💠On Problem Solving: On Elegance

"It is true that not every geometric construction is elegant, for each particular problem has its own refinements. It is also true that [that construction] is preferred to any other that makes clear not the structure of a work from an equation but the equation from the structure; thus the structure demonstrates itself. So a skillful geometer, although thoroughly versed in analysis, conceals the fact and, while thinking about the accomplishment of his work, sheds light on and explains his problem Then, as an aid to the arithmeticians, he sets out and demonstrates his theorem with the equation or proportion he sees in it." (François Viète, "On the Meaning and Components of Analysis and on Matters Useful to Zetetics", 1591)

"The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors, is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and difficult that even for numbers that do not exceed the limits of tables constructed by estimable men, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers. […] Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated." (Carl F Gauss, "Disquisitiones Arithmeticae” [“Arithmetical Researches”], 1801)

"To appreciate the living spirit rather than the dry bones of mathematics, it is necessary to inspect the work of a master at first hand.Textbooks and treatises are an unavoidable evil [...] The very crudities of the first attack on a significant problem by a master are more illuminating than all the pretty elegance of the standard texts which has been won at the cost of perhaps centuries of finicky polishing." (Eric T Bell, "Mathematics: Queen and Servant of Science", 1951)

"Mathematical problems, or puzzles, are important to real mathematics (like solving real-life problems), just as fables, stories, and anecdotes are important to the young in understanding real life. Mathematical problems are ‘sanitized’ mathematics, where an elegant solution has already been found (by someone else, of course), the question is stripped of all superfluousness and posed in an interesting and (hopefully) thought-provoking way. If mathematics is likened to prospecting for gold, solving a good mathematical problem is akin to a ‘hide-and-seek’ course in gold-prospecting: you are given a nugget to find, and you know what it looks like, that it is out there somewhere, that it is not too hard to reach, that it is unearthing within your capabilities, and you have conveniently been given the right equipment (i.e. data) to get it. It may be hidden in a cunning place, but it will require ingenuity rather than digging to reach it." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)

"This is what it means to do mathematics. To make a discovery (by whatever means, including playing around with physical models like paper, string, and rubber bands), and then to explain it in the simplest and most elegant way possible. This is the art of it, and this is why it is so challenging and fun." (Paul Lockhart, "Measurement", 2012)

"What makes a great mathematical problem great? Intellectual depth, combined with simplicity and elegance. Plus: it has to be hard. Anyone can climb a hillock; Everest is another matter entirely. A great problem is usually simple to state, although the terms required may be elementary or highly technical." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

💠On Problem Solving: On Paths

"Having gotten, with God’s help, to the very desired place, i.e. the mother of all cases called by the people 'the rule of the thing' or the 'Greater Art', i.e. speculative practice; otherwise called Algebra and Almucabala in the Arab language or Chaldean according to some, which in our [language] amounts to saying 'restaurationis et oppositionis', Algebra id est Restau­ ratio. Almucabala id est Oppositio vel contemptio et Solutio, because by this path one solves infinite questions. And one picks out those which cannot yet be solved." (Luca Pacioli, "Summa de arithmetica geometria proportioni et proportionalita", 1494)

"Systems in physical science […] are no more than appropriate instruments to aid the weakness of our organs: they are, properly speaking, approximate methods which put us on the path to the solution of the problem; these are the hypotheses which, successively modified, corrected, and changed in proportion as they are found false, should lead us infallibly one day, by a process of exclusion, to the knowledge of the true laws of nature." (Antoine L Lavoisier, "Mémoires de l’Académie Royale des Sciences", 1777)

"The insights gained and garnered by the mind in its wanderings among basic concepts are benefits that theory can provide. Theory cannot equip the mind with formulas for solving problems, nor can it mark the narrow path on which the sole solution is supposed to lie by planting a hedge of principles on either side. But it can give the mind insight into the great mass of phenomena and of their relationships, then leave it free to rise into the higher realms of action." (Carl von Clausewitz, "On War", 1832)

"The philosophical emphasis here is: to solve a geometrical problem of a global nature, one first reduces it to a homotopy theory problem; this is in turn reduced to an algebraic problem and is solved as such. This path has historically been the most fruitful one in algebraic topology." (Brayton Gray, "Homotopy Theory", Pure and Applied Mathematics Vol. 64, 1975)

"In the real world, none of these assumptions are uniformly valid. Often people want to know why mathematics and computers cannot be used to handle the meaningful problems of society, as opposed, let us say, to the moon boondoggle and high energy-high cost physics. The answer lies in the fact that we don't know how to describe the complex systems of society involving people, we don't understand cause and effect, which is to say the consequences of decisions, and we don't even know how to make our objectives reasonably precise. None of the requirements of classical science are met. Gradually, a new methodology for dealing with these 'fuzzy' problems is being developed, but the path is not easy." (Richard E Bellman, "Eye of the Hurricane: An Autobiography", 1984)

"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)

"We wrote down all the states and legal moves (here it turned out to be helpful to have a systematic notation, but that's not essential). Then we formed a graph whose nodes correspond to states and whose edges correspond to legal moves. The solution of the puzzle is then a path through the graph that joins the start to the finish. Such a path is usually obvious to the eye, provided the puzzle is sufficiently simple for the entire graph to be drawn. Puzzles of this type are really mazes, for a maze is just a graph drawn in a slightly different fashion. Metaphorically, they are logical mazes - you have to find the right sequence of moves to solve them. The graph turns the logical maze into a genuine maze, turning the metaphor into reality. The fact that solving the real maze also solves the logical maze is one of the magical features of the maze that is mathematics." (Ian Stewart, "The Magical Maze: Seeing the World Through Mathematical Eyes", 1997)

"When a book is being written, it is a maze of possibilities, most of which are never realised. Reading the resulting book, once all decisions have been taken, is like tracing one particular path through that maze. The writer's job is to choose that path, define it clearly, and make it as smooth as possible for those who follow. Mathematics is much the same. Mathematical ideas form a network. The interconnections between ideas are logical deductions. If we assume this, then that must follow - a logical path from this to that. When mathematicians try to understand a problem, they have to thread a maze of logic. The body of knowledge that we call mathematics is a catalogue of interesting excursions through the logical maze."(Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"An algorithm refers to a successive and finite procedure by which it is possible to solve a certain problem. Algorithms are the operational base for most computer programs. They consist of a series of instructions that, thanks to programmers’ prior knowledge about the essential characteristics of a problem that must be solved, allow a step-by-step path to the solution." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Monte Carlo is able to discover practical solutions to otherwise intractable problems because the most efficient search of an unmapped territory takes the form of a random walk. Today’s search engines, long descended from their ENIAC-era ancestors, still bear the imprint of their Monte Carlo origins: random search paths being accounted for, statistically, to accumulate increasingly accurate results. The genius of Monte Carlo - and its search-engine descendants - lies in the ability to extract meaningful solutions, in the face of overwhelming information, by recognizing that meaning resides less in the data at the end points and more in the intervening paths." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"The genius of Monte Carlo - and its search-engine descendants - lies in the ability to extract meaningful solutions, in the face of overwhelming information, by recognizing that meaning resides less in the data at the end points and more in the intervening paths." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

