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 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)

"Solvable Lie algebras are close to both upper triangular matrices and commutative Lie algebras. In contrast to this, semisimple Lie algebras are as far as possible from being commutative. By Levi’s decomposition theorem, any Lie algebra is built out of a solvable and a semisimple one. The nontrivial prototype of a solvable Lie algebra is the Heisenberg algebra. [3](Eberhard Zeidler, "Quantum Field Theory I: Gauge Theory", 2006)

"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 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 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 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 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)