❄️Systems Thinking: On Problem Solving (Quotes)

"Even these humble objects reveal that our reality is not a mere collocation of elemental facts, but consists of units in which no part exists by itself, where each part points beyond itself and implies a larger whole. Facts and significance cease to be two concepts belonging to different realms, since a fact is always a fact in an intrinsically coherent whole. We could solve no problem of organization by solving it for each point separately, one after the other; the solution had to come for the whole. Thus we see how the problem of significance is closely bound up with the problem of the relation between the whole and its parts. It has been said: The whole is more than the sum of its parts. It is more correct to say that the whole is something else than the sum of its parts, because summing is a meaningless procedure, whereas the whole-part relationship is meaningful." (Kurt Koffka, "Principles of Gestalt Psychology", 1935)

"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)

"Systems engineering embraces every scientific and technical concept known, including economics, management, operations, maintenance, etc. It is the job of integrating an entire problem or problem to arrive at one overall answer, and the breaking down of this answer into defined units which are selected to function compatibly to achieve the specified objectives. [...] Instrument and control engineering is but one aspect of systems engineering - a vitally important and highly publicized aspect, because the ability to create automatic controls within overall systems has made it possible to achieve objectives never before attainable, While automatic controls are vital to systems which are to be controlled, every aspect of a system is essential. Systems engineering is unbiased, it demands only what is logically required. Control engineers have been the leaders in pulling together a systems approach in the various technologies." (Instrumentation Technology, 1957)

"Systems engineering is the name given to engineering activity which considers the overall behavior of a system, or more generally which considers all factors bearing on a problem, and the systems approach to control engineering problems is correspondingly that approach which examines the total dynamic behavior of an integrated system. It is concerned more with quality of performance than with sizes, capacities, or efficiencies, although in the most general sense systems engineering is concerned with overall, comprehensive appraisal." (Ernest F Johnson, "Automatic process control", 1958)

"[System dynamics] is an approach that should help in important top-management problems [...] The solutions to small problems yield small rewards. Very often the most important problems are but little more difficult to handle than the unimportant. Many [people] predetermine mediocre results by setting initial goals too low. The attitude must be one of enterprise design. The expectation should be for major improvement [...] The attitude that the goal is to explain behavior; which is fairly common in academic circles, is not sufficient. The goal should be to find management policies and organizational structures that lead to greater success." (Jay W Forrester, "Industrial Dynamics", 1961)

"Systems engineering is most effectively conceived of as a process that starts with the detection of a problem and continues through problem definition, planning and designing of a system, manufacturing or other implementing section, its use, and finally on to its obsolescence. Further, Systems engineering is not a matter of tools alone; It is a careful coordination of process, tools and people." (Arthur D. Hall, "Systems Engineering from an Engineering Viewpoint" In: Systems Science and Cybernetics. Vol.1 Issue.1, 1965)

"System theory is basically concerned with problems of relationships, of structure, and of interdependence rather than with the constant attributes of objects. In general approach it resembles field theory except that its dynamics deal with temporal as well as spatial patterns. Older formulations of system constructs dealt with the closed systems of the physical sciences, in which relatively self-contained structures could be treated successfully as if they were independent of external forces. But living systems, whether biological organisms or social organizations, are acutely dependent on their external environment and so must be conceived of as open systems." (Daniel Katz, "The Social Psychology of Organizations", 1966)

"Only if mathematical rigor is adhered to, can systems problems be dealt with effectively, and so it is that the systems engineer must, at least, develop an appreciation for mathematical rigor if not also considerable mathematical competence." (A Wayne Wymore, "A Mathematical Theory of Systems Engineering", 1967)

"Solving a problem simply means representing it so as to make the solution transparent." (Herbert A Simon, "The Sciences of the Artificial", 1968)

"The systems approach to problems focuses on systems taken as a whole, not on their parts taken separately. Such an approach is concerned with total - system performance even when a change in only one or a few of its parts is contemplated because there are some properties of systems that can only be treated adequately from a holistic point of view. These properties derive from the relationship between parts of systems: how the parts interact and fit together." (Russell L Ackoff, "Towards a System of Systems Concepts", 1971)