💠On Problem Solving: On Simplicity

"The difficult problems in life always start off being simple. Great affairs always start off being small." (Lao Tzu, cca 400 BC)

"There is the very real danger that a number of problems which could profitably be subjected to analysis, and so treated by simpler and more revealing techniques. will instead be routinely shunted to the computing machines [...] The role of computing machines as a mathematical tool is not that of a panacea for all computational ills." (Richard E Bellman & Paul Brock, "On the Concepts of a Problem and Problem-Solving", American Mathematical Monthly 67, 1960)

"The first step is to simplify the problem, by reducing it to its essential features. Mathematics always works best when its raw materials are as simple as possible: extraneous information [...] should be resolutely excluded. [...] The essential features can be captured by introducing a nice, compact notation. Symbolism isn't everything in mathematics, but it often suggests useful ideas - and here the right symbolism pays dividends." (Ian Stewart, "The Magical Maze: Seeing the World Through Mathematical Eyes", 1997)

"The basic idea behind all of these techniques is to simplify problem solving by concentrating on its essentials. Consolidate and simplify the objectives. Focus on the things with the highest impact, things that determine other things. Put to one side minor issues likely to be resolved by the resolution of major ones. Discard the nonessentials. Model (abstract) the system at as high a level as possible, then progressively reduce the level of abstraction. In short: Simplify!" (Mark W Maier, "The Art Systems of Architecting" 3rd Ed., 2009)

"What makes a great mathematical problem great? Intellectual depth, combined with simplicity and elegance. Plus: it has to be hard. Anyone can climb a hillock; Everest is another matter entirely. A great problem is usually simple to state, although the terms required may be elementary or highly technical." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"Calculus succeeds by breaking complicated problems down into simpler parts. That strategy, of course, is not unique to calculus. All good problem-solvers know that hard problems become easier when they’re split into chunks. The truly radical and distinctive move of calculus is that it takes this divide-and-conquer strategy to its utmost extreme - all the way out to infinity." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

"A problem thoroughly understood is always fairly simple." (Charles Kettering)

"A problem thoroughly understood is always fairly simple. Found your opinions on facts, not prejudices. We know too many things that are not true." (Charles F Kettering)

"I do believe in simplicity. It is astonishing as well as sad, how many trivial affairs even the wisest thinks he must attend to in a day; how singular an affair he thinks he must omit. When the mathematician would solve a difficult problem, he first frees the equation of all encumbrances, and reduces it to its simplest terms. So simplify the problem of life, distinguish the necessary and the real. Probe the earth to see where your main roots run." (Henry D Thoreau)

💠On Problem Solving 21: On Statistics

"Difficulties in identifying problems have delayed statistics far more than difficulties in solving problems." (John W Tukey, Unsolved Problems of Experimental Statistics, 1954)

"Stepwise regression is probably the most abused computerized statistical technique ever devised. If you think you need stepwise regression to solve a particular problem you have, it is almost certain that you do not. Professional statisticians rarely use automated stepwise regression." (Leland Wilkinson, "SYSTAT", 1984)

"We have to teach non-statisticians to recognize where statistical expertise is required. No one else will. We teach students how to solve simple statistical problems, but how often do we make any serious effort to teach them to recognize situations that call for statistical expertise that is beyond the technical content of the course." (Christopher J Wild, "Embracing the ‘Wider view’ of Statistics", The American Statistician 48, 1994)

"The goals in statistics are to use data to predict and to get information about the underlying data mechanism. Nowhere is it written on a stone tablet what kind of model should be used to solve problems involving data. To make my position clear, I am not against models per se. In some situations they are the most appropriate way to solve the problem. But the emphasis needs to be on the problem and on the data. Unfortunately, our field has a vested interest in models, come hell or high water." (Leo Breiman, "Statistical Modeling: The Two Cultures, Statistical Science" Vol. 16(3), 2001)

"Teaching statistical thinking means giving people tools for problem solving in the real world. It should not be taught as pure mathematics. Instead of mechanically solving a dozen problems with the help of a particular formula, children and adolescents should be asked to find solutions to real-life problems. That’s what teaches them how to solve problems, and also shows that there may be more than one good answer in the first place. Equally important is encouraging curiosity, such as asking for answers to questions by doing experiments." (Gerd Gigerenzer, "Risk Savvy: How to make good decisions", 2014)

💠On Problem Solving 31: On Challenges

"A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery." (George Polya, "How to solve it", 1944)

"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)

"[...] all problems can be reperceived as challenges, or 'opportunities' to change, grow or learn." (Robert B Dilts, "Sleight of Mouth: The Magic of Conversational Belief Change", 1999

"The real raison d'etre for the mathematician's existence is simply to solve problems. So what mathematics really consists of is problems and solutions. And it is the 'good' problems, the ones that challenge the greatest minds for decades, if not centuries, that eventually become enshrined as mathematical mountaintops." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001) 

"Many of the great mathematical problems stem from deep and difficult questions in well-established areas of the subject. They are the big challenges that emerge when a major area has been thoroughly explored. They tend to be quite technical, and everyone in the area knows they’re hard to answer, because many experts have tried and failed. The area concerned will already possess many powerful techniques, massive mathematical machines whose handles can be cranked if you’ve done your homework - but if the problem is still open, then all of the plausible ways to use those techniques have already been tried, and they didn’t work. So either there is a less plausible way to use the tried-and-tested techniques of the area, or you need new techniques." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery." (George Polya)

"I hope that seeing the excitement of solving this problem will make young mathematicians realize that there are lots and lots of other problems in mathematics which are going to be just as challenging in the future." (Andrew Wiles)

"The greatest challenge to any thinker is stating the problem in a way that will allow a solution." (Bertrand Russell)

💠On Problem Solving 29: On Solvability (2000-2009)

"Theorems are fun especially when you are the prover, but then the pleasure fades. What keeps us going are the unsolved problems." (Carl Pomerance, 2000)"

"The existence of the tipping point means it is theoretically possible to completely eradicate a disease. Eradication does not require a perfect vaccine and universal immunization but only the weaker condition that the reproduction rate of the disease fall and remain below one so that new cases arise at a lower rate than old cases are resolved." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world. ", 2000)

