On Perspective IV

"Our generational perspective contributes to the mental models we hold about ourselves, the world, and the way things ‘should’ be. These beliefs create blind spots that can become our undoing as we pursue our values and seek to accomplish our goals. Likewise, they can have a powerful effect on our culture.” (Deborah Gilburg, “Empowering Multigenerational Collaboration in the Workplace”, The Systems Thinker Vol. 18 No. 4, 2007)

"A heuristic is a rule applied to an existing solution represented in a perspective that generates a new (and hopefully better) solution or a new set of possible solutions." (Scott E Page, "The Difference: How the Power of Diversity Creates Better Groups, Firms, Schools and Societies", 2008)

"Every problem has a solution; it may sometimes just need another perspective." (Rebecca Mallery et al, "NLP for Rookies", 2009) 

"Also known as worldview, mental model, or mind-set, our perspective of the world is based on the sum total of our knowledge and experiences. It defines us, shaping our thoughts and actions because it represents the way we see ourselves and situations, how we judge the relative importance of things, and how we establish a meaningful relationship with everything around us." (Navi Radjou, Prasad Kaipa, "From Smart to Wise: Acting and Leading with Wisdom", 2013)

"Although good ethical decision-making requires us carefully to take into account as much relevant information as is available to us, we have good reason to think that we commonly fall well short of this standard – either by overlooking relevant facts completely or by underestimating their significance. The mental models we employ can contribute to this problem. As we have explained, mental models frame our experiences in ways that both aid and hinder our perceptions. They enable us to focus selectively on ethically relevant matters. By their very nature, they provide incomplete perspectives, resulting in bounded awareness and bounded ethicality. Insofar as our mental modeling practices result in unwarranted partiality, or even ethical blindness, the desired reflective process is distorted. This distortion is aggravated by the fact that our mental models can have this distorting effect without our consciously realizing it. Thus, although we cannot do without mental models, they leave us all vulnerable to blindness and, insofar as we are unaware of this, self-deception." (Patricia H Werhane et al, "Obstacles to Ethical: Decision-Making Mental Models, Milgram and the Problem of Obedience", 2013)

"These framing perspectives or mental models construe the data of our experiences, and it is the construed data that we call 'facts'. What we often call reality, or the world, is constructed or socially construed in certain ways such that one cannot get at the source of the data except through these construals." (Patricia H Werhane et al, "Obstacles to Ethical: Decision-Making Mental Models, Milgram and the Problem of Obedience", 2013)

"Using random processes in our models allows economists to capture the variability of time series data, but it also poses challenges to model builders. As model builders, we must understand the uncertainty from two different perspectives. Consider first that of the econometrician, standing outside an economic model, who must assess its congruence with reality, inclusive of its random perturbations. An econometrician’s role is to choose among different parameters that together describe a family of possible models to best mimic measured real world time series and to test the implications of these models. I refer to this as outside uncertainty. Second, agents inside our model, be it consumers, entrepreneurs, or policy makers, must also confront uncertainty as they make decisions. I refer to this as inside uncertainty, as it pertains to the decision-makers within the model. What do these agents know? From what information can they learn? With how much confidence do they forecast the future? The modeler’s choice regarding insiders’ perspectives on an uncertain future can have significant consequences for each model’s equilibrium outcomes." (Lars P Hansen, "Uncertainty Outside and Inside Economic Models", [Nobel lecture] 2013)

"A worldview consists of observations of the individual and other people with respect to the self, time and space, the natural and the supernatural and the sacred and profane. […] Beliefs about the world do not reside in the human mind in chaotic disorder; rather they form a latent system. A worldview cannot, however, be viewed as a well-organised network of cognitive models or a static collection of values; instead it should be regarded as the product of a process shaped by historical, cultural and social perspectives and contexts." (Helena Helve, "A longitudinal perspective on worldviews, values and identities", 2016)

"We should conclude nothing because that pair of numbers alone tells us very little. If we want to understand what’s happening, we need to step back and take in a broader perspective." (Tim Harford, "The Data Detective: Ten easy rules to make sense of statistics", 2020)

