of the original produce its own distinct set of unpredictable outcomes? Then we would be replicating the original’s most important relevant property of being governed by chance.

The problem is that the brain’s 1/f noise is wildly random. So suppose we emulate some person’s brain perfectly. And suppose further that the original person and her emulation have an identity merger where each one somehow mistakes themselves for the other. Yet, if both minds are subject to wild variations, then their consciousness and identity might come to differ more than just slightly. They could veer-off wildly. Perhaps our very effort to emulate a specific human brain results in our producing an entirely different mind altogether.

Whether this technology succeeds or fails, it still can advance a number of philosophical debates. It could tell us if our minds emerge from our brains; if the philosophy of artificial intelligence should take analog more seriously. We might learn whether our brain’s randomness is responsible for creativity, adaptation, and free choice; or, if this randomness is the reason our personal identities cannot be duplicated. The only failure, as I see it, is if we neglect this technology’s philosophical potential.


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14 Jun 2009

Even While Men’s Minds are Wild?; Bostrom and Sandberg's Brain Emulation, Examined and Critiqued. Section 5


by Corry Shores
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[The following is tentative material for my presentation at the Society for Philosophy & Technology Conference this summer.]




Corry Shores


Do Posthumanists Dream of Pixilated Sheep?

Bostrom and Sandberg's Brain Emulation,

Examined and Critiqued


Section 5:


Even While Men’s Minds are Wild?



Neural noise can result from external interferences like magnetic fields. Or internal random fluctuations might make the signals unpredictable. (Ward 116-117) In both cases, chance & chaos reign our brains. According to Steven Rose, our brain is an “uncertain” system on account of “random, indeterminate, and probabilistic” events that are essential to its functioning (Rose 93). Alex Pouget and his research team recently found that the mind's ability to compute complex calculations has much to do with its noise. The word noise is misleading, he says, because it implies something goes wrong. But these unpredictable irregularities are the mind’s way of running at optimum performance. Our mind produces noisy signals to represent the uncertainty of the world around us. Pouget explains,

if we want to do something, such as jump over a stream, we need to extract data that is not inherently part of that information. We need to process all the variables we see, including how wide the stream appears, what the consequences of falling in might be, and how far we know we can jump.

In this way, the brain is flooded with countless variables. And the neurons transmit various signal patterns for the same stimulus. This allows us to estimate margins of error. We then use a probabilistic inference to make what is most likely to be the best decision. (Pouget, interview with Science Daily) So we might jump the stream, if probably we can cross it, even though we can never be certain about such matters.

Some also theorize that noise is essential to the human brain’s creativity. Johnson-Laird claims that creative mental processes are never predictable. (Johnson-Laird, The Computer and the Mind 256) He hypothesizes that we could make a machine creative by programming it to alter its own functioning according to generated random variations. (Human and Machine Thinking, 119-120) This would produce what Ben Goertzel refers to as “a complex combination of random chance with strict, deterministic rules.” (Goertzel 119) And according to Daniel Dennett, this indeterminism is precisely what endows us with what we call free will. (Dennett 295, cited in Dartnall 37) Likewise, Bostrom & Sandberg suggest we introduce random noise into our simulation by using pseudo-random number generators. They are not truly random, because eventually the pattern will repeat. But if it takes a very long time before the repetitions appear, then probably it would be sufficiently close to real randomness (Bostrom & Sandberg 38-39). Also, there might be random variations that are hidden to our observations, and thus would not be properly represented in the simulation. They recognize the profound difficulty in incorporating true randomness or hidden variables into the simulation. Yet they believe these sorts of randoms most likely will be unnecessary for whole brain emulation.

But perhaps there is more to consider. Lawrence Ward reviews findings that demonstrate neural noise is pink noise, or what is called 1/f noise. (Ward 145-153, citing research by Lundström and McQueen, and Novikov, Shannonhoff-Khalsa, Schwartz, and Wright) On account of its fractal nature, 1/f noises are always parts of similar larger-orders of variation happening on much longer time-scales. We might have to wait weeks or months to see larger-scale variations that were varying the randomness of the more local noisy events. (Anderson & Mandell 78-79) These are what Gregory Bateson calls metarandom variables. They are hidden to us, because we never see the whole picture (Bateson Steps to an Ecology 418). It’s why live lobsters never notice their cooking water gradually increase to boil (Mind and Nature 109). They only notices alterations on a local level, so nothing really seems to be changing. In a similar way, if all we are observing is randomness on a smaller scale, we might be missing the larger scale variations. It would be like randomly adjusting a radio to pick up different bands of radio static. If all we knew was the randomness of radio static at each moment, we might not also notice the higher order randomness that varies the lower one that we are listening-to. Because these uncontrollable unpredictabilities are essential to all the random changes happening around us, Bateson calls them wild variables. (Mind and Nature 49-50)

