"Statisticians can calculate the probability that such random samples represent the population; this is usually expressed in terms of sampling error [...]. The real problem is that few samples are random. Even when researchers know the nature of the population, it can be time-consuming and expensive to draw a random sample; all too often, it is impossible to draw a true random sample because the population cannot be defined. This is particularly true for studies of social problems. [...] The best samples are those that come as close as possible to being random." (Joel Best, "Damned Lies and Statistics: Untangling Numbers from the Media, Politicians, and Activists", 2001)
"[...] a general-purpose universal optimization strategy is theoretically impossible, and the only way one strategy can outperform another is if it is specialized to the specific problem under consideration." Yu-Chi Ho & David L Pepyne, "Simple explanation of the no-free-lunch theorem and its implications", Journal of Optimization Theory and Applications 115, 2002)
"The diversity of networks in business and the economy is mindboggling. There are policy networks, ownership networks, collaboration networks, organizational networks, network marketing-you name it. It would be impossible to integrate these diverse interactions into a single all-encompassing web. Yet no matter what organizational level we look at, the same robust and universal laws that govern nature's webs seem to greet us. The challenge is for economic and network research alike to put these laws into practice." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)
"At the basis of the impossibility of making reliable predictions for systems such as the atmosphere, there is a phenomenon known today as the butterfly effect. This deals with the progressive limitless magnification of the slightest imprecision (error) present in the measurement of the initial data (the incomplete knowledge of the current state of each molecule of air), which, although in principle negligible, will increasingly expand during the course of the model’s evolution, until it renders any prediction on future states (atmospheric weather conditions when the forecast refers to more than a few days ahead) completely insignificant, as these states appear completely different from the calculated ones." (Cristoforo S Bertuglia & Franco Vaio, "Nonlinearity, Chaos, and Complexity: The Dynamics of Natural and Social Systems", 2003)
"How is it that -1 can have a square root? The square of a positive number is always positive, and the square of a negative number is again positive (and the square of 0 is just 0 again, so that is hardly of use to us here). It seems impossible that we can find a number whose square is actually negative." (Sir Roger Penrose, "The Road to Reality: A Complete Guide to the Laws of the Universe", 2004)
"What was impossible, inconceivable, and incoherent based on literal vocabulary becomes possible, conceivable, and coherent through metaphoric redescription. Combinations of terms that were incoherent, in relation to the conventional rules of meaning, become meaningful. Metaphoric description arises from a momentary suspension of the rules for literal vocabulary. The semantics of a metaphor convey an alternative realm of conceptual possibilities, through a new set of possible attributes. Of course, not all scientific language is metaphoric. But when unexpected empirical findings raise serious doubts about a familiar scientific theory, a satisfactory resolution occur through the use of metaphoric vocabulary." (Daniel Rothbart [Ed.], "Modeling: Gateway to the Unknown", 2004)
"Apparent Impossibilities that Are New Truths […] irrational numbers, imaginary numbers, points at infinity, curved space, ideals, and various types of infinity. These ideas seem impossible at first because our intuition cannot grasp them, but they can be captured with the help of mathematical symbolism, which is a kind of technological extension of our senses." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)
"It is impossible for √-1 to be a real number, since its square is negative. This implies that √-1 is neither greater nor less than zero, so we cannot see √-1 on the real line. However, √-1 behaves like a number with respect to + and x. This prompts us to look elsewhere for it, and indeed we find it on another line (the imaginary axis) perpendicular to the real line." (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)
"Once you understand a problem, many aspects of it suddenly become much simpler. As mathematicians the world over say, everything is either impossible or trivial." (Ian Stewart, "Letters to a Young Mathematician", 2006)
"The model theory postulates that mental models are parsimonious. They represent what is possible, but not what is impossible, according to assertions. This principle of parsimony minimizes the load on working memory, and so it applies unless something exceptional occurs to overrule it." (Philip N Johnson-Laird, Mental Models, Sentential Reasoning, and Illusory Inferences, [in "Mental Models and the Mind"], 2006)
"People don’t need to know all the details of how a complex mechanism actually works in order to use it, so they create a cognitive shorthand for explaining it, one that is powerful enough to cover their interactions with it, but that doesn’t necessarily reflect its actual inner mechanics. […] In the digital world, however, the differences between a user’s mental model and the implementation model are often quite distinct. The discrepancy between implementation and mental models is particularly stark in the case of software applications, where the complexity of implementation can make it nearly impossible for the user to see the mechanistic connections between his actions and the program’s reactions." (Alan Cooper et al, "About Face 3: The Essentials of Interaction Design", 2007)
"Yet, with the discovery of the butterfly effect in chaos theory, it is now understood that there is some emergent order over time even in weather occurrence, so that weather prediction is not next to being impossible as was once thought, although the science of meteorology is far from the state of perfection." (Peter Baofu, "The Future of Complexity: Conceiving a Better Way to Understand Order and Chaos", 2007)
"Algebraic symbols carry a universality of interpretation that allows them to be manipulated in a way that words cannot. Indeed, this was the key breakthrough that allowed mathematics to flourish in a way that was not possible until the advent of algebra. All higher mathematics relies on constant use of algebraic manipulation and would be impossible without it." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)
"It is impossible to construct a model that provides an entirely accurate picture of network behavior. Statistical models are almost always based on idealized assumptions, such as independent and identically distributed (i.i.d.) interarrival times, and it is often difficult to capture features such as machine breakdowns, disconnected links, scheduled repairs, or uncertainty in processing rates." (Sean Meyn, "Control Techniques for Complex Networks", 2008)
"Prior to the discovery of the butterfly effect it was generally believed that small differences averaged out and were of no real significance. The butterfly effect showed that small things do matter. This has major implications for our notions of predictability, as over time these small differences can lead to quite unpredictable outcomes. For example, first of all, can we be sure that we are aware of all the small things that affect any given system or situation? Second, how do we know how these will affect the long-term outcome of the system or situation under study? The butterfly effect demonstrates the near impossibility of determining with any real degree of accuracy the long term outcomes of a series of events." (Elizabeth McMillan, Complexity, "Management and the Dynamics of Change: Challenges for practice", 2008)
"Topology allows the possibility of making qualitative predictions when quantitative ones are impossible." (Timothy Gowers, "The Princeton Companion to Mathematics", 2008)
"Perhaps the simplest way to explain symmetry is to follow the operational approach used by mathematicians: a symmetry is a motion. That is, suppose you have an object and pick it up, move it around, and set it down. If it is impossible to distinguish between the object in its original and final positions, we say that it has a symmetry." (Michael Field & Martin Golubitsky, "Symmetry in Chaos: A Search for Pattern in Mathematics, Art, and Nature" 2nd Ed, 2009)
"With Kurt Gödel, we find in the twentieth century the idea that formal systems are incomplete, a concept that is perhaps important to chess theory. If undecidable statements exist in chess, then it is impossible to solve them completely with a computer chess program." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)
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