"A system may be called complex here if its dimension" (order) is too high and its model" (if available) is nonlinear, interconnected, and information on the system is uncertain such that classical techniques can not easily handle the problem." (M Jamshidi, "Autonomous Control on Complex Systems: Robotic Applications", Current Advances in Mechanical Design and Production VII, 2000)
"For string theory to make sense, the universe should have nine spatial dimensions and one time dimension, for a total of ten dimensions." (Brian Greene, "The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory", 2000)
"If string theory is right, the microscopic fabric of our universe is a richly intertwined multidimensional labyrinth within which the strings of the universe endlessly twist and vibrate, rhythmically beating out the laws of the cosmos." (Brian Greene, "The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory", 2000)
"One measure of the depth of a physical theory is the extent to which it poses serious challenges to aspects of our worldview that had previously seemed immutable." (Brian Greene, "The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest, for the Ultimate Theory", 2000)
"Sometimes attaining the deepest familiarity with a question is our best substitute for actually having the answer." (Brian Greene, "The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest, for the Ultimate Theory", 2000)
"String theory [...] resolves the central dilemma confronting contemporary physics - the incompatibility between quantum mechanics and general relativity - and that unifies our understanding of all of nature's fundamental material constituents and forces. But to accomplish these feats, [...] string theory requires that the universe have extra space dimensions. " (Brian Greene, "The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory", 2000)
"The fruitful generalization in mathematics often involves starting from a commonsense concept such as a point on a line. A mathematical framework is then developed within which the particular example of a point in space is seen to be just a very special case of a much broader structure, say a point in three-dimensional space. Further generalizations then show this new structure itself to be only a special case of an even broader framework, the notion of a point in a space of n dimensions. And so it goes, one generalization piled atop another, each element leading to a deeper understanding of how the original object fits into a bigger picture." (John L Casti, "Five More Golden Rules : Knots, Codes, Chaos, and Other Great Theories of 20th Century Mathematics", 2000)
"The greatest plus of data modeling is that it produces a simple and understandable picture of the relationship between the input variables and responses [...] different models, all of them equally good, may give different pictures of the relation between the predictor and response variables [...] One reason for this multiplicity is that goodness-of-fit tests and other methods for checking fit give a yes–no answer. With the lack of power of these tests with data having more than a small number of dimensions, there will be a large number of models whose fit is acceptable. There is no way, among the yes–no methods for gauging fit, of determining which is the better model." (Leo Breiman, "Statistical modeling: The two cultures" Statistical Science 16(3), 2001)
"Engineering is quite different from science. Scientists try to understand nature. Engineers try to make things that do not exist in nature. Engineers stress invention. To embody an invention the engineer must put his idea in concrete terms, and design something that people can use. That something can be a device, a gadget, a material, a method, a computing program, an innovative experiment, a new solution to a problem, or an improvement on what is existing. Since a design has to be concrete, it must have its geometry, dimensions, and characteristic numbers. Almost all engineers working on new designs find that they do not have all the needed information. Most often, they are limited by insufficient scientific knowledge. Thus they study mathematics, physics, chemistry, biology and mechanics. Often they have to add to the sciences relevant to their profession. Thus engineering sciences are born." (Yuan-Cheng Fung & Pin Tong, "Classical and Computational Solid Mechanics", 2001)
"Heisenberg’s principle must be considered a special case of the complementarity principle […]. This states that an experiment on one aspect of a system" (of atomic dimensions) destroys the possibility of learning about a complementarity aspect of the same system. Together these principles have shocking consequences for the comprehension of entropy and determinism." (Lars Skyttner, "General Systems Theory: Ideas and Applications", 2001)
"The greatest plus of data modeling is that it produces a simple and understandable picture of the relationship between the input variables and responses [...] different models, all of them equally good, may give different pictures of the relation between the predictor and response variables [...] One reason for this multiplicity is that goodness-of-fit tests and other methods for checking fit give a yes–no answer. With the lack of power of these tests with data having more than a small number of dimensions, there will be a large number of models whose fit is acceptable. There is no way, among the yes–no methods for gauging fit, of determining which is the better model." (Leo Breiman, "Statistical modeling: The two cultures" Statistical Science 16(3), 2001)
"A theory makes certain predictions and allows calculations to be made that can be tested directly through experiments and observations. But a theory such as superstrings talks about quantum objects that exist in a multidimensional space and at incredibly short distances. Other grand unified theories would require energies close to those experienced during the creation of the universe to test their predictions." (F David Peat, "From Certainty to Uncertainty", 2002)
"Complex numbers are really not as complex as you might expect from their name, particularly if we think of them in terms of the underlying two dimensional geometry which they describe. Perhaps it would have been better to call them 'nature's numbers'. Behind complex numbers is a wonderful synthesis between two dimensional geometry and an elegant arithmetic in which every polynomial equation has a solution." (David Mumford, Caroline Series & David Wright, "Indra’s Pearls: The Vision of Felix Klein", 2002)
"Networks do not offer a miracle drug, a strategy that makes you invincible in any business environment. The truly important role networks play is in helping existing organizations adapt to rapidly changing market conditions. The very concept of network implies a multidimensional approach." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)
"Scientists often invent words to fill the holes in their understanding.These words are meant as conveniences until real understanding can be found. […] Words such as dimension and field and infinity […] are not descriptions of reality, yet we accept them as such because everyone is sure someone else knows what the words mean." (Scott Adams, "God's Debris: A Thought Experiment", 2004"