"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)

"In sharp contrast (with the traditional social planning) the systems design approach seeks to understand a problem situation as a system of interconnected, interdependent, and interacting issues and to create a design as a system of interconnected, interdependent, interacting, and internally consistent solution ideas." (Béla H Bánáthy, "Designing Social Systems in a Changing World", 1996)

"It [system dynamics] focuses on building system dynamics models with teams in order to enhance team learning, to foster consensus and to create commitment with a resulting decision […] System dynamics can be helpful to elicit and integrate mental models into a more holistic view of the problem and to explore the dynamics of this holistic view […] It must be understood that the ultimate goal of the intervention is not to build a system dynamics model. The system dynamics model is a means to achieve other ends […] putting people in a position to learn about a messy problem … create a shared social reality […] a shared understanding of the problem and potential solutions … to foster consensus within the team [..]" (Jac A M Vennix, "Group Model Building: Facilitating Team Learning Using System Dynamics", 1996)

"It is, however, fair to say that very few applications of swarm intelligence have been developed. One of the main reasons for this relative lack of success resides in the fact that swarm-intelligent systems are hard to 'program', because the paths to problem solving are not predefined but emergent in these systems and result from interactions among individuals and between individuals and their environment as much as from the behaviors of the individuals themselves. Therefore, using a swarm-intelligent system to solve a problem requires a thorough knowledge not only of what individual behaviors must be implemented but also of what interactions are needed to produce such or such global behavior." (Eric Bonabeau et al, "Swarm Intelligence: From Natural to Artificial Systems", 1999)

"True systems thinking, on the other hand, studies each problem as it relates to the organization’s objectives and interaction with its entire environment, looking at it as a whole within its universe. Taking your organization from a partial systems to a true systems state requires effective strategic management and backward thinking." (Stephen G Haines, "The Systems Thinking Approach to Strategic Planning and Management", 2000)

"Systems thinking means the ability to see the synergy of the whole rather than just the separate elements of a system and to learn to reinforce or change whole system patterns. Many people have been trained to solve problems by breaking a complex system, such as an organization, into discrete parts and working to make each part perform as well as possible. However, the success of each piece does not add up to the success of the whole. to the success of the whole. In fact, sometimes changing one part to make it better actually makes the whole system function less effectively." (Richard L Daft, "The Leadership Experience", 2002)

"Self-organization can be seen as a spontaneous coordination of the interactions between the components of the system, so as to maximize their synergy. This requires the propagation and processing of information, as different components perceive different aspects of the situation, while their shared goal requires this information to be integrated. The resulting process is characterized by distributed cognition: different components participate in different ways to the overall gathering and processing of information, thus collectively solving the problems posed by any perceived deviation between the present situation and the desired situation." (Carlos Gershenson & Francis Heylighen, "How can we think the complex?", 2004)

"System Thinking is a common concept for understanding how causal relationships and feedbacks work in an everyday problem. Understanding a cause and an effect enables us to analyse, sort out and explain how changes come about both temporarily and spatially in common problems. This is referred to as mental modelling, i.e. to explicitly map the understanding of the problem and making it transparent and visible for others through Causal Loop Diagrams (CLD)." (Hördur V. Haraldsson, "Introduction to System Thinking and Causal Loop Diagrams", 2004)

"In engineering, a self-organizing system would be one in which elements are designed to dynamically and autonomously solve a problem or perform a function at the system level. In other words, the engineer will not build a system to perform a function explicitly, but elements will be engineered in such a way that their behaviour and interactions will lead to the system function. Thus, the elements need to divide, but also to integrate, the problem." (Carlos Gershenson, "Design and Control of Self-organizing Systems", 2007)

"Swarm intelligence can be effective when applied to highly complicated problems with many nonlinear factors, although it is often less effective than the genetic algorithm approach [...]. Swarm intelligence is related to swarm optimization […]. As with swarm intelligence, there is some evidence that at least some of the time swarm optimization can produce solutions that are more robust than genetic algorithms. Robustness here is defined as a solution’s resistance to performance degradation when the underlying variables are changed. (Michael J North & Charles M Macal, Managing Business Complexity: Discovering Strategic Solutions with Agent-Based Modeling and Simulation, 2007) 