On Perspective III

"What is the difference between merely knowing (or remembering, or memorizing) and understanding? […] A thing or idea seems meaningful only when we have several different ways to represent it - different perspectives and different associations […]. Then we can turn it around in our minds, so to speak: however it seems at the moment, we can see it another way and we never come to a full stop. In other words, we can 'think' about it. If there were only one way to represent this thing or idea, we would not call this representation thinking." (Marvin Minsky, "Music, Mind, and Meaning", 1981)

"Wisdom is your perspective on life, your sense of balance, your understanding of how the various parts and principles apply and relate to each other." (Stephen R Covey, "The 7 Habits of Highly Effective People", 1989)

"Symmetry breaking in psychology is governed by the nonlinear causality of complex systems (the 'butterfly effect'), which roughly means that a small cause can have a big effect. Tiny details of initial individual perspectives, but also cognitive prejudices, may 'enslave' the other modes and lead to one dominant view." (Klaus Mainzer, "Thinking in Complexity", 1994)

"One of the strongest benefits of the systems thinking perspective is that it can help you learn to ask the right questions. This is an important first step toward understanding a problem. […] Much of the value of systems thinking comes from the different framework that it gives us for looking at problems in new ways." (Virginia Anderson & Lauren Johnson, "Systems Thinking Basics: From Concepts to Causal Loops", 1997)

"The beauty of this [systems thinking] mindset is that its mental models are based on natural laws, principles of interrelationship, and interdependence found in all living systems. They give us a new view of ourselves and our many systems, from the tiniest cell to the entire earth; and as our organizations are included in that great range, they help us define organizational problems as systems problems, so we can respond in more productive ways. The systems thinking mindset is a new orientation to life. In many ways it also operates as a worldview - an overall perspective on, and understanding of, the world." (Stephen G Haines, "The Managers Pocket Guide to Systems Thinking & Learning", 1998)

"There are many modes of thinking about the world around us and our place in it. I like to consider all the angles from which we might gain perspective on our amazing universe and the nature of existence." (John A Wheeler & Kenneth W Ford, "Geons, Black Holes, and Quantum Foam: A Life in Physics", 1998)

"Often a successful problem-solver is one who creates a new context in which to view the problem. This can often be done by directing one's attention away from the distracting details of the difficulty. From a detached perspective, we may examine the situation in a new or different light and, after exploring information and options, choose an appropriate course of action." (John Templeton, "Wisdom From World Religions: Pathways Toward Heaven on Earth", 2002)

"The smartest groups, then, are made up of people with diverse perspectives who are able to stay independent of each other." (James Surowiecki, "The Wisdom of Crowds", 2005)

On Perspective I

"Everything we hear is an opinion, not a fact. Everything we see is a perspective, not the truth." (Marcus Aurelius, "Meditations", cca. 2nd century)

"Among all the studies of natural causes and reasoning, Light chiefly delights the beholder; and among the great features of mathematics the certainty of its demonstrations is what preeminently tends to elevate the mind of the investigator: Perspective, therefore, might be preferred to all the discourses and systems of the human learning." (Leonardo da Vinci, 1497)

"Perspective is nothing more than a rational demonstration applied to the consideration of how objects in front of the eye transmit their image to it, by means of a pyramid of lines." (Leonardo da Vinci, "The Notebooks of Leonardo da Vinci", 1888)

"Reduced to their most pregnant difference, empiricism means the habit of explaining wholes by parts, and rationalism means the habit of explaining parts by wholes. Rationalism thus preserves affinities with monism, since wholeness goes with union, while empiricism inclines to pluralistic views. No philosophy can ever be anything but a summary sketch, a picture of the world in abridgment, a foreshortened bird's-eye view of the perspective of events. And the first thing to notice is this, that the only material we have at our disposal for making a picture of the whole world is supplied by the various portions of that world of which we have already had experience. We can invent no new forms of conception, applicable to the whole exclusively, and not suggested originally by the parts." (William James, "A Pluralistic Universe", 1908)

"Methods, when classified and separated, acquire their true bearing and perspective as a means to an end, not as ends in themselves." (Benjamin N Cardozo, "The Growth of the Law", 1924)

"A system in one perspective is a subsystem in another. But the systems view always treats systems as integrated wholes of their subsidiary components and never as the mechanistic aggregate of parts in isolable causal relations." (Ervin László, "Introduction to Systems Philosophy", 1972)

