Hunting the Hidden Dimension

Determined to understand the repeating patterns he was finding in nature, French mathematician Benoit Mandelbrot used an early form of computer imagery to produce his own versions, coining the recurring shapes fractals. This installment of the PBS series "Nova" examines the rules of these self-similar patterns and explores the ways these fascinating geometric configurations can be applied in the fields of science, medicine and the arts. "The Fractal Geometry of Nature".

This is an excellent hour documentary that NOVA is known for. Everything is simply described. First you'll find out what the heck a fractal is. Then you learn its applications; from finding tumors; mapping the oxygen flow of a rainforest; weaving designs on shirts; and computer graphics. The computer graphics show how lava was simulated by fractal. That example is my favorite part of the entire video.

You will be opened to another subject of mathematics much like when you first learn algebra, trigonometry, or calculus. Fractals are a subject of their own. Prepare for a "mind expanding" video. And at the same time help support PBS.

NOVA: Secrets of the Mind  NOVA - The Elegant Universe  NOVA - Genius: The Science of Einstein, Newton, Darwin, and Galileo  Decoding Reality: The Universe as Quantum Information

PBS Nova - Fractals - Hunting the Hidden Dimension

 

Fermat's Last Theorem

Documentary - Mathematics
Simon Singh and John Lynch’s film tells the enthralling and emotional story of Andrew Wiles. A quiet English mathematician, he was drawn into maths by Fermat’s puzzle, but at Cambridge in the ’70s, FLT was considered a joke, so he set it aside. Then, in 1986, an extraordinary idea linked this irritating problem with one of the most profound ideas of modern mathematics: the Taniyama-Shimura Conjecture, named after a young Japanese mathematician who tragically committed suicide.

The link meant that if Taniyama was true then so must be FLT. When he heard, Wiles went after his childhood dream again. “I knew that the course of my life was changing.” For seven years, he worked in his attic study at Princeton, telling no one but his family. “My wife has only known me while I was working on Fermat”, says Andrew.

In June 1993 he reached his goal. At a three-day lecture at Cambridge, he outlined a proof of Taniyama – and with it Fermat’s Last Theorem. Wiles’ retiring life-style was shattered. Mathematics hit the front pages of the world’s press. Then disaster struck. His colleague, Dr Nick Katz, made a tiny request for clarification. It turned into a gaping hole in the proof. As Andrew struggled to repair the damage, pressure mounted for him to release the manuscript – to give up his dream. So Andrew Wiles retired back to his attic.

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Links; Mathworld.wolfram.com | Wikipedia

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The Mind, Machines, and Mathematics

Creativity: The Mind, Machines, and Mathematics: A Public Debate
November 30, 2006 Running Time: 0:59:10
About the Lecture
Two of the sharpest minds in the computing arena spar gamely, but neither scores a knockdown in one of the oldest debates around: whether machines may someday achieve consciousness. (NB: Viewers may wish to brush up on the work of computer pioneer Alan Turing and philosopher John Searle in preparation for this video.)

Ray Kurzweil confidently states that artificial intelligence will, in the not distant future, “master human intelligence.” He cites the “exponential power of growth in technology” that will enable both a minute, detailed understanding of the human brain, and the capacity for building a machine that can at least simulate original thought. The “frontier” such a machine must cross is emotional intelligence—“being funny, expressing loving sentiment…” And when this occurs, says Kurzweil, it’s not entirely clear that the entity will have achieved consciousness, since we have no “consciousness detector” to determine if it is capable of subjective experiences.

Acknowledging that his position will prove unpopular, David Gelernter launches his attack: “We won’t even be able to build super-intelligent zombies unless we approach the problem right.” This means admitting that a continuum of cognitive styles exists among humans. As for building a conscious machine, he sees no possibility of one emerging from even the most sophisticated software. “Consciousness means the presence of mental states strictly private with no visible functions or consequences. A conscious entity can call on a thought or memory merely to feel happy, be inspired, soothed, feel anger…” Software programs, by definition, can be separated out, peeled away and run in a logically identical way on any computing platform. How could such a program spontaneously give rise to “a new node of consciousness?”

Kurzweil concedes the difficulty of defining consciousness, but does not want to wish away the concept, since it serves as the basis for our moral and ethical systems. He maintains his argument that reverse engineering of the human brain will enable machines that can act with a level of complexity, from which somehow consciousness will emerge.

Gelernter replies that believing this “seems a completely arbitrary claim. Anything might be true, but I don’t see what makes the claim plausible.” Ultimately, he says, Kurzweil must explain objectively and scientifically what consciousness is -- “how it’s created and got there.” Kurzweil stakes his claim on our future capacity to model digitally the actions of billions of neurons and neurotransmitters, which in humans somehow give rise to consciousness. Gelernter believes such a machine might simulate mental states, but not actually pass muster as a conscious entity. Ultimately, he questions the desirability of building such computers: “We might reach the state some day when we prefer the company of a robot from Walmart to our next-door neighbor or roommates.”