"Theorems are fun especially when you are the prover, but then the pleasure fades. What keeps us going are the unsolved problems." (Carl Pomerance, MAA, 2000)

"And although mathematical ideas and thought are constantly evolving, you will also see that the most basic fundamental problems never go away. Many of these problems go back to the ancient Greeks, and maybe even to ancient Sumer, although we may never know for sure. The fundamental philosophical questions like the continuous versus the discrete or the limits of knowledge are never definitively solved. Each generation formulates its own answer, strong personalities briefly impose their views, but the feeling of satisfaction is always temporary, and then the process continues, it continues forever." (Gregory Chaitin, "Meta Math: The Quest for Omega", 2005)

"A model does not always predict all the features of the data. Nature has no privileged tendency to present me with solvable challenges." (Eliezer S Yudkowsky, "A Technical Explanation of Technical Explanation", 2005)

"Knowing a solution is at hand is a huge advantage; it’s like not having a 'none of the above' option. Anyone with reasonable competence and adequate resources can solve a puzzle when it is presented as something to be solved. We can skip the subtle evaluations and move directly to plugging in possible solutions until we hit upon a promising one. Uncertainty is far more challenging." (Garry Kasparov, "How Life Imitates Chess", 2007)

"Social ecology is based on the conviction that nearly all of our present ecological problems originate in deep-seated social problems. It follows, from this view, that these ecological problems cannot be understood, let alone solved, without a careful understanding of our existing society and the irrationalities that dominate it. To make this point more concrete: economic, ethnic, cultural, and gender conflicts, among many others, lie at the core of the most serious ecological dislocations we face today - apart, to be sure, from those that are produced by natural catastrophes." (Murray Bookchin, "Social Ecology and Communalism", 2007)

"When in the sciences or techniques one states that a certain problem is unsolvable, a rigorous demonstration of such unsolvability is required. And when a scientist submits an article to publication, the least that its referees demand is that it be intelligible. Why? Because rational beings long for understanding and because only clear statements are susceptible to be put to examination to verify whether they are true or false. In the Humanities it is the same, or it should be, but it is not always so." (Mario Bunge, "Xenius, Platón y Manolito", La Nación, 2008)

"An algorithm refers to a successive and finite procedure by which it is possible to solve a certain problem. Algorithms are the operational base for most computer programs. They consist of a series of instructions that, thanks to programmers’ prior knowledge about the essential characteristics of a problem that must be solved, allow a step-by-step path to the solution." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Chess also offers a modality that includes an exercise of totally free creation - compositions. These artificial positions are created for didactic reasons to illustrate a certain subject or to propose a problem that has to be solved following a series of indications" (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Chess, as a game of zero sum and total information is, theoretically, a game that can be solved. The problem is the immensity of the search tree: the total number of positions surpasses the number of atoms in our galaxy. When there are few pieces on the board, the search space is greatly reduced, and the problem becomes trivial for computers’ calculation capacity." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Mental acuity of any kind comes from solving problems yourself, not from being told how to solve them." (Paul Lockhart,"A Mathematician's Lament", 2009)

"What literally defines social ecology as ‘social’ is its recognition of the often overlooked fact that nearly all our present ecological problems arise from deep-seated social problems. Conversely, present ecological problems cannot be clearly understood, much less resolved, without resolutely dealing with problems within society." (Murray Bookchin, "What is Social Ecology", 2009)

💠On Problem Solving 30: On Solvability (2010-)

"Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the ‘three-fold way’ […] This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics." (John C Baez, "Division Algebras and Quantum Theory", 2011)

"Under normal conditions the research scientist is not an innovator but a solver of puzzles, and the puzzles upon which he concentrates are just those which he believes can be both stated and solved within the existing scientific tradition." (Thomas S Kuhn, "The Essential Tension: Selected Studies in Scientific Tradition and Change", 2011)

"Mathematicians approach problems the way rock climbers do cliffs: the more difficult the pitch, the more exhilarating the ascent. After a climb has been solved, others look for new routes, or try equipment that no one else has used, simply for the joy of pioneering." (David Perkins, "Calculus and Its Origins", 2012)

"Complex systems defy intuitive solutions. Even a third-order, linear differential equation is unsolvable by inspection. Yet, important situations in management, economics, medicine, and social behavior usually lose reality if simplified to less than fifth-order nonlinear dynamic systems. Attempts to deal with nonlinear dynamic systems using ordinary processes of description and debate lead to internal inconsistencies. Underlying assumptions may have been left unclear and contradictory, and mental models are often logically incomplete. Resulting behavior is likely to be contrary to that implied by the assumptions being made about' underlying system structure and governing policies." (Jay W Forrester, "Modeling for What Purpose?", The Systems Thinker Vol. 24 (2), 2013)

"Systems always contain problematic elements in a sense that is usually not clear until the implications of the new system are sufficiently explored. Often problems can be stated in the language of the initial system but can only be resolved by creating a new system. [...] Problems that can be stated in the language of one system often cannot be solved within that system because the solution depends upon the development of a new system." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"The subject of computational complexity theory is focused on classifying problems by how hard they are. […] (1) P problems are those that can be solved by a Turing machine" (TM)" (deterministic) in polynomial time. (‘P’ stands for polynomial). P problems form a class of problems that can be solved efficiently. (2) NP problems are those that can be solved by non-deterministic TM in polynomial time. A problem is in NP if you can quickly" (in polynomial time) test whether a solution is correct" (without worrying about how hard it might be to find the solution). NP problems are a class of problems that cannot be solved efficiently. NP does not stand for 'non-polynomial'. There are many complexity classes that are much harder than NP. (3) Undecidable problems: For some problems, we can prove that there is no algorithm that always solves them, no matter how much time or space is allowed." (K V N Sunitha & N Kalyani, "Formal Languages and Automata Theory", 2015)

"When we extend the system of natural numbers and counting to embrace infinite cardinals, the larger system need not have all of the properties of the smaller one. However, familiarity with the smaller system leads us to expect certain properties, and we can become confused when the pieces don’t seem to fit. Insecurity arose when the square of a complex number violated the real number principle that all squares are positive. This was resolved when we realised that the complex numbers cannot be ordered in the same way as their subset of reals." (Ian Stewart & David Tall, "The Foundations of Mathematics" 2nd Ed., 2015)

"Judgments made in difficult circumstances can be based on a limited number of simple, rapidly-arrived-at rules ('heuristics'), rather than formal, extensive algorithmic calculus and programs. Often, even complex problems can be solved quickly and accurately using such 'quick and dirty' heuristics. However, equally often, such heuristics can be beset by systematic errors or biases." (Jérôme Boutang & Michel De Lara, "The Biased Mind", 2016)