"A systems approach is one that focuses on the system as a whole, specifically linking value judgments (what is desired) and design decisions (what is feasible). A true systems approach means that the design process includes the 'problem' as well as the solution. The architect seeks a joint problem–solution pair and understands that the problem statement is not fixed when the architectural process starts. At the most fundamental level, systems are collections of different things that together produce results unachievable by the elements alone."  (Mark W Maier, "The Art Systems of Architecting" 3rd Ed., 2009)

"Taking a systems approach means paying close attention to results, the reasons we build a system. Architecture must be grounded in the client’s/user’s/customer’s purpose. Architecture is not just about the structure of components. One of the essential distinguishing features of architectural design versus other sorts of engineering design is the degree to which architectural design embraces results from the perspective of the client/user/customer. The architect does not assume some particular problem formulation, as 'requirements'  is fixed. The architect engages in joint exploration, ideally directly with the client/user/customer, of what system attributes will yield results worth paying for."  (Mark W Maier, "The Art Systems of Architecting" 3rd Ed., 2009)

"DevOps is about team play and a collaborative problem-solving approach. If a service goes down, everyone must know what procedures to follow to diagnose the problem and get the system up and running again. Additionally, all of the roles and skills necessary to perform these tasks must be available and able to work together well. Training and effective collaboration are critical here." (Michael Hüttermann et al, "DevOps for Developers", 2013)

🦋Science: On Problem Solving (Quotes)

"He who seeks for methods without having a definite problem in mind seeks for the most part in vain." (David Hilbert, 1902)

"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, 1945)

"In picking that problem be sure to analyze it carefully to see that it is worth the effort. It takes just as much effort to solve a useless problem as a useful one." (Charles F. Kettering, 1955)

"Solving a problem simply means representing it so as to make the solution transparent." (Herbert A Simon, "The Sciences of the Artificial", 1968)

"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)

"An important symptom of an emerging understanding is the capacity to represent a problem in a number of different ways and to approach its solution from varied vantage points; a single, rigid representation is unlikely to suffice." (Howard Gardner, "The Unschooled Mind", 1991)

"[By understanding] I mean simply a sufficient grasp of concepts, principles, or skills so that one can bring them to bear on new problems and situations, deciding in which ways one’s present competencies can suffice and in which ways one may require new skills or knowledge." (Howard Gardner, "The Unschooled Mind", 1991)

"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)

"This does not mean that the representation has to be 'complete' before any problem solving can take place. If you had a 'complete' representation of a problem then you wouldn't have a problem, as you would know exactly how to get from where you are now to where you want to be. A problem only exists when it is not immediately obvious how to get from where you are now to your goal. An adequate representation should at least allow you to see what moves you can possibly make and allow you to start heading towards your goal." (S Ian Robertson, "Problem Solving", 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)

"[...] a general-purpose universal optimization strategy is theoretically impossible, and the only way one strategy can outperform another is if it is specialized to the specific problem under consideration." Yu-Chi Ho & David L Pepyne, "Simple explanation of the no-free-lunch theorem and its implications", Journal of Optimization Theory and Applications 115, 2002)

"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)

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

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

"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)

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

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

"An expert problem solver must be endowed with two incompatible qualities, a restless imagination and a patient pertinacity." (Howard W Eves)

"Finding the right answer is important, of course. But more important is developing the ability to see that problems have multiple solutions, that getting from X to Y demands basic skills and mental agility, imagination, persistence, patience." (Mary H Futrell)

"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)

"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)

"Man is not born to solve the problems of the universe, but to find out where the problems begin, and then to take his stand within the limits of the intelligible." (Johann Wolfgang von Goethe)

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

"One measure of our understanding is the number of independent ways we are able to get to the same result." (Richard P Feynman) 