Perhaps it is for similar reasons that Benoit Mandelbrot classifies 1/f noise under what he terms “wild randomness” and “wild variation.” (Mandelbrot The (mis)Behavior of Markets 39-41) This sort of random might not be so easily simulated. Mandelbrot gives two reasons for this.

1) In wild randomness, there are events that defy the normal random distribution of the bell curve. He cites a number of stock market events that are astronomically improbable. But such events in fact happen quite frequently in natural systems despite their seeming impossibility. There is no way to predict when they will happen or how drastic they will be. (The (mis)Behavior of Markets 4)

2) Each event is random and yet it is not independent from the rest, like each toss of a coin is. One seemingly small anomalous event will echo like reverberations at unpredictable intervals into the future. (The (mis)Behavior of Markets 181-185)

For these reasons, he considers wild variation to be a qualitatively different state of indeterminism than the usual mild variations we encounter at the casino. For, there is infinite variance in the distributions of wild randomness. Anything can happen at any time (Mandelbrot, Fractals and Scaling 128). He says, “the fluctuation from one value to the next is limitless and frightening.” (Mandelbrot (mis)Behavior of Markets 39-41) This is the wildness of our brains.

Paul Shepard considers our minds to be wild in an even more literal sense: we are wild animals. He distinguishes tameness from domestication. Cows are domesticated. They have been bred to suit our needs. And now their genes would probably not prepare them to live in the wild without human protections. But the human species has merely been tamed by culture and not domesticated like cows. Genetically, we are still the same wild creatures who hunted the Pleistocene savannas. So to emulate the human brain is to simulate the workings not of a rational machine, but of a wild animal. (Shepard 132-133) He writes, “The savage mind is ours! ... as a species we have in us the call of the wild.” (143)

Shepard’s characterizes the wild, like Bateson and Mandelbrot do, as being too complex for any simulation. But he offers his own theory to explain why. He notes the fractal nature of reality. Within every scale is another smaller scale, and so on to infinity. He says that every layer of complexity operates according to deterministic principles. But, there is no lowest level of complexity. Hence there is no way to get to the bottom of what is happening now. It’s turtles all the way down. Thus there is no way to fully understand why things are the way they are now. And thus we can never know how things will be in the future. (146-147)


But let’s suppose that the brain’s wild randomness can be adequately simulated. Will brain emulation still attain its fullest success of perfectly replicating a specific person’s own identity? Bostrom & Sandberg recognize that neural noise will prevent precise one-to-one emulation. However, they think that the noise will not prevent the simulation from producing meaningful brain states (
Bostrom & Sandberg 7). But to pursue further the personal identity question, let’s imagine that we want to emulate a certain slot machine. A relevant property is its unpredictability. Consider these two possibilities. 1) We set the original and the simulation to the same starting position. We give both handles a number of pulls. Each time, they both show the same outcomes, because we replicated the mechanics perfectly. But then, we cannot say that we have preserved its relevant essential property of being unpredictable. For, we can just run the simulator by itself and that will predict the original’s future outcomes. Or, 2) instead the emulation produced its own different random series of outcomes. Then in fact we would be replicating the original’s property of unpredictability.

The problem is that the brain’s 1/f noise is wildly random. So suppose we emulate some person’s brain perfectly. And suppose further that the original person and her emulation have an identity merger where each one thinks they are talking to their very own selves when really they are talking to the other. They confuse themselves with one another. They are completely aware of what is in the other’s mind at that first moment, because they can tell it is the same as what is in their own mind. And suppose further that the original person loses her fear of death, knowing that something she cannot distinguish from himself will carry on after her body dies. But if both minds are subject to wild variations, then their consciousness and identity might come to differ more than just slightly. They could veer-off wildly. The original person and her emulation might become so mistrustful of each other, that they want to end the other’s existence.

So we might need to emulate this wild neural randomness. But that seems to remove the possibility that the emulation will continue on as the original person. Perhaps our very effort to emulate a specific human brain results in our producing an entirely different brain altogether.