💠On Problem Solving 28: On Solvability (1990-1999)

"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)

💠On Problem Solving 27: On Solvability (1970-1979)

"A problem adequately stated is a problem well on its way to being solved." (R Buckminster Fuller, "I Seem to be a Verb", 1970)

"Deep in the human nature there is an almost irresistible tendency to concentrate physical and mental energy on attempts at solving problems that seem to be unsolvable." (Ragnar Frisch, "From Utopian Theory to Practical Applications", [Nobel lecture] 1970)

"The definition of a problem and the action taken to solve it largely depend on the view which the individuals or groups that discovered the problem have of the system to which it refers. A problem may thus find itself defined as a badly interpreted output, or as a faulty output of a faulty output device, or as a faulty output due to a malfunction in an otherwise faultless system, or as a correct but undesired output from a faultless and thus undesirable system. All definitions but the last suggest corrective action; only the last definition suggests change, and so presents an unsolvable problem to anyone opposed to change." (Herbert Brün, "Technology and the Composer", 1971)

"The matter of the normalcy or non-normalcy of π will never, of course, be resolved by electronic computers. We have here an example of a theoretical problem which requires profound mathematical talent and cannot be solved by computations alone. The existence of such problems ought to furnish at least a partial antidote to the disease of computeritis, which seems so rampant today." (Howard Eves, "Mathematical Circles Revisited", 1971)

"Extrema is the generic term for the concepts 'maximum' and 'minimum' , like 'parents' is the generic term for 'father' and 'mother'. Extremal problems have to do with finding maxima and minima. We encounter them everywhere. It is hardly an exaggeration to say that all problems solved by living organisms are those involving a search for extrema." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"The synthetic mode of thought, when applied to systems problems, is called the systems approach. In this approach a problem is not solved by taking it apart but by viewing it as a part of a larger problem." (Russell L Ackoff, "Redesigning the future", 1974)

"It is one of our most exciting discoveries that local discovery leads to a complex of further discoveries. Corollary to this we find that we no sooner get a problem solved than we are overwhelmed with a multiplicity of additional problems in a most beautiful payoff of heretofore unknown, previously unrecognized, and as-yet unsolved problems." (Buckminster Fuller, "Synergetics: Explorations in the Geometry of Thinking", 1975)"

"A solution of newly appearing economic problems, and in particular those connected with the scientific-technical revolution often cannot be based on existing methods but needs new ideas and approaches. Such one is the problem of the protection of nature. The problem of economic valuation of technical innovations efficiency and rates of their spreading cannot be solved only by the long-term estimation of direct outcomes and results without accounting peculiarities of new industrial technology, its total contribution to technical progress." (Leonid V Kantorovich, "Mathematics in Economics: Achievements, Difficulties, Perspectives", [Nobel lecture]1975)

"The philosophical emphasis here is: to solve a geometrical problem of a global nature, one first reduces it to a homotopy theory problem; this is in turn reduced to an algebraic problem and is solved as such. This path has historically been the most fruitful one in algebraic topology." (Brayton Gray, Homotopy Theory", Pure and Applied Mathematics Vol. 64, 1975)

"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", 1976)

"All mathematical problems are solved by reasoning within a deductive system in which basic laws of logic are embedded." (Martin Gardner,"Aha! Insight", 1978)

💠On Problem Solving 26: On Solvability (1980-1989)

"Metaphors deny distinctions between things: problems often arise from taking structural metaphors too literally. Because unexamined metaphors lead us to assume the identity of unidentical things, conflicts can arise which can only be resolved by understanding the metaphor (which requires its recognition as such), which means reconstructing the analogy on which it is based. […] The unexplained extension of concepts can too often result in the destruction rather than the expansion of meaning." (David Pimm,"Metaphor and Analogy in Mathematics", For the Learning of Mathematics Vol. 1" (3), 1981)

"In various fields of knowledge the problem of the relationship between cause and condition is solved in different ways, depending mainly on the complexity of the relationships that are being studied, their uniformity or, on the contrary, the distinctness and comparative importance of separate factors." (Alexander Spirkin, "Dialectical Materialism", 1983)

"There are, roughly speaking, two kinds of mathematical creativity. One, akin to conquering a mountain peak, consists of solving a problem which has remained unsolved for a long time and has commanded the attention of many mathematicians. The other is exploring new territory." (Mark Kac, "Enigmas Of Chance", 1985)

"To learn mathematics is to learn mathematical problem solving." (Patrik W Thompson, 1985)

"One never knows how hard a problem is until it has been solved. You don’t necessarily know that you will succeed if you work harder or longer." (Linus Pauling, [interview] 1986)

"Linear relationships are easy to think about: the more the merrier. Linear equations are solvable, which makes them suitable for textbooks. Linear systems have an important modular virtue: you can take them apart and put them together again - the pieces add up. Nonlinear systems generally cannot be solved and cannot be added together. [...] Nonlinearity means that the act of playing the game has a way of changing the rules. [...] That twisted changeability makes nonlinearity hard to calculate, but it also creates rich kinds of behavior that never occur in linear systems." (James Gleick, "Chaos: Making a New Science", 1987)

"A scientific problem can be illuminated by the discovery of a profound analogy, and a mundane problem can be solved in a similar way." (Philip Johnson-Laird, "The Computer and the Mind", 1988)

"The successes of the differential equation paradigm were impressive and extensive. Many problems, including basic and important ones, led to equations that could be solved. A process of self-selection set in, whereby equations that could not be solved were automatically of less interest than those that could." (Ian Stewart, "Does God Play Dice? The Mathematics of Chaos", 1989)

💠On Problem Solving 25: On Solvability (1960-1969)

"A problem that is located and identified is already half solved!" (Bror R Carlson, "Managing for Profit", 1961)"

"No mathematical idea has ever been published in the way it was discovered. Techniques have been developed and are used, if a problem has been solved, to turn the solution procedure upside down, or if it is a larger complex of statements and theories, to turn definitions into propositions, and propositions into definitions, the hot invention into icy beauty. This then if it has affected teaching matter, is the didactical inversion, which as it happens may be anti-didactical." (Hans Freudenthal, "The Concept and the Role of the Model in Mathematics and Natural and Social Sciences", 1961)

"Because engineering is science in action - the practice of decision making at the earliest moment - it has been defined as the art of skillful approximation. No situation in engineering is simple enough to be solved precisely, and none worth evaluating is solved exactly. Never are there sufficient facts, sufficient time, or sufficient money for an exact solution, for if by chance there were, the answer would be of academic and not economic interest to society. These are the circumstances that make engineering so vital and so creative." (Ronald B Smith, "Engineering Is…", Mechanical Engineering Vol. 86" (5), 1964)