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

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

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

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

"The measure of our intellectual capacity is the capacity to feel less and less satisfied with our answers to better and better problems." (Charles W Churchman)

"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)

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

"There is no such thing as a problem without a gift. We seek problems because we need their gifts." (Richard Bach)

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

"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) 

"You are never sure whether or not a problem is good unless you actually solve it." (Mikhail Gromov)

❄️Systems Thinking: On Tipping Points (Quotes)

"But the world of the Tipping Point is a place where the unexpected becomes expected, where radical change is more than possibility. It is - contrary to all our expectations - a certainty." (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 2000)

"For any given population of susceptibles, there is some critical combination of contact frequency, infectivity, and disease duration just great enough for the positive loop to dominate the negative loops. That threshold is known as the tipping point. Below the tipping point, the system is stable: if the disease is introduced into the community, there may be a few new cases, but on average, people will recover faster than new cases are generated. Negative feedback dominates and the population is resistant to an epidemic. Past the tipping point, the positive loop dominates .The system is unstable and once a disease arrives, it can spread like wildfire that is, by positive feedback-limited only by the depletion of the susceptible population." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"If the contact rate, infectivity, and duration of infection are small enough, the system is below the tipping point and stable. Such a situation is known as herd immunity because the arrival of an infected individual does not produce an epidemic (though a few unlucky individuals may come in contact with any infectious arrivals and contract the disease, the group as a community is protected). However, changes in the contact rate, infectivity, or duration of illness can push a system past the tipping point." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"In the real world, advertising is expensive and does not persist indefinitely. The marketing plan for most new products includes a certain amount for a kickoff ad campaign and other initial marketing efforts. If the product is successful, further advertising can be supported out of the revenues the product generates. If, however, the product does not take off, the marketing budget is soon exhausted and external sources of adoption fall. Advertising is not exogenous, as in the Bass model, but is part of the feedback structure of the system. There is a tipping point for ideas and new products no less than for diseases." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"Simply by finding and reaching those few special people who hold so much social power, we can shape the course of social epidemics. In the end, Tipping Points are a reaffirmation of the potential for change and the power of intelligent action." (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 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)

"The sharp boundary between an epidemic and stability defined by the tipping point in the deterministic models becomes a probability distribution characterizing the chance an epidemic will occur for any given average rates of interaction, infectivity, and recovery. Likewise, the SI and SIR models assume a homogeneous and well-mixed population, while in reality it is often important to represent subpopulations and the spatial diffusion of an epidemic." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"The theory of Tipping Points requires, however, that we reframe the way we think about the world. [...] We have trouble estimating dramatic, exponential change. [...] There are abrupt limits to the number of cognitive categories we can make and the number of people we can truly love and the number of acquaintances we can truly know. We throw up our hands at a problem phrased in an abstract way, but have no difficulty at all solving the same problem rephrased as a social dilemma. All of these things are expressions of the peculiarities of the human mind and heart, a refutation of the notion that the way we function and communicate and process information is straightforward and transparent. It is not. It is messy and opaque." (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 2000)

"The tipping point is that magic moment when an idea, trend, or social behavior crosses a threshold, tips, and spreads like wildfire." (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 2000)

"These three characteristics - one, contagiousness; two, the fact that little causes can have big effects; and three, that change happens not gradually but at one dramatic moment - are the same three principles that define how measles moves through a grade-school classroom or the flu attacks every winter. Of the three, the third trait - the idea that epidemics can rise or fall in one dramatic moment - is the most important, because it is the principle that makes sense of the first two and that permits the greatest insight into why modern change happens the way it does. The name given to that one dramatic moment in an epidemic when everything can change all at once is the Tipping Point." (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 2000)

"This possibility of sudden change is at the center of the idea of the Tipping Point and might well be the hardest of all to accept. [...] The Tipping Point is the moment of critical mass, the threshold, the boiling point." (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 2000)

"Tipping points involve three notions: that events and phenomena are contagious, that little causes can have big effects, and that changes can happen not in a gradual linear way but dramatically at a moment when the system switches." (John Urry, "Global Complexity", 2003)

