"A mature science, with respect to the matter of errors in variables, is not one that measures its variables without error, for this is impossible. It is, rather, a science which properly manages its errors, controlling their magnitudes and correctly calculating their implications for substantive conclusions." (Otis D Duncan, "Introduction to Structural Equation Models", 1975)
"Even the simplest calculation in the purest mathematics can have terrible consequences. Without the invention of the infinitesimal calculus most of our technology would have been impossible." (Stanislaw M Ulam, "Adventures of a Mathematician", 1976)
"Numbers are the product of counting. Quantities are the product of measurement. This means that numbers can conceivably be accurate because there is a discontinuity between each integer and the next. Between two and three there is a jump. In the case of quantity there is no such jump, and because jump is missing in the world of quantity it is impossible for any quantity to be exact. You can have exactly three tomatoes. You can never have exactly three gallons of water. Always quantity is approximate." (Gregory Bateson, "Number is Different from Quantity", CoEvolution Quarterly, 1978)
"[Human consciousness] depends wholly on our seeing the outside world in such categories. And the problems of consciousness arise from putting reconstitution beside internalization, from our also being able to see ourselves as if we were objects in the outside world. That is in the very nature of language; it is impossible to have a symbolic system without it." (Jacob Bronowski, "The origins of knowledge and imagination", 1978)
"The most complex system imaginable is the mind - by definition, since the mind must be at least one degree more complex than whatever it imagines. Catastrophe theory proposes that qualitative stability is a necessary attribute of thought; without it, recognition and memory would be impossible." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)
"Two assumptions are needed to apply catastrophe theory as it now stands: first, that the system described be governed by a potential, and second, that its behavior depend on a limited number of control factors. Without these assumptions, the classification of the elementary catastrophes is impossible." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)
"The world's present industrial civilization is handicapped by the coexistence of two universal, overlapping, and incompatible intellectual systems: the accumulated knowledge of the last four centuries of the properties and interrelationships of matter and energy; and the associated monetary culture which has evolved from folkways of prehistoric origin. […] Despite their inherent incompatibilities, these two systems during the last two centuries have had one fundamental characteristic in common, namely exponential growth, which has made a reasonably stable coexistence possible. But, for various reasons, it is impossible for the matter-energy system to sustain exponential growth for more than a few tens of doublings, and this phase is by now almost over. The monetary system has no such constraints, and according to one of its most fundamental rules, it must continue to grow by compound interest." (Marion K Hubbert, "Two Intellectual Systems: Matter-energy and the Monetary Culture", [seminar] 1981)
"It is actually impossible in theory to determine exactly what the hidden mechanism is without opening the box, since there are always many different mechanisms with identical behavior. Quite apart from this, analysis is more difficult than invention in the sense in which, generally, induction takes more time to perform than deduction: in induction one has to search for the way, whereas in deduction one follows a straightforward path." (Valentino Braitenberg, "Vehicles: Experiments in Synthetic Psychology", 1984)
"If we want to solve problems effectively [...] we must keep in mind not only many features but also the influences among them. Complexity is the label we will give to the existence of many interdependent variables in a given system. The more variables and the greater their interdependence, the greater the system's complexity. Great complexity places high demands on a planner's capacity to gather information, integrate findings, and design effective actions. The links between the variables oblige us to attend to a great many features simultaneously, and that, concomitantly, makes it impossible for us to undertake only one action in a complex system." (Dietrich Dorner, "The Logic of Failure: Recognizing and Avoiding Error in Complex Situations", 1989)
"Clearly, however, a zero probability is not the same thing as an impossibility; […] In systems that are now called chaotic, most initial states are followed by nonperiodic behavior, and only a special few lead to periodicity. […] In limited chaos, encountering nonperiodic behavior is analogous to striking a point on the diagonal of the square; although it is possible, its probability is zero. In full chaos, the probability of encountering periodic behavior is zero." (Edward N Lorenz, "The Essence of Chaos", 1993)
"In practice, it may be impossible to purge a real system of its actual randomness and observe the consequences, but often we can guess what these would be by turning to theory. Most theoretical studies of real phenomena are studies of approximations." (Edward N Lorenz, "The Essence of Chaos", 1993)