"Engineering is a method and a philosophy for coping with that which is uncertain at the earliest possible moment and to the ultimate service to mankind. It is not a science struggling for a place in the sun. Engineering is extrapolation from existing knowledge rather than interpolation between known points. Because engineering is science in action - the practice of decision making at the earliest moment - it has been defined as the art of skillful approximation. No situation in engineering is simple enough to be solved precisely, and none worth evaluating is solved exactly. Never are there sufficient facts, sufficient time, or sufficient money for an exact solution, for if by chance there were, the answer would be of academic and not economic interest to society. These are the circumstances that make engineering so vital and so creative." (Ronald B Smith, "Engineering Is…", Mechanical Engineering Vol. 86" (5), 1964)

"No branch of number theory is more saturated with mystery than the study of prime numbers: those exasperating, unruly integers that refuse to be divided evenly by any integers except themselves and 1. Some problems concerning primes are so simple that a child can understand them and yet so deep and far from solved that many mathematicians now suspect they have no solution. Perhaps they are 'undecideable'. Perhaps number theory, like quantum mechanics, has its own uncertainty principle that makes it necessary, in certain areas, to abandon exactness for probabilistic formulations." (Martin Gardner, "The remarkable lore of the prime numbers", Scientific American, 1964)

"It is a commonplace of modern technology that there is a high measure of certainty that problems have solutions before there is knowledge of how they are to be solved." (John K Galbraith, "The New Industrial State", 1967)

"The problem that still remains to be solved is that of the orderable matrix, that needs the use of imagination […] When the two components of a data table are orderable, the normal construction is the orderable matrix. Its permutations show the analogy and the complementary nature that exist between the algorithmic treatments and the graphical treatments." (Jacques Bertin, Semiology of graphics [Semiologie Graphique], 1967)

"It is sheer nonsense to expect that any human being has yet been able to attain such insight into the problems of society that he can really identify the central problems and determine how they should be solved. The systems in which we live are far too complicated as yet for our intellectual powers and technology to understand." (C West Churchman, 1968). 

"Combinatorial theory has been slowed in its theoretical development by the very success of the few men who have solved some of the outstanding combinatorial problems of their day, for, just as the man of action feel little need to philosophize, so the successful problem-solver in mathematics feels little need for designing theories, that would unify, ant therefore enable the less-talented worker to solve, problems of comparable and similar difficulty. But the sheer number and the rapidly increasing complexity of combinatorial problems has made the situation no longer tolerable. It is doubtful that one man alone can solve any of the major combinatorial problems of our day." (Gian-Carlo Rota, "Discrete Thoughts", 1969)

💠On Problem Solving 24: On Solvability (1950-1959)

"The problems are solved, not by giving new information, but by arranging what we have known since long." (Ludwig Wittgenstein, "Philosophical Investigations", 1953)

"It is interesting to consider what a thinking machine will be like. It seems clear that as soon as the machines become able to solve intellectual problems of the highest difficulty which can be solved by humans they will be able to solve most of the problems enormously faster than a human." (John F Nash, "Parallel Control", 1954)

"Within the confines of my abstraction, for instance, it is clear that the problem of truth and validity cannot be solved completely, if what we mean by the truth of an image is its correspondence with some reality in the world outside it. The difficulty with any correspondence theory of truth is that images can only be compared with images. They can never be compared with any outside reality. The difficulty with the coherence theory of truth, on the other hand, is that the coherence or consistency of the image is simply not what we mean by its truth." (Kenneth E Boulding, "The Image: Knowledge in life and society", 1956)

"By some definitions 'systems engineering' is suggested to be a new discovery. Actually it is a common engineering approach which has taken on a new and important meaning because of the greater complexity and scope of problems to be solved in industry, business, and the military. Newly discovered scientific phenomena, new machines and equipment, greater speed of communications, increased production capacity, the demand for control over ever-extending areas under constantly changing conditions, and the resultant complex interactions, all have created a tremendously accelerating need for improved systems engineering. Systems engineering can be complex, but is simply defined as 'logical engineering within physical, economic and technical limits' - bridging the gap from fundamental laws to a practical operating system." (Instrumentation Technology, 1957)

"A change in science, whether novelty or discovery, when properly understood, when the linguistic problem is adequately solved, will even then provide only a hunch, a starting point for looking at an area of experience other than the science in which it was nourished and born." (J Robert Oppenheimer, "The Growth of Science and the Structure of Culture", Daedalus, 1958)

"The answer to the question ‘Can there be a general method for solving all mathematical problems?’ is no! Perhaps, in a world of unsolved and apparently unsolvable problems, we would have thought that the desirable answer to this question from any point of view, would be yes. But from the point of view of mathematicians a yes would have been far less satisfying than a no is. […] Not only are the problems of mathematics infinite and hence inexhaustible, but mathematics itself is inexhaustible." (Constance Reid, "Introduction to Higher Mathematics for the General Reader", 1959)

💠On Problem Solving 22: On Solvability (-1899)

"Those who devised the eccentrics seen thereby in large measure to have solved the problem of apparent motions with approximate calculations. But meanwhile they introduced a good many ideas which apparently contradict the first principles of uniform motion. Nor could they elicit or deduce from the eccentrics the principal consideration, that is, the structure of the universe and the true symmetry of its parts. " (Nicolaus Copernicus, "De revolutionibus orbium coelestium", 1543)

"Having gotten, with God’s help, to the very desired place, i.e. the mother of all cases called by the people 'the rule of the thing' or the 'Greater Art', i.e. speculative practice; otherwise called Algebra and Almucabala in the Arab language or Chaldean according to some, which in our [language] amounts to saying 'restaurationis et oppositionis', Algebra id est Restau­ ratio. Almucabala id est Oppositio vel contemptio et Solutio, because by this path one solves infinite questions. And one picks out those which cannot yet be solved." (Luca Pacioli, "Summa de arithmetica geometria proportioni et proportionalita", 1494)

"Each problem that I solved became a rule which served afterwards to solve other problems. [...] thus each truth discovered was a rule available in the discovery of subsequent ones." (René Descartes,"Discourse on Method", 1637)

"Thus, you see, most noble Sir, how this type of solution bears little relationship to mathematics, and I do not understand why you expect a mathematician to produce it, rather than anyone else, for the solution is based on reason alone, and its discovery does not depend on any mathematical principle. Because of this, I do not know why even questions which bear so little relationship to mathematics are solved more quickly by mathematicians than by others." (Leonhard Euler, [letter to Carl Leonhard Gottlieb Ehler, mayor of Danzig] 1736)