[Next entry in this series.]


Anderson, Carl M. & Arnold J. Mandell. Fractal Time and the Foundations of Consciousness: Vertical Convergence of 1/fPhenomena from Ion Channels to Behavior States. in Fractals of Brain, Fractals of Mind. Ed. Earl Mac Cormac & Maxim I. Stamenov. Amsterdam: John Benjamins Publishing Company, 1996. More information and limited preview available at: http://books.google.be/books?id=WdERazd7Ik4C&hl=en


Bateson, Gregory. "Effects of Conscious Purpose on Human Adaptation." in Steps to an Ecology of Mind. London: Granada Publishing, 1972. . More information and limited preview available at: http://books.google.be/books?id=FQvfqk31zFQC&hl=en


Bateson, Gregory. Mind and Nature: A Necessary Unity. London: Fontana, 1979. More information available at: http://books.google.be/books?id=aQtHAAAAMAAJ&hl=en&pgis=1


Sandberg, A. & Bostrom, N. (2008): Whole Brain Emulation: A Roadmap, Technical Report #20083, Future of Humanity Institute, Oxford University. Available online at:http://www.fhi.ox.ac.uk/Reports/2008-3.pdf


Dartnall, Terry. "Introduction: On Having a Mind of Your Own." in Artificial Intelligence and Creativity: An Interdisciplinary Approach. Ed. Terry Dartnall. Dordrecht: Kluwer Academic Publishers, 1994. More information and limited preview available at: http://books.google.be/books?id=4phC9RwvC8YC&hl=en


Dennett, Daniel. Brainstorms. Hassocks: Harvester Press, 1978. More information available at: http://books.google.be/books?id=s3V-AAAAMAAJ&hl=en&pgis=1


Goertzel, Ben. Chaotic Logic: Language, Thought, and Reality from the Perspective of Complex Systems Science. London: Plenum Press, 1994. More information and limited preview available at: http://books.google.be/books?id=zVOWoXDunp8C&hl=en

Johnson-Laird, R. N. The Computer and the Mind: An Introduction to Cognitive Science. Cambridge: Harvard University Press, 1988. More information and limited preview available at: http://books.google.be/books?id=Tf5gRFgVuegC&hl=en


Johnson-Laird, Philip. Human and Machine Thinking. London: Lawrence Erlbaum Associates, Publishers, 1993. More information and limited preview available at: http://books.google.be/books?id=sPbdQjtkIhkC&hl=en


Mandelbrot, Benoit B., & Richard L. Hudson. The (mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward. New York: Basic Books, 2004. More information available at: http://books.google.be/books?id=DPwBTj99a7UC&hl=en


Science Daily. "Mysterious 'Neural Noise' Actually Primes Brain For Peak Performance." Nov. 13, 2006. Available online at: http://www.sciencedaily.com/releases/2006/11/061112094812.htm


Shepard, Paul. Coming Home to the Pleistocene. Washington, D.C.: Island Press, 1998. More information and limited preview available at: http://books.google.be/books?id=5b18NqLB8LMC&hl=en


Ward, Lawrence M. Dynamical Cognitive Science. London: MIT Press, 2002. More information and limited preview available at: http://books.google.be/books?id=g1ZMAoWGYesC&hl=en


Also mentioned:

(Lundström and McQueen, 1974, "A proposed 1/f noise mechanism in nerve cell membranes," Journal of Theoretical Biology, 45, 405-409).


(Novikov E., A. Novikov, Shannonhoff-Khalsa, Schwartz, and Wright, 1997, "Scale-similar activity in the brain," Physical Review E, 56, R2387-R2389,)



9 Jun 2009

Mandelbrot's Folds within Fractal Folds, in The (mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward

by Corry Shores
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[The following is quotation.]





Folds within Fractal Folds


Benoit Mandelbrot & Richard Hudson

The (mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward

Chapter VII
Studies in Roughness:
A Fractal Primer

Perhaps the most striking idea in fractal geometry is its peculiar view of dimension. Since Euclid’s day, an imaginary mathematical point has had no dimension, a line has had one, a plane, two, and the familiar space we live in, three. Einstein added a fourth, time. Mathematics can generalize the idea, and imagine higher dimensions – purely fictitious, but useful for solving a problem in engineering, economics, or physics. Topology, the mathematical study of surfaces, adds some interesting new twists. From a topological point of view, a cucumber is the same as an orange because on can be remolded into the same shape as the other without having to cut the surface. And the circumference of a circle has the same dimension, one, as a jagged coastline on a shipping map. They are both continuous lines; one can be transformed into the other just by bending, folding and stretching – without cutting.