"Systems Are Seductive. They promise to do a hard job faster, better, and more easily than you could do it by yourself. But if you set up a system, you are likely to find your time and effort now being consumed in the care and feeding of the system itself. New problems are created by its very presence. Once set up, it won't go away, it grows and encroaches. It begins to do strange and wonderful things. Breaks down in ways you never thought possible. It kicks back, gets in the way, and opposes its own proper function. Your own perspective becomes distorted by being in the system. You become anxious and push on it to make it work. Eventually you come to believe that the misbegotten product it so grudgingly delivers is what you really wanted all the time. At that point encroachment has become complete. You have become absorbed. You are now a systems person." (John Gall, "General Systemantics: How systems work, and especially how they fail", 1975)

"The thinking person goes over the same ground many times. He looks at it from varying points of view - his own, his arch-enemy’s, others’. He diagrams it, verbalizes it, formulates equations, constructs visual images of the whole problem, or of troublesome parts, or of what is clearly known. But he does not keep a detailed record of all this mental work, indeed could not. […] Deep understanding of a domain of knowledge requires knowing it in various ways. This multiplicity of perspectives grows slowly through hard work and sets the state for the re-cognition we experience as a new insight." (Howard E Gruber, "Darwin on Man", 1981)

Terence Tao - Collected Quotes

"Algebra is what most people associate with mathematics. In a sense, this is justified. Mathematics is the study of abstract objects, numerical, logical, or geometrical, that follow a set of several carefully chosen axioms. And basic algebra is about the simplest meaningful thing that can satisfy the above definition of mathematics. There are only a dozen or so postulates, but that is enough to make the system beautifully symmetric." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)

"Analysis is also a heavily explored subject, and it is just as general as algebra: essentially, analysis is the study of functions and their properties. The more complicated the properties, the higher the analysis." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)

"Data is there to be used, so one should pick up the data and play with it. Can it produce more meaningful data?" (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)

"It is also a good idea to not apply any given technique or method blindly, but to think ahead and see where one could hope such a technique to take one; this can allow one to save enormous amounts of time by eliminating unprofitable directions of inquiry before sinking lots of effort into them, and conversely to give the most promising directions priority."(Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)

"Like and unlike the proverb above, the solution to a problem begins (and continues, and ends) with simple, logical steps. But as long as one steps in a firm, clear direction, with long strides and sharp vision, one would need far, far less than the millions of steps needed to journey a thousand miles. And mathematics, being abstract, has no physical constraints; one can always restart from scratch, try new avenues of attack, or backtrack at an instant’s notice. One does not always have these luxuries in other forms of problem-solving (e.g. trying to go home if you are lost)." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006) 

"Mathematics is a multifaceted subject, and our experience and appreciation of it changes with time and experience." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)

"Mathematics is sometimes thought of as great entity, like a tree, branching off into several large chunks of mathematics, which themselves branch off into specialized fields, until you reach the very ends of the tree, where you find the blossoms and the fruit. But it is not easy to classify all of mathematics into such neat compartments: there are always fuzzy regions in between branches and also extra bits outside all the classical branches." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)

"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)

"Number theory may not necessarily be divine, but it still has an aura of mystique about it. Unlike algebra, which has as its backbone the laws of manipulating equations, number theory seems to derive its results from a source unknown." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)

"Understand the data. What is given in the problem? Usually, a question talks about a number of objects which satisfy some special requirements. To understand the data, one needs to see how the objects and requirements react to each other. This is important in focusing attention on the proper techniques and notation to handle the problem." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006) 

"Understand the objective. What do we want? One may need to find an object, prove a statement, determine the existence of an object with special properties, or whatever. Like the flip side of this strategy, ‘understand the data’, knowing the objective helps focus attention on the best weapons to use. Knowing the objective also helps in creating tactical goals which we know will bring us closer to solving the question." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006) 

"Write down what you know in the notation selected; draw a diagram. Putting everything down on paper helps in three ways: (a) you have an easy reference later on; (b) the paper is a good thing to stare at when you are stuck; (c) the physical act of writing down of what you know can trigger new inspirations and connections." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)

"At a purely formal level, one could call probability theory the study of measure spaces with total measure one, but that would be like calling number theory the study of strings of digits which terminate." (Terence Tao, "Topics in Random Matrix Theory", 2012)