"All that can be said upon the number and nature of elements is, in my opinion, confined to discussions entirely of a metaphysical nature. The subject only furnishes us with indefinite problems, which may be solved in a thousand different ways, not one of which, in all probability, is consistent with nature." (Antoine-Laurent Lavoisier,"Elements of Chemistry", 1790)

"Concerning the theory of equations, I have tried to find out under what circumstances equations are solvable by radicals, which gave me the opportunity of investigating thoroughly, and describing, all transformations possible on an equation, even if it is the case that it is not solvable by radicals." (Évariste Galois, [letter to Auguste Chevalier] 1832)

💠On Problem Solving 23: On Solvability (1900-1949)

"It has long seemed obvious - and is, in fact, the characteristic tone of European science - that 'science' means breaking up complexes into their component elements. Isolate the elements, discover their laws, then reassemble them, and the problem is solved. All wholes are reduced to pieces and piecewise relations between pieces. The fundamental 'formula' of Gestalt theory might be expressed in this way. There are wholes, the behaviour of which is not determined by that of their individual elements, but where the part-processes are themselves determined by the intrinsic nature of the whole. It is the hope of Gestalt theory to determine the nature of such wholes." (Max Wertheimer, "Gestalt Theory," 1924)

"I recall my own emotions: I had just been initiated into the mysteries of the complex number. I remember my bewilderment: here were magnitudes patently impossible and yet susceptible of manipulations which lead to concrete results. It was a feeling of dissatisfaction, of restlessness, a desire to fill these illusory creatures, these empty symbols, with substance. Then I was taught to interpret these beings in a concrete geometrical way. There came then an immediate feeling of relief, as though I had solved an enigma, as though a ghost which had been causing me apprehension turned out to be no ghost at all, but a familiar part of my environment." (Tobias Dantzig, "The Two Realities", 1930)

"In reality the cycles we have the occasion to observe are generally not damped. How can the maintenance of the swings be explained? Have theses dynamic laws deduced from theory and showing damped oscillations no value in explaining the real phenomena, or in what respect do the dynamic laws need to be completed in order to explain the real happenings? They (dynamic laws) only form one element of the explanation: they solve the propagation problem. But the impulse problem remains." (Ragnar Frisch, "Propagation problems and impulse problems in dynamic economics", 1933)

"The unsolved problems of Nature have a distinctive fascination, though they still far outnumber those which have even approximately been resolved.(Henry N Russell, "The Solar System and Its Origin", 1935)

"It is time, therefore, to abandon the superstition that natural science cannot be regarded as logically respectable until philosophers have solved the problem of induction. The problem of induction is, roughly speaking, the problem of finding a way to prove that certain empirical generalizations which are derived from past experience will hold good also in the future." (Alfred J Ayer, "Language, Truth and Logic", 1936)

"The mistakes and unresolved difficulties of the past in mathematics have always been the opportunities of its future; and should analysis ever appear to be without or blemish, its perfection might only be that of death." (Eric T Bell, "The Development of Mathematics", 1940)

"We can scarcely imagine a problem absolutely new, unlike and unrelated to any formerly solved problem; but if such a problem could exist, it would be insoluble. In fact, when solving a problem, we should always profit from previously solved problems, using their result or their method, or the experience acquired in solving them." (George Pólya, 1945)

💠On Problem Solving 21: On Solvability (Unsourced)

"A good problem should be more than a mere exercise; it should be challenging and not too easily solved by the student, and it should require some ‘dreaming’ time." (Howard W Eves)

"A problem is not solved in a laboratory. It is solved in some fellow's head. All the apparatus is for is to get his head turned around so that he can see the thing right." (Charles F Kettering)

"A problem well-defined is a problem half solved." (John Dewey)

"Each problem that I solved became a rule which served afterwards to solve other problems." (Descartes, Oeuvres, vol. VI)

"I knew nothing, except how to think, how to grapple with a problem and then go on grappling with it until you had solved it." (Sir Barnes Wallis)

"Indeed, when in the course of a mathematical investigation we encounter a problem or conjecture a theorem, our minds will not rest until the problem is exhaustively solved and the theorem rigorously proved; or else, until we have found the reasons which made success impossible and, hence, failure unavoidable. Thus, the proofs of the impossibility of certain solutions plays a predominant role in modern mathematics; the search for an answer to such questions has often led to the discovery of newer and more fruitful fields of endeavour." (David Hilbert)

"This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by reason, for in mathematics there is no ignorabimus." (David Hilbert)

"What I really am is a mathematician. Rather than being remembered as the first woman this or that, I would prefer to be remembered, as a mathematician should, simply for the theorems I have proved and the problems I have solved." (Julia Robinson)

💠On Problem Solving 18: On Practice

"The insights gained and garnered by the mind in its wanderings among basic concepts are benefits that theory can provide. Theory cannot equip the mind with formulas for solving problems, nor can it mark the narrow path on which the sole solution is supposed to lie by planting a hedge of principles on either side. But it can give the mind insight into the great mass of phenomena and of their relationships, then leave it free to rise into the higher realms of action." (Carl von Clausewitz, "On War", 1832)

"One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles." (George Pólya, "How to Solve It", 1945)

"We acquire any practical skill by imitation and practice. […] Trying to solve problems, you have to observe and to imitate what other people do when solving problems and, finally, you learn to do problems by doing them." (George Pólya, "How to Solve It", 1945)

"The trouble with mathematics is that, however diligent you are in going to lectures and in learning the theory, you may still not be able to do the problems." (Maurice Wilkes, "Memoirs of a Computer Pioneer", 1985)

"Design thinking taps into capacities we all have but that are overlooked by more conventional problem-solving practices. It is not only human-centered; it is deeply human in and of itself. Design thinking relies on our ability to be intuitive, to recognize patterns, to construct ideas that have emotional meaning as well as functionality, to express ourselves in media other than words or symbols." (Tim Brown, "Change by Design: How Design Thinking Transforms Organizations and Inspires Innovation", 2009)

"Solving problems is a practical skill like, let us say, swimming. We acquire any practical skill by imitation and practice." (George Pólya)

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💠On Problem Solving 20: On Life

"The difficult problems in life always start off being simple. Great affairs always start off being small." (Lao Tzu, cca 400 BC)

"Man was not born to solve the problems of the universe, but rather to seek to lay bare the heart of the problem and then confine himself within the limits of what is amenable to understanding." (Johann Wolfgang von Goethe, 1825)

"The greatest art, both in teaching and in life itself, consists in transforming the problem into a postulate." (Johann Wolfgang von Goethe, 1928)

"The meaning and design of a problem seem not to lie in its solution, but in our working at it incessantly." (Carl G Jung, "Modern Man in Search of a Soul", 1933)