But is that all there is to dimension? Look at a ball of thread, and think about it first from the idealized viewpoint of Euclid. Assume it is five inches in diameter, made of fiber a fraction of an inch thick. From a long distance away, you can barely see the ball; it is, effectively, a point – of no dimension, according to classical geometry. Hold it in your hand, and it resolves to a normal, three-dimensional ball. Bring it up closer: You see it is a tangle of one-dimensional fibers. Closer still, and the fibers are clearly three-dimensional strands. Keep going until the atoms resolve in an electron microscope: Back to zero-dimensional points again. So what is this ball of thread, anyway? Zero, one, or three dimensions? It depends on your point of view. For a complex natural shape, dimension is relative. It varies with the observer. The same object can have more than one dimension, depending on how you measure it and what you want to do with it. And dimension need not be a whole number; it can be fractional. Now an ancient concept dimension, becomes thoroughly modern. (129)

Think of dimension, not as an inherent property, but as a tool of measurement. So how do you actually measure something? If you want to measure a straight line, you get a ruler. If you want to measure a curved line, you could use a smaller ruler, inching it along the curve and counting how many times you moved it. You could get a more accurate, if tedious, measure by using a still-smaller ruler; its measurement will be a bit longer than the first, crude one. Eventually, as the ruler keeps shirking, the measurement settles down to one number that you call the curve’s length. But what if the curve is jagged and irregular? What if it is the coast of Scotland? You can start off with a surveyor’s glass – a big ruler – and measure from promontory to promontory. Then a long tape might measure point to point. Then a yardstick, then calipers, then a microscope. But this is useless: Unlike the smooth curve, the rocky coastline never provides just one “best” estimate of length. It depends on the scale of the map you want to draw – or your political motives. One researcher, Lewis Fry Richardson, who investigated this paradox nearly a century ago, looked in official references for the surveyed length of political borders between countries. Spanish authorities reckoned their border with Portugal to be 987 kilometers long, whereas the plucky Portuguese counted 1,214 kilometers. The Netherlands measured its border with smaller, poorer Belgium at 380 kilometers, whereas the Belgians counted 449 kilometers.

So how long is it? A useless question, as we have seen. But one way around the problem is to plot on graph paper the measurement you get for each size ruler you use. Of course, the measurements increase as the rulers shrink. But – happy surprise – they often do so at a near-steady rate. Start with a trivial example, a straight line. Say the first ruler you use happens to be exactly the length of the line. Now try a smaller ruler, half as big: it measures the line as two of its lengths. Another ruler, half again as big as the last; the line is four of its lengths. You get the picture. But now try measuring that jagged coastline mentioned earlier. Something unusual develops as you use ever-smaller rulers: The length you measure is growing faster than the rulers are shrinking. (130) And that phenomenon is measured by a quantity called fractal dimension. Begin simply. For a straight line, the fractal dimension is 1. And one dimension is exactly what we expect a straight line to have. But the British coastline, it turns out, has a fractal dimension of about 1.25. Does that make sense? Certainly. A rugged coast is more intricate than a one-dimensional straight line; but however numerous its crags and bays, its outline would not be so intensely convoluted as to fill a two dimensional square.

That is not all. The Australian coastline, less rugged than the Cornish, turns out to have a fractal dimension of 1.13. By contrast, the smooth South African shore has a dimension 1.02, only slightly rougher than a straight line. Another example: rivers. A U.S. Geological Survey study of the course of large American rivers found they have a typical fractal dimension of 1.3 in the East; but in the wilder West, it is 1.4. Again, the measurement fits our intuition of the difference between the rugged Colorado and the placid Charles. Other examples: If you measure the immensely intricate surface area inside the lungs, through which a network of branching bronchia stretch, you find that the total area is vast – something like that of a tennis court. But the fractal dimension is very close to 3. The lining is so convoluted and folded in upon itself that it partakes something of a three-dimensional nature.

What have we here? A new tool to measure, not how long, heavy, hot, or loud something is, but how convoluted and irregular it is. It provides science with its first yardstick for roughness. (131)




Mandelbrot, Benoit B., & Richard L. Hudson. The (mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward. New York: Basic Books, 2004.