"Why is it so important to find primes, or to show that a certain integer is one? A very practical application in cryptography rests on the fact that since it is extremely hard to factor very large numbers, a two-hundred-digit number that was the product of two primes could govern text encoding: It would be virtually impossible to guess what the two numbers were if you didn't know them in advance, and out of the question (save perhaps on a state-of-the-art supercomputer) to go at it by trial and error." (Alexander Humez et al, "Zero to Lazy Eight: The romance of numbers", 1993)
"The impossibility of constructing a complete, accurate quantitative description of a complex system forces observers to pick which aspects of the system they most wish to understand." (Thomas Levenson, "Measure for Measure: A musical history of science", 1994)
"It is impossible to understand the true meaning of an equation, or to appreciate its beauty, unless it is read in the delightfully quirky language in which it was penned." (Michael Guillen, "Five Equations That Changed the World", 1995)
"How surprising it is that the laws of nature and the initial conditions of the universe should allow for the existence of beings who could observe it. Life as we know it would be impossible if any one of several physical quantities had slightly different values." (Steven Weinberg, Life in the Quantum Universe", Scientific American, 1995)
"The Law of Entropy Nonconservation required that life be lived forward, from birth to death. […] To wish for the reverse was to wish for the entropy of the universe to diminish with time, which was impossible. One might as well wish for autumn leaves to assemble themselves in neat stacks just as soon as they had fallen from trees or for water to freeze whenever it was heated." (Michael Guillen," Five Equations That Changed the World", 1995)
"No other theory known to science [other than superstring theory] uses such powerful mathematics at such a fundamental level. […] because any unified field theory first must absorb the Riemannian geometry of Einstein’s theory and the Lie groups coming from quantum field theory. […] The new mathematics, which is responsible for the merger of these two theories, is topology, and it is responsible for accomplishing the seemingly impossible task of abolishing the infinities of a quantum theory of gravity." (Michio Kaku, "Hyperspace", 1995)
"Small changes in the initial conditions in a chaotic system produce dramatically different evolutionary histories. It is because of this sensitivity to initial conditions that chaotic systems are inherently unpredictable. To predict a future state of a system, one has to be able to rely on numerical calculations and initial measurements of the state variables. Yet slight errors in measurement combined with extremely small computational errors (from roundoff or truncation) make prediction impossible from a practical perspective. Moreover, small initial errors in prediction grow exponentially in chaotic systems as the trajectories evolve. Thus, theoretically, prediction may be possible with some chaotic processes if one is interested only in the movement between two relatively close points on a trajectory. When longer time intervals are involved, the situation becomes hopeless." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)
"There are no hard problems, only problems that are hard to a certain level of intelligence. Move the smallest bit upwards, and some problems will suddenly move from 'impossible' to 'obvious'. Move a substantial degree upwards, and all of them will become obvious. Move a huge distance upwards [...]" (Eliezer S Yudkowsky, "Staring into the Singularity", 1996)
"All things which are proved to be impossible must obviously rest on some assumptions, and when one or more of these assumptions are not true then the impossibility proof fails - but the expert seldom remembers to carefully inspect the assumptions before making their 'impossible' statements." (Richard Hamming, "The Art of Doing Science and Engineering: Learning to Learn", 1997)
"Another limit imposed by reality is its sheer complexity, which makes it impossible to predict some ordinary things (like weather) at the same time that it’s possible to predict truly extraordinary things (like the fate of the universe)." (K C Cole, "The Universe and the Teacup: The Mathematics of Truth and Beauty", 1997)
"The most common instance of beauty in mathematics is a brilliant step in an otherwise undistinguished proof. […] A beautiful theorem may not be blessed with an equally beautiful proof; beautiful theorems with ugly proofs frequently occur. When a beautiful theorem is missing a beautiful proof, attempts are made by mathematicians to provide new proofs that will match the beauty of the theorem, with varying success. It is, however, impossible to find beautiful proofs of theorems that are not beautiful." (Gian-Carlo Rota, "The Phenomenology of Mathematical Beauty", 1997)
"Topology studies those characteristics of figures which are preserved under a certain class of continuous transformations. Imagine two figures, a square and a circular disk, made of rubber. Deformations can convert the square into the disk, but without tearing the figure it is impossible to convert the disk by any deformation into an annulus. In topology, this intuitively obvious distinction is formalized." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)
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