"The field of consciousness is tiny. It accepts only one problem at a time. Get into a fist fight, put your mind on the strategy of the fight, and you will not feel the other fellow's punches." (Antoine de Saint-Exupéry, "Flight to Arras", 1942)

"Life ultimately means taking the responsibility to find the right answer to its problems and to fulfill the tasks which it constantly sets for each individual." (Viktor E Frankl, "Man's Search for Meaning", 1946)

"We are built to conquer environment, solve problems, achieve goals, and we find no real satisfaction or happiness in life without obstacles to conquer and goals to achieve." (Maxwell Maltz, "Psycho-Cybernetics", 1960)

"The mystery of life isn't a problem to solve, but a reality to experience." (Frank Herbert, "Dune", 1965)

"Sometimes the situation is only a problem because it is looked at in a certain way. Looked at in another way, the right course of action may be so obvious that the problem no longer exists." (Edward de Bono, "The use of lateral thinking", 1967)

"There are problems in this universe for which there are no answers." (Frank Herbert, "Dune Messiah", 1969)

"The easiest way to solve a problem is to deny it exists." (Isaac Asimov, "The Gods Themselves", 1972)

"When a decision is made to cope with the symptoms of a problem, it is generally assumed that the corrective measures will solve the problem itself. They seldom do." (Masanobu Fukuoka, "The One-Straw Revolution", 1975)

"If you go through the world looking for excellence, you will find excellence. If you go through the world looking for problems you will find problems." (Joseph O'Connor & John Seymour, "Introducing Neuro-Linguistic Programming: Psychological Skills for Understanding and Influencing People", 1990)

"Every problem has a solution, although it may not be the outcome that was originally hoped for or expected."  (Alice Hoffman, "Practical Magic", 1995)

"Pain is a relatively objective, physical phenomenon; suffering is our psychological resistance to what happens. Events may create physical pain, but they do not in themselves create suffering. Resistance creates suffering. Stress happens when your mind resists what is... The only problem in your life is your mind's resistance to life as it unfolds." (Dan Millman, "Everyday Enlightenment: The Twelve Gateways to Personal Growth", 1998)

"[...] all problems can be reperceived as challenges, or 'opportunities' to change, grow or learn." (Robert B Dilts, "Sleight of Mouth: The Magic of Conversational Belief Change", 1999)

"All problems are illusions of the mind." (Eckhart Tolle, "Practicing the Power of Now: Essential Teachings, Meditations, and Exercises", 2001)

"We humans have two great problems: the first is knowing when to begin; the second is knowing when to stop." (Paulo Coelho, "The Zahir: A Novel of Obsession", 2005)

"Most problems we face in life, as I have said already, happen in our minds. Furthermore, problems generally exist in our concept of the past and the future. The past and the future don’t exist except in our minds." (Richard Bandler, "Get the Life You Want: The Secrets to Quick and Lasting Life Change with Neuro-Linguistic Programming", 2008)

"One of the most important aspects of what human beings do is build beliefs. Beliefs are what trap most people in their problems. Unless you believe you can get over something, get through something, or get to something, there is little likelihood you will be able to do it. Your beliefs refer to your sense of certainty on some of your thoughts." (Richard Bandler, "Get the Life You Want: The Secrets to Quick and Lasting Life Change with Neuro-Linguistic Programming", 2008)

"When you can take on board new, positive suggestions and disbelieve the old, limiting suggestions, you will be ready to tackle the rest of your problems, especially your fears." (Richard Bandler, "Get the Life You Want: The Secrets to Quick and Lasting Life Change with Neuro-Linguistic Programming", 2008)

"Our most important problems cannot be solved; they must be outgrown." (Wayne Dyer, "Excuses Begone!: How to Change Lifelong, Self-Defeating Thinking Habits", 2009)

"A problem is a difference between things as desired and things as perceived. […] Seen in this way, the problem could be solved either by changing desires or changing perceptions." (Donald C Gause & Gerald M Weinberg, "Are Your Lights On?", 2011)

"As a practical matter, it is impossible to define natural, day-to-day problems in a single, unique, totally unambiguous fashion. On the other hand, without some common understanding of the problem, a solution will almost invariably be to the wrong problem." (Donald C Gause & Gerald M Weinberg, "Are Your Lights On?", 2011)

"The really important thing in dealing with problems is to know that the question is never answered, but that it doesn't matter, as long as you keep asking. It's only when you fool yourself into thinking you have the final problem definition - the final, true answer - that you can be fooled into thinking you have the final solution. And if you think that, you're always wrong, because there is no such thing as a 'final solution'." (Donald C Gause & Gerald M Weinberg, "Are Your Lights On?", 2011)

💠On Problem Solving 12: On Theories

"The insights gained and garnered by the mind in its wanderings among basic concepts are benefits that theory can provide. Theory cannot equip the mind with formulas for solving problems, nor can it mark the narrow path on which the sole solution is supposed to lie by planting a hedge of principles on either side. But it can give the mind insight into the great mass of phenomena and of their relationships, then leave it free to rise into the higher realms of action." (Carl von Clausewitz, "On War", 1832)

"No mathematical idea has ever been published in the way it was discovered. Techniques have been developed and are used, if a problem has been solved, to turn the solution procedure upside down, or if it is a larger complex of statements and theories, to turn definitions into propositions, and propositions into definitions, the hot invention into icy beauty. This then if it has affected teaching matter, is the didactical inversion, which as it happens may be anti-didactical." (Hans Freudenthal, "The Concept and the Role of the Model in Mathematics and Natural and Social Sciences", 1961)

"The final test of a theory is its capacity to solve the problems which originated it." (George Dantzig, "Linear Programming and Extensions", 1963)

"If we deal with our problem not knowing, or pretending not to know the general theory encompassing the concrete case before us, if we tackle the problem 'with bare hands', we have a better chance to understand the scientist's attitude in general, and especially the task of the applied mathematician." (George Pólya, "Mathematical Methods in Science", 1977)

"[...] two related deficiencies have prevented real progress in understanding insight and its role in problem solving. First, we do not yet have a system of classifying problems into those in which insight occurs versus those in which it does not. However, only if we can isolate problems in which insight occurs will we be able to set on a firm base our theories of the mechanisms underlying insight. Second, formulation of such a taxonomic system requires that we agree on a definition of insight." (Robert W Weisberg, "Prolegomena to theories of insight in problem solving: a taxonomy of problems", 1995)