Bas C van Fraassen - Collected Quotes

"A map is a graphical representation of geographical or astronomical features, but this may range from a sketch of a subway system, to an interactive, zoomable, or animated map on a computer which constantly changes in front of the eyes."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"A map is designed to help one get around in the landscape it depicts. [...]"  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"A model often contains much that does not correspond to any observable feature in the domain. Then, from an empiricist point of view, the model’s structure must be taken to reveal structure in the observable phenomena, while the rest of the model must be serving that purpose indirectly. It may be practically as well as theoretically useful to think of the phenomena as embedded in a larger - and largely unobservable - structure."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"A scale model represents, and yields information about what it is a model of, by selective resemblance. [...] Scale models can be produced for the sheer aesthetic pleasure of it, but more typically they serve in studies meant to design the very things of which they are meant to be the scaled down versions."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"A science presents us with representations of the phenomena through artifacts, both abstract, such as theories and mathematical models, and concrete such as graphs, tables, charts, and ‘table-top’ models. These representations do not form a haphazard compilation though any unity, in the range of representations made available, is visible mainly at the more abstract levels."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"An algebra is a mathematical structure consisting of a set of elements and a collection of operators on those elements - though the term is variously defined in different contexts in mathematics, so as to narrow the meaning (e.g. an algebra is a vector space with a bilinear multiplication operation)."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"[...] construction of a data model is precisely the selective relevant depiction of the phenomena by the user of the theory required for the possibility of representation of the phenomenon."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"For unqualified adequacy of the theory, what is required is that the surface models of phenomena fit properly with or into the theoretical models. The surface models will provide probability functions for events that are classified as outcomes in situations classified as measurements of given observables."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"In maps we have scale models of terrain, but projected onto a plane, thus producing occlusion of a sort not inherent to three-dimensional imaging. Maps do not usually have an obvious perspective; but we see perspectivity when, for example, the curvature of the earth makes marginal distortion inevitable as a result of this projection that lowers the dimensionality. A map too is the product of a measuring procedure, but they bring to light a much more important point about ‘point of view’, essentially independent of these limitations in cartography. The point extends to all varieties of modeling, but is made salient by the sense in which use enters the concept of ‘map’ from the beginning. A map is not only an object used to represent features of a terrain, it is an object for the use of the industrial designer, the navigator, and most of all the traveler, to plan and direct action. This brings us to an aspect of scientific representation not touched on so far, though implicit in the discussion of perspective, crucial to its overall understanding: its indexicality."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"The general concept of a map is not so different from that of a model, though the one is extrapolated from a graph with spatial similarity to certain features of a landscape, and the other from a table-top contraption."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

'Model' is a metaphor, whose base is the simply constructed table-top model."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"Scale modeling displays the characteristics of picturing, by relying on selective resemblance to achieve its aim, but in a way that is subject to inevitable occlusion or distortion."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"Scientific theories represent how things are, doing so mainly by presenting a range of models as candidate representations of the phenomena. [...] A theory provides, in essence, a set of models. The 'in essence' signals much that must be delicately expanded and qualified; [...] These models - the theoretical models - are provided in the first instance to fit observed and observable phenomena. Since the description of these phenomena is in practice already by means of models - the ‘data models’ or ‘surface models’, we can put the requirement as follows: the data or surface models must ideally be isomorphically embeddable in theoretical models."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"The activity of representation is successful in that case only if the recipients are able to receive that information through their ‘viewing’ of the representation. [...] In science the original creation of a model may have been a purely theoretical activity, but eventually it provides input for an application, where conditional predictions made on the basis of that model feed into planning and action."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"The observable phenomenon makes its appearance to us first of all in the outcome of a specific measurement, or large set of such measurements - or at slight remove, in a data model constructed from these individual outcomes, or at a slightly further remove yet, in the surface model constructed by extrapolating the patterns in the data model to something finer than our instruments can register."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"The physical sciences give us representations of nature, and scientific representation is in general three-faceted. From a purely foundational point of view, the theoretical models that depict the ‘underlying reality’ are the main thing. But some elements or substructures of those models are meant to represent the observable phenomena - the empirical substructures."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"Theories represent the phenomena just in case their models, in some sense, 'share the same structure' with those phenomena - that, in slogan form, is what is called the semantic view of theories. [...] Embedding, that means displaying an isomorphism to selected parts of those models. Here is the argument to present the first challenge. For a phenomenon to be embeddable in a model, that means that it is isomorphic to a part of that model. So the two, the phenomenon and the relevant model part must have the same structure. Therefore, the phenomenon must have a structure, and this shared structure is obviously not itself a physical, concrete individual - so what is implied here is something of the order of realism about universals."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"[...] when a theoretical model is said to represent certain phenomena, there is indeed reference to a matching, namely between parts of the theoretical models and the relevant data models - both of them abstract entities. Note now, the crucial word in this sentence: the punch comes in the word 'relevant'. There is nothing in an abstract structure itself that can determine that it is the relevant data model, to be matched by the theory. A particular data model is relevant because it was constructed on the basis of results gathered in a certain way, selected by specific criteria of relevance, on certain occasions, in a practical experimental or observational setting, designed for that purpose."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

