"Humans have a strong tendency to guard their opinion of their own competence in acting. To a certain extent this makes sense, as someone who considers himself to be incapable of acting will hardly act. Guarding one's opinion of one's competence is an important motivation. But it can lead to deformations in the thought process. To maintain a high opinion of one's own competence, people fail to take notice of data that show that their hypotheses are wrong. Or they act 'ballistically' and do not check the effects of their actions so as to maintain the illusion of having solved the corresponding problems by means of their action. The underlying reasons for dispensing with self-reflection may also lie in the tendency to avoid looking at one's own mistakes so as not to endanger one's estimation of one's own competence." (Dietrich Dörner, "The Logic of Failure", Philosophical Transactions of the Royal Society of London" (B), 1990)
"[...] mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It’s the nature of mathematics to pose more problems than it can solve. Indeed, mathematics itself may be built on small islands of truth comprising the pieces of mathematics that can be validated by relatively short proofs. All else is speculation." (Ivars Peterson, „Islands of Truth: A Mathematical Mystery Cruise", 1990)
"Although a system may exhibit sensitive dependence on initial condition, this does not mean that everything is unpredictable about it. In fact, finding what is predictable in a background of chaos is a deep and important problem." (Which means that, regrettably, it is unsolved.) In dealing with this deep and important problem, and for want of a better approach, we shall use common sense." (David Ruelle, "Chance and Chaos", 1991)
"One of the lessons that the history of mathematics clearly teaches us is that the search for solutions to unsolved problems, whether solvable or unsolvable, invariably leads to important discoveries along the way." (Carl B Boyer & Uta C Merzbach, "A History of Mathematics", 1991)
"The most persuasive positive argument for mental images as objects is [that] whenever one thinks one is seeing something there must be something one is seeing. It might be an object directly, or it might be a mental picture. [This] point is so plausible that it is deniable only at the peril of becoming arbitrary. One should concede that the question whether mental images are entities of some sort is not resolvable by logical or linguistic analysis, and believe what makes sense of experience." (Eva T H Brann,"The World of Imagination", 1991)
"Throughout the evolution of mathematics, problems have acted as catalysts in the discovery and development of mathematical ideas. In fact, the history of mathematics can probably be traced by studying the problems that mathematicians have tried to solve over the centuries. It is almost disheartening when an old problem is finally solved, for it will no longer be around to challenge and stimulate mathematical thought." (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)
"Dynamical systems that vary in discrete steps […] are technically known as mappings. The mathematical tool for handling a mapping is the difference equation. A system of difference equations amounts to a set of formulas that together express the values of all of the variables at the next step in terms of the values at the current step. […] For mappings, the difference equations directly express future states in terms of present ones, and obtaining chronological sequences of points poses no problems. For flows, the differential equations must first be solved. General solutions of equations whose particular solutions are chaotic cannot ordinarily be found, and approximations to the latter are usually determined by numerical methods." (Edward N Lorenz, "The Essence of Chaos", 1993)
"Mathematicians typically do not feel that they have completely solved a system of differential equations until they have written down a general solution - a set of formulas giving the value of each variable at every time, in terms of the supposedly known values at some initial time." (Edward N Lorenz, "The Essence of Chaos", 1993)
"Perhaps the greatest strength of graph theory is the abundance of natural and beautiful problems waiting to be solved. […] Paradoxically, much of what is wrong with graph theory is due to this richness of problems. It is all too easy to find new problems based on no theory whatsoever, and to solve the first few cases by straightforward methods. Unfortunately, in some instances the problems are unlikely to lead anywhere […]" (Béla Bollobás, "The Future of Graph Theory", [in "Quo Vadis, Graph Theory?"] 1993)
"To understand what kinds of problems are solvable by the Monte Carlo method, it is important to note that the method enables simulation of any process whose development is influenced by random factors. Second, for many mathematical problems involving no chance, the method enables us to artificially construct a probabilistic model (or several such models), making possible the solution of the problems." (Ilya M Sobol, "A Primer for the Monte Carlo Method", 1994)
"It remains an unhappy fact that there is no best method for finding the solution to general nonlinear optimization problems. About the best general procedure yet devised is one that relies upon imbedding the original problem within a family of problems, and then developing relations linking one member of the family to another. If this can be done adroitly so that one family member is easily solvable, then these relations can be used to step forward from the solution of the easy problem to that of the original problem. This is the key idea underlying dynamic programming, the most flexible and powerful of all optimization methods." (John L Casti, "Five Golden Rules", 1995)
"General relativity, one of the most famous theories, is formulated in terms of a nonlinear equation. This makes us wonder if some of the phenomena described by general relativity, namely black holes, objects orbiting black holes, and even the universe itself, can become chaotic under certain circumstances. [...] The problem is the equation itself, namely the equation of general relativity; it is so complex that the most general solution has never been obtained. It has, of course, been solved for many simple systems; if the system has considerable symmetry (e.g., it is spherical) the equation reduces to a number of ordinary equations that can be solved, but chaos does not occur in these cases. In more realistic cases - situations that actually occur in nature - chaos may occur, but the equations are either unsolvable or very difficult to solve. This presents a dilemma. If we try to model the system using many simplifications it won't exhibit chaos, but if we model it realistically we can't solve it." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)
"Our sole responsibility is to produce something smarter than we are; any problems beyond that are not ours to solve." (Eliezer S Yudkowsky, "Staring into the Singularity", 1996)
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