"Mathematics is not a matter of 'anything goes', and every mathematician is guided by explicit or unspoken assumptions as to what counts as legitimate – whether we choose to view these assumptions as the product of birth, experience, indoctrination, tradition, or philosophy. At the same time, mathematicians are primarily problem solvers and theory builders, and answer first and foremost to the internal exigencies of their subject." (Jeremy Avigad, "Methodology and Metaphysics in the Development of Dedekind’s Theory of Ideals", 2006)

"Therefore, although the notion of insight as a distinct process has a long history in the psychological study of problem solving, it might be useful at this point to refrain from using analytic and insight as theoretical terms applied a priori to problems." (Jason M Chein et al, "Working memory and insight in the nine-dot problem", Memory & Cognition 38, 2010)

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💠On Problem Solving 19: On Insight

"The insights gained and garnered by the mind in its wanderings among basic concepts are benefits that theory can provide. Theory cannot equip the mind with formulas for solving problems, nor can it mark the narrow path on which the sole solution is supposed to lie by planting a hedge of principles on either side. But it can give the mind insight into the great mass of phenomena and of their relationships, then leave it free to rise into the higher realms of action." (Carl von Clausewitz, "On War", 1832)

"Some problems are just too complicated for rational logical solutions. They admit of insights, not answers." (Jerome B Wiesner, The New Yorker, 1963)

"[...] two related deficiencies have prevented real progress in understanding insight and its role in problem solving. First, we do not yet have a system of classifying problems into those in which insight occurs versus those in which it does not. However, only if we can isolate problems in which insight occurs will we be able to set on a firm base our theories of the mechanisms underlying insight. Second, formulation of such a taxonomic system requires that we agree on a definition of insight." (Robert W Weisberg, "Prolegomena to theories of insight in problem solving: a taxonomy of problems", 1995)

"Therefore, although the notion of insight as a distinct process has a long history in the psychological study of problem solving, it might be useful at this point to refrain from using analytic and insight as theoretical terms applied a priori to problems." (Jason M Chein et al, "Working memory and insight in the nine-dot problem", Memory & Cognition 38, 2010)

"A feeling of insight is a kind of 'Aha!' characterized by suddenness and obviousness (and often relief!) - like a revelation. You are relatively confident that your solution is correct without having to check it. In contrast, you experienced no Aha! if the solution occurs to you slowly and stepwise. As an example, imagine a light bulb that is switched on all at once in contrast to slowly dimming it up." (Amory H Danek et al, "Working wonders? Investigating insight with magic tricks", Cognition 130, 2014)

"If insight involves an abrupt change from a state of not knowing how to solve the problem to a state of knowing [...], with no conscious awareness of what caused the change, the person experiencing the insight can still be expected to be able to report that the change occurred." (Edward Bowden & Kristin Grunewald, "Whose insight is it anyway?", [in "Insight], 2018)

"In the problem-solving literature, the term 'insight' is used to designate the clear and sudden understanding of how to solve a problem. Insight is thought to arise as the result of a solver breaking free of unwarranted assumptions, or forming novel, task-related connections between existing concepts or skills. Although it would be preferable to have a single, clear, universally accepted definition of insight, this may not be possible – and almost certainly will not be possible until we start to assess on a regular basis the single feature that seems to be widely acknowledged as distinguishing insight solutions from other solutions: the 'Aha!'" (Edward Bowden & Kristin Grunewald, "Whose insight is it anyway?", [in "Insight], 2018)

"If insight involves an abrupt change from a state of not knowing how to solve the problem to a state of knowing how to solve the problem (or, in some cases, knowing the solution), with no conscious awareness of what caused the change, the person experiencing the insight can still be expected to be able to report that the change occurred." (Edward Bowden & Kristin Grunewald, "Whose insight is it anyway?", [in "Insight], 2018)

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💠On Problem Solving 17: On Heuristics

"Devising the plan of the solution, we should not be too afraid of merely plausible, heuristic reasoning. Anything is right that leads to the right idea. But we have to change this standpoint when we start carrying out the plan and then we should accept only conclusive, strict arguments." (George Pólya, "How to solve it", 1945)

"Heuristic reasoning is reasoning not regarded as final and strict but as provisional and plausible only, whose purpose is to discover the solution of the present problem. We are often obliged to use heuristic reasoning. We shall attain complete certainty when we shall have obtained the complete solution, but before obtaining certainty we must often be satisfied with a more or less plausible guess. We may need the provisional before we attain the final. We need heuristic reasoning when we construct a strict proof as we need scaffolding when we erect a building." (George Pólya, "How to solve it", 1945)

"Fast and frugal heuristics employ a minimum of time, knowledge, and computation to make adaptive choices in real environments. They can be used to solve problems of sequential search through objects or options, as in satisficing. They can also be used to make choices between simultaneously available objects, where the search for information (in the form of cues, features, consequences, etc.) about the possible options must be limited, rather than the search for the options themselves. Fast and frugal heuristics limit their search of objects or information using easily computable stopping rules, and they make their choices with easily computable decision rules." (Gerd Gigerenzer & Peter M Todd, "Fast and Frugal Heuristics: The Adaptive Toolbox" [in "Simple Heuristics That Make Us Smart"], 1999)

"Heuristics are rules of thumb that help constrain the problem in certain ways (in other words they help you to avoid falling back on blind trial and error), but they don't guarantee that you will find a solution. Heuristics are often contrasted with algorithms that will guarantee that you find a solution - it may take forever, but if the problem is algorithmic you will get there. However, heuristics are also algorithms." (S Ian Robertson, "Problem Solving", 2001)

"Heuristics are needed in situations where the world does not permit optimization. For many real-world problems (as opposed to optimization-tuned textbook problems), optimal solutions are unknown because the problems are computationally intractable or poorly defined." (Christoph Engel & Gerd Gigerenzer, "Law and Heuristics: An interdisciplinary venture" [in "Heuristics and the Law", 2006)

"Less apparent is that qualitatively different problem-solving techniques are required at high levels of complexity than at low ones. Purely analytical techniques, powerful for the lower levels, can be overwhelmed at the higher ones. At higher levels, architecting methods, experience-based heuristics, abstraction, and integrated modeling must be called into play."  (Mark W Maier, "The Art Systems of Architecting" 3rd Ed., 2009)

"Many of the great mathematical problems stem from deep and difficult questions in well-established areas of the subject. They are the big challenges that emerge when a major area has been thoroughly explored. They tend to be quite technical, and everyone in the area knows they’re hard to answer, because many experts have tried and failed. The area concerned will already possess many powerful techniques, massive mathematical machines whose handles can be cranked if you’ve done your homework - but if the problem is still open, then all of the plausible ways to use those techniques have already been tried, and they didn’t work. So either there is a less plausible way to use the tried-and-tested techniques of the area, or you need new techniques." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013) 

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