Cognitive Neuroscience of Aging

Review

"This is an ambitious undertaking...chapters dense in information, but actually it works..."--The Psychologist
"This excellent book marks the advent of a new discipline, the cognitive neuroscience of aging. It comprehensively covers measurement tools, empirical findings, and theoretical models. Editors and authors are leading scholars of this evolving discipline. I highly recommend this book to everyone interested in the intriguing dynamic between brain and cognition in old age." -Ulman Lindenberger, Professor of Psychology, Max Planck Institute for Human Development and Director, Center for Lifespan Development
"This is the right book, by the right authors, at the right time. The editors have assembled most of the leading investigators taking a neuroscience approach to the study of cognitive aging, and have asked them to write integrative reviews of the existing literature and to speculate about productive directions for future research. The result is not only a compendium of, in the editors' words "state-of-the-art knowledge about the cognitive neuroscience of aging in 2004," but a valuable source of ideas for research over the next 5 to 10 years." -Timothy Salthouse, Brown-Forman Professor of Psychology, University of Virginia


Product Description
Until very recently, what we knew about the neural basis of cognitive aging was based on two disciplines that had very little contact with each other. Whereas the neuroscience of aging investigated the effects of aging on the brain independently of age-related changes in cognition, the cognitive psychology of aging investigated the effects of aging on cognition independently of age related changes in the brain. Because an increasing number of studies have focused on the relationships between cognitive aging and cerebral aging, these two disciplines have begun to interact. This rapidly growing body of research has come to constitute a new discipline: cognitive neuroscience of aging. The goal of this book is to introduce this new discipline at a level that is useful to both professionals and students in cognitive neuroscience, cognitive psychology, neuroscience, neuropsychology, neurology, and related areas. The book is divided into four main sections. The first section describes noninvasive measures of cerebral aging, including structural (e.g., volumetric MRI), chemical, (e.g., dopamine PET), electrophysiological (e.g., ERP's), and hemodynamic measures (e.g. fMRI), and discusses how they can be linked to behavioral measures of cognitive aging. The second section reviews evidence for the effects of aging on neural activity during different cognitive functions, including perception and attention, use of imagery, working memory, long-term memory, and prospective memory. The third section focuses on clinical and applied concerns, such as the distinction between health aging and aging with Alzheimer's disease, and the use of cognitive training to ameliorate age-related cognitive decline. The final section describes theories that relate cognitive and cerebral aging, including models accounting for functional neuroimaging evidence and models supported by computer simulations. Taken together, the chapters in this volume provide the first unified and comprehensive overview of the new discipline of cognitive neuroscience of aging.

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Product Details

* Format: Kindle Edition
* Print Length: 408 pages
* Publisher: Oxford University Press, USA; 1 edition (October 22, 2004)
* Sold by: Amazon Digital Services
* Language: English
* ASIN: B000TRH2RS

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Uncertain Principles

"Uncertain Principles" - this documentary features the miscellaneous ramblings of a physicist at a small liberal arts college. Physics, politics, pop culture, and occasional conversations with his dog.

Werner Heisenberg formulated the uncertainty principle in Niels Bohr's institute at Copenhagen, while working on the mathematical foundations of quantum mechanics.

In 1925, following pioneering work with Hendrik Kramers, Heisenberg developed matrix mechanics, which replaced the ad-hoc old quantum theory with modern quantum mechanics. The central assumption was that the classical motion was not precise at the quantum level, and electrons in an atom did not travel on sharply defined orbits. Rather, the motion was smeared out in a strange way: the time Fourier transform only involving those frequencies that could be seen in quantum jumps.



Heisenberg's paper did not admit any unobservable quantities like the exact position of the electron in an orbit at any time; he only allowed the theorist to talk about the Fourier components of the motion. Since the Fourier components were not defined at the classical frequencies, they could not be used to construct an exact trajectory, so that the formalism could not answer certain overly precise questions about where the electron was or how fast it was going.

Watch Documentary; Uncertain Principles

About the Uncertainty Principle
In quantum mechanics, the Heisenberg uncertainty principle states that certain pairs of physical properties, like position and momentum, cannot both be known to arbitrary precision. That is, the more precisely one property is known, the less precisely the other can be known. It is impossible to measure simultaneously both position and velocity of a microscopic particle with any degree of accuracy or certainty. This is not only a statement about the limitations of a researcher's ability to measure particular quantities of a system, but once the wave-nature of matter is accepted, the general properties of waves cause the uncertainty principle to be a statement about the nature of the system itself.

In quantum mechanics, a particle is described by a wave. The position is where the wave is concentrated and the momentum is determined by the wavelength. The position is uncertain to the degree that the wave is spread out, and the momentum is uncertain to the degree that the wavelength is ill-defined.

The only kind of wave with a definite position is concentrated at one point, and such a wave has an indefinite wavelength. Conversely, the only kind of wave with a definite wavelength is an infinite regular periodic oscillation over all space, which has no definite position. So in quantum mechanics, there are no states that describe a particle with both a definite position and a definite momentum. The more precise the position, the less precise the momentum.

The uncertainty principle can be restated in terms of measurements, which involves collapse of the wavefunction. When the position is measured, the wavefunction collapses to a narrow bump near the measured value, and the momentum wavefunction becomes spread out. The particle's momentum is left uncertain by an amount inversely proportional to the accuracy of the position measurement. The amount of left-over uncertainty can never be reduced below the limit set by the uncertainty principle, no matter what the measurement process.

This means that the uncertainty principle is related to the observer effect, with which it is often conflated. The uncertainty principle sets a lower limit to how small the momentum disturbance in an accurate position experiment can be, and vice versa for momentum experiments.

Wikipedia Link
Hyperphysics - the uncertainty Principle