"[…] the laws of physics, carefully constructed after thousands of years of experimentation, are nothing but the laws of harmony one can write down for strings and membranes." (Michio Kaku, "Parallel Worlds: A journey through creation, higher dimensions, and the future of the cosmos", 2004)
"Chaos theory, for example, uses the metaphor of the ‘butterfly effect’. At critical times in the formation of Earth’s weather, even the fluttering of the wings of a butterfly sends ripples that can tip the balance of forces and set off a powerful storm. Even the smallest inanimate objects sent back into the past will inevitably change the past in unpredictable ways, resulting in a time paradox." (Michio Kaku, "Parallel Worlds: A journey through creation, higher dimensions, and the future of the cosmos", 2004)
"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." (Benoît B Mandelbrot, "The" (Mis)Behavior of Markets", 2004)
"Scientists often invent words to fill the holes in their understanding. These words are meant as conveniences until real understanding can be found. […] Words such as dimension and field and infinity […] are not descriptions of reality, yet we accept them as such because everyone is sure someone else knows what the words mean." (Scott Adams, "God's Debris: A Thought Experiment", 2004)
"The important thing is to understand that frequentist and Bayesian methods are answering different questions. To combine prior beliefs with data in a principled way, use Bayesian inference. To construct procedures with guaranteed long run performance, such as confidence intervals, use frequentist methods. Generally, Bayesian methods run into problems when the parameter space is high dimensional." (Larry A Wasserman, "All of Statistics: A concise course in statistical inference", 2004)
"Minkowski calls a spatial point existing at a temporal point a world point. These coordinates are now called 'space-time coordinates'. The collection of all imaginable value systems or the set of space-time coordinates Minkowski called the world. This is now called the manifold. The manifold is four-dimensional and each of its space-time points represents an event." (Friedel Weinert," The Scientist as Philosopher: Philosophical Consequences of Great Scientific Discoveries", 2005)
"A five-dimensional space is not a strange deformation of ordinary space, one that only mathematicians can see, but a place where numbers are collected in ordered sets. When string theorists talk of the eleven dimensions required by their latest theory, they are not encouraging one another to search for eight otherwise familiar spatial dimensions that have somehow become lost. They are saying only that for their purposes, eleven numbers are needed to specify points. Where they are is no one’s business." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)
"When real numbers are used as coordinates, the number of coordinates is the dimension of the geometry. This is why we call the plane two-dimensional and space three-dimensional. However, one can also expect complex numbers to be useful, knowing their geometric properties […] What is remarkable is that complex numbers are if anything more appropriate for spherical and hyperbolic geometry than for Euclidean geometry. With hindsight, it is even possible to see hyperbolic geometry in properties of complex numbers that were studied as early as 1800, long before hyperbolic geometry was discussed by anyone." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)
"Mathematical language is littered with pejorative and mystical terms - such as irrational, imaginary, surd, transcendental - that were once used to ridicule supposedly impossible objects. And these are just terms applied to numbers. Geometry also has many concepts that seem impossible to most people, such as the fourth dimension, finite universes, and curved space - yet geometers" (and physicists) cannot do without them. Thus there is no doubt that mathematics flirts with the impossible, and seems to make progress by doing so." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)
"The word 'complex' was introduced m a well-meaning attempt to dispel the mystery surrounding 'imaginary' or 'impossible' numbers, and" (presumably) because two dimensions are more complex than one Today, 'complex' no longer seems such a good choice of word. It is usually interpreted as 'complicated', and hence is almost as prejudicial as its predecessors. Why frighten people unnecessarily? If you are not sure what 'analysis' is, you won't want to know about 'complex analysis' - but it is the best part of analysis." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)
"When real numbers are used as coordinates, the number of coordinates is the dimension of the geometry. This is why we call the plane two-dimensional and space three-dimensional. However, one can also expect complex numbers to be useful, knowing their geometric properties." (John Stillwell,"Yearning for the impossible: the surpnsing truths of mathematics", 2006)
"Art and music make manifest, by bringing into conscious awareness, that which has previously been felt only tentatively and internally. Art, in its widest sense, is a form of play that lies at the origin of all making, of language, and of the mind's awareness of its place within the world. Art, in all its forms, makes manifest the spiritual dimension of the cosmos, and expresses our relationship to the natural world. This may have been the cause of that natural light which first illuminated the preconscious minds of early hominids." (F David Peat, "Pathways of Chance", 2007)
"In maps we have scale models of terrain, but projected onto a plane, thus producing occlusion of a sort not inherent to three-dimensional imaging. Maps do not usually have an obvious perspective; but we see perspectivity when, for example, the curvature of the earth makes marginal distortion inevitable as a result of this projection that lowers the dimensionality. A map too is the product of a measuring procedure, but they bring to light a much more important point about ‘point of view’, essentially independent of these limitations in cartography. The point extends to all varieties of modeling, but is made salient by the sense in which use enters the concept of ‘map’ from the beginning. A map is not only an object used to represent features of a terrain, it is an object for the use of the industrial designer, the navigator, and most of all the traveler, to plan and direct action. This brings us to an aspect of scientific representation not touched on so far, though implicit in the discussion of perspective, crucial to its overall understanding: its indexicality. " (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)
"As art, chess speaks to us of the personal decisions that are made in the course of a game. Looking at this facet of the game, the essential protagonist is the aesthetic sense rather than the capacity for calculation, which thus moves us closer to the human dimension and farther from mathematical algorithms." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)
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