Mental Models XXXV

"Here I am at the limit which God and nature has assigned to my individuality. I am compelled to depend upon word, language and image in the most precise sense, and am wholly unable to operate in any manner whatever with symbols and numbers which are easily intelligible to the most highly gifted minds." (Johann Wolfgang von Goethe, [Letter to Naumann] 1826)

"It may sound quite strange, but for me, as for other scientists on whom these kinds of imaginative images have a greater effect than other poems do, no science is at its very heart more closely related to poetry, perhaps, than is chemistry." (Just Liebig, 1854)

"But I thoroughly believe myself, and hope to prove to you, that science is full of beautiful pictures, of real poetry, and of wonder-working fairies; and what is more […] though they themselves will always remain invisible, yet you will see their wonderful power at work everywhere around you. […] There is only one gift we must have before we can learn to know them - we must have imagination. I do not mean mere fancy, which creates unreal images and impossible monsters, but imagination, the power of making pictures or images in our mind, of that which is, though it is invisible to us." (Arabella Buckley, Fairyland, 1879)

"Ask your imagination if it will accept a vibrating multiple proportion - a numerical ratio in a state of oscillation? I do not think it will. You cannot crown the edifice with this abstraction. The scientific imagination, which is here authoritative, demands, as the origin and cause of a series of ether-waves, a particle of vibrating matter quite as definite, though it may be excessively minute, as that which gives origin to a musical sound. Such a particle we name an atom or a molecule. I think the intellect, when focused so as to give definition without penumbral haze, is sure to realize this image at the last." (John Tyndall, "Fragments of Science", 1892)

"The mathematical formula is the point through which all the light gained by science passes in order to be of use to practice; it is also the point in which all knowledge gained by practice, experiment, and observation must be concentrated before it can be scientifically grasped. The more distant and marked the point, the more concentrated will be the light coming from it, the more unmistakable the insight conveyed. All scientific thought, from the simple gravitation formula of Newton, through the more complicated formulae of physics and chemistry, the vaguer so called laws of organic and animated nature, down to the uncertain statements of psychology and the data of our social and historical knowledge, alike partakes of this characteristic, that it is an attempt to gather up the scattered rays of light, the different parts of knowledge, in a focus, from whence it can be again spread out and analyzed, according to the abstract processes of the thinking mind. But only when this can be done with a mathematical precision and accuracy is the image sharp and well-defined, and the deductions clear and unmistakable. As we descend from the mechanical, through the physical, chemical, and biological, to the mental, moral, and social sciences, the process of focalization becomes less and less perfect, - the sharp point, the focus, is replaced by a larger or smaller circle, the contours of the image become less and less distinct, and with the possible light which we gain there is mingled much darkness, the sources of many mistakes and errors. But the tendency of all scientific thought is toward clearer and clearer definition; it lies in the direction of a more and more extended use of mathematical measurements, of mathematical formulae." (John T Merz, "History of European Thought in the 19th Century" Vol. 1, 1904)

"Many people believe that reasoning, and therefore science, is a different activity from imagining. But this is a fallacy […] Reasoning is constructed with movable images just as certainly as poetry is." (Jacob Bronowski, "Visionary Eye", 1978)

"The thinking person goes over the same ground many times. He looks at it from varying points of view - his own, his arch-enemy’s, others’. He diagrams it, verbalizes it, formulates equations, constructs visual images of the whole problem, or of troublesome parts, or of what is clearly known. But he does not keep a detailed record of all this mental work, indeed could not. […] Deep understanding of a domain of knowledge requires knowing it in various ways. This multiplicity of perspectives grows slowly through hard work and sets the state for the re-cognition we experience as a new insight." (Howard E Gruber, "Darwin on Man", 1981)

"Mathematicians have always needed to ‘see’ the complex concepts they work with in order to reason with them effectively. In the past, they conjured up mental images as best they could, but the wonders of computer graphics provide them with far more detailed pictures to think with." (Richard Palais and Luc Bernard, "2006 Visualization Project", 2006)

"Time is to eternity as an image is to its exemplar, and those things which are temporal bear a resemblance to those things which are eternal." (Nicholas of Cusa)

"You know how the divine Simplicity enfolds all things. Mind is the image of this enfolding Simplicity. If, then, you called this divine Simplicity infinite Mind, it will be the exemplar of our mind. If you called the divine mind the totality of the truth of things, you will call our mind the totality of the assimilation of things, so that it may be a totality of ideas. In the divine Mind conception is the production of things; in our mind conception is the knowledge of things. If the divine Mind is absolute Being, then its conception is the creation of beings; and conception in the human mind is the assimilation of beings." (Nicholas of Cusa)

On Problem Solving IV: Solvability

"A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted. There remains always something to do; with sufficient study and penetration, we could improve any solution, and, in any case, we can always improve our understanding of the solution." (George Pólya, "How to Solve It", 1945)

"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 Polya, "How to Solve It", 1945)

"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)

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

"A problem will be difficult if there are no procedures for generating possible solutions that are guaranteed (or at least likely) to generate the actual solution rather early in the game. But for such a procedure to exist, there must be some kind of structural relation, at least approximate, between the possible solutions as named by the solution-generating process and these same solutions as named in the language of the problem statement." (Herbert A Simon, "The Logic of Heuristic Decision Making", [in "The Logic of Decision and Action"], 1966)

"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)

"In general, complexity and precision bear an inverse relation to one another in the sense that, as the complexity of a problem increases, the possibility of analysing it in precise terms diminishes. Thus 'fuzzy thinking' may not be deplorable, after all, if it makes possible the solution of problems which are much too complex for precise analysis." (Lotfi A Zadeh, "Fuzzy languages and their relation to human intelligence", 1972)

"A great many problems are easier to solve rigorously if you know in advance what the answer is." (Ian Stewart, "From Here to Infinity", 1987)

"A common mistake in problem solving is to encompass too much territory, which dilutes any solutions chance of success. [...] However, the opposite error occurs more frequently." (Terry Richey, "The Marketer's Visual Tool Kit", 1994)

"[…] the meaning of the word 'solve' has undergone a series of major changes. First that word meant 'find a formula'. Then its meaning changed to 'find approximate numbers'. Finally, it has in effect become 'tell me what the solutions look like'. In place of quantitative answers, we seek qualitative ones." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"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)

"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)

"Solving a problem for which you know there’s an answer is like climbing a mountain with a guide, along a trail someone else has laid. In mathematics, the truth is somewhere out there in a place no one knows, beyond all the beaten paths. And it’s not always at the top of the mountain. It might be in a crack on the smoothest cliff or somewhere deep in the valley." (Yōko Ogawa, "The Housekeeper and the Professor", 2003)

"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)

"Mathematical good taste, then, consists of using intelligently the concepts and results available in the ambient mathematical culture for the solution of new problems. And the culture evolves because its key concepts and results change, slowly or brutally, to be replaced by new mathematical beacons." (David Ruelle, "The Mathematician's Brain", 2007)

"The beauty of mathematics is that clever arguments give answers to problems for which brute force is hopeless, but there is no guarantee that a clever argument always exists!" (David Ruelle, "The Mathematician's Brain", 2007)

"Every problem has a solution; it may sometimes just need another perspective." (Rebecca Mallery et al, "NLP for Rookies", 2009)

"Don't mistake a solution method for a problem definition - especially if it’s your own solution method." (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)

"I have not seen any problem, however complicated, which, when you looked at it in the right way, did not become still more complicated." (Paul Anderson)

"It is efficient to look for beautiful solutions first and settle for ugly ones only as a last resort. [...] It is a good rule of thumb that the more beautiful the guess, the more likely it is to survive." (Timothy Gowers)

"One is always a long way from solving a problem until one actually has the answer." (Stephen Hawking)

"The best way to escape from a problem is to solve it." (Brendan Francis)

"The worst thing you can do to a problem is solve it completely." (Daniel Kleitman)

"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)

"We can not solve our problems with the same level of thinking that created them." (Albert Einstein) 

"When the answer to a mathematical problem cannot be found, then the reason is frequently that we have not recognized the general idea from which the given problem only appears as a link in a chain of related problems." (David Hilbert) 

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On Problem Solving I (Stating the Problem)

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

"We call a problem well-defined if there is a test which can be applied to a proposed solution. In case the proposed solution is a solution, the test must confirm this in a finite number of steps." (John McCarthy, "The Inversion of Functions Denned by Turing Machines", 1956)

"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)

"A problem will be difficult if there are no procedures for generating possible solutions that are guaranteed (or at least likely) to generate the actual solution rather early in the game. But for such a procedure to exist, there must be some kind of structural relation, at least approximate, between the possible solutions as named by the solution-generating process and these same solutions as named in the language of the problem statement." (Herbert A Simon, "The Logic of Heuristic Decision Making", [in "The Logic of Decision and Action"], 1966)

"Definitions, like questions and metaphors, are instruments for thinking. Their authority rests entirely on their usefulness, not their correctness. We use definitions in order to delineate problems we wish to investigate, or to further interests we wish to promote. In other words, we invent definitions and discard them as suits our purposes." (Neil Postman, "Language Education in a Knowledge Context", 1980)

"Define the problem before you pursue a solution." (John Williams, Inc. Magazine's Guide to Small Business Success, 1987)

"There are many things you can do with problems besides solving them. First you must define them, pose them. But then of course you can also refine them, depose them, or expose them or even dissolve them! A given problem may send you looking for analogies, and some of these may lead you astray, suggesting new and different problems, related or not to the original. Ends and means can get reversed. You had a goal, but the means you found didn’t lead to it, so you found a new goal they did lead to. It’s called play. Creative mathematicians play a lot; around any problem really interesting they develop a whole cluster of analogies, of playthings." (David Hawkins, "The Spirit of Play", Los Alamos Science, 1987)

"Most people would rush ahead and implement a solution before they know what the problem is." (Q T Wiles, Inc. Magazine, 1988)

"An internal model corresponds to a specific concrete situation in the external world and allows us to reason about the external situation. To do so you used information about the problem presented in the problem statement. The process of understanding, then, refers to constructing an initial mental representation of what the problem is, based on the information in the problem statement about the goal, the initial state, what you are not allowed to do, and what operator to apply, as well as your own personal past experience." (S Ian Robertson, "Problem Solving", 2001)

"Problem solving starts off from an initial given situation or statement of a problem (known as the initial state of the problem). Based on the problem situation and your prior knowledge you have to work towards a solution. When you reach it you are in the goal state of the problem. On the way from the initial state to the goal state you pass through a number of intermediate problem states." (S Ian Robertson, "Problem Solving", 2001)

"The way a problem is defined determines how we attempt to solve it. […] If the definition is wrong, you will develop the right solution to the wrong problem." (James P Lewis, "Project Planning, Scheduling, and Control" 3rd Ed., 2001)

"Understanding a problem means building some kind of representation of the problem in one's mind, based on what the situation is or what the problem statement says and on one's prior knowledge. It is then possible to reason about the problem within this mental representation. Generating a useful mental representation is therefore the most important single factor for successful problem solving." (S Ian Robertson, "Problem Solving", 2001)

"Don't mistake a solution method for a problem definition - especially if it’s your own solution method." (Donald C Gause & Gerald M Weinberg, "Are Your Lights On?", 2011)

"The fledgling problem solver invariably rushes in with solutions before taking time to define the problem being solved. Even experienced solvers, when subjected to social pressure, yield to this demand for haste. When they do, many solutions are found, but not necessarily to the problem at hand." (Donald C Gause & Gerald M Weinberg, "Are Your Lights On?", 2011)

"Framing the right problem is equally or even more important than solving it." (Pearl Zhu, "Change, Creativity and Problem-Solving", 2017)

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

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

"The mere formulation of a problem is often far more essential than its solution. To raise new questions, new possibilities, to regard old problems from a new angle, requires creative imagination and marks real advances in science." (Albert Einstein)

"To ask the right question is harder than to answer it." (Georg Cantor)

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