"But in mathematics there is a kind of threshold effect, an intellectual tipping point. If a student can just get over the first few humps, negotiate the notational peculiarities of the subject, and grasp that the best way to make progress is to understand the ideas, not just learn them by rote, he or she can sail off merrily down the highway, heading for ever more abstruse and challenging ideas, while an only slightly duller student gets stuck at the geometry of isosceles triangles." (Ian Stewart, "Why Beauty is Truth: A history of symmetry", 2007)

"The product that first gets over its own tipping point attracts many consumers and this may make the competing product less attractive. Being the first to reach this tipping point is very important - more important than being the 'best' in an abstract sense." (David Easley & Jon Kleinberg, "Networks, Crowds, and Markets: Reasoning about a Highly Connected World", 2010)

"Stochastic variability and tipping points in the catch are two different dynamical phenomena. Yet they are both compatible with real-world data [...]" (John D W Morecroft, "Strategic Modelling and Business Dynamics: A Feedback Systems Approach", 2015)

🦉Knowledge Management: On Critical Thinking (Quotes)

"Reflection upon situationality is reflection about the very condition of existence: critical thinking by means which people discover each other to be 'in a situation'. (Paulo Freire, "Pedagogia do oprimido" ["Pedagogy of the Oppressed"], 1968)

"Critical thinking, in short, offers the way to keep science and technology in harness." (Kenneth Ludmerer, "Learning to Heal: The Development of American Medical Education", 1985)

"Critical thinking is a type of thinking pattern that requires people to be reflective, and pay attention to decision-making which guides their beliefs and actions. Critical thinking allows people to deduct with more logic, to process sophisticated information and look at various sides of an issue so they can produce more solid conclusions." (Joan Baron & Robert Sternberg, "Book Reviews and Notes : Teaching Thinking Skills: Theory and Practice", 1987)

"Critical thinking does seem a superior sort of thinking because it seems as though the critic is actually going beyond the scope of what is being criticized in order to criticize it. That is only rarely a true assumption because, most often, the critic will seize on some little aspect that he or she understands and tackle only that." (Edward de Bono, "I Am Right, You are Wrong", 1990)

"Thinking about one's thinking in a manner designed to organize and clarify, raise the efficiency of, and recognize errors and biases in one's own thinking. Critical thinking is not 'hard' thinking nor is it directed at solving problems (other than 'improving' one's own thinking). Critical thinking is inward-directed with the intent of maximizing the rationality of the thinker. One does not use critical thinking to solve problems- one uses critical thinking to improve one's process of thinking." (Kirby Carmichael, [letter] 1997)

"The purpose of critical thinking [...] is rethinking: that is, reviewing, evaluating, and revising thought." (Jon Stratton, Critical Thinking for College Students, 1999)

[critical thinking:] "The evaluation of ideas, sources, or solutions that are proposed as potential resources or solutions to unusual problems or to product design." (Ruth C Clark, "Building Expertise: Cognitive Methods for Training and Performance Improvement", 2008)

"Critical thinking is essentially a questioning, challenging approach to knowledge and perceived wisdom. It involves ideas and information from an objective position and then questioning this information in the light of our own values, attitudes and personal philosophy." Brenda Judge et al, "Critical Thinking Skills for Education Students", 2009)

[critical thinking:] "Evaluation of products and ideas, such as critiquing an e-learning course or preparing an argument for a position." (Ruth C Clark & Richard E Mayer, "e-Learning and the Science of Instruction", 2011)

[critical thinking:] "Purposeful, self-regulatory judgment which results in interpretation, analysis, evaluation, and inference, as well as explanation of the evidential, conceptual, methodological, criteriological, or contextual considerations upon which that judgment is based" (Peter A Facione, "Critical Thinking: What It is and Why It Counts", 2011)

[critical thinking:] "A process in which one applies observation, analysis, inference, context, reflective thinking, and the like, in order to reach judgments. Such judgments should be open to alternative perspectives that may not normally be otherwise considered." (Project Management Institute, "Navigating Complexity: A Practice Guide", 2014)

[critical thinking:] "The ability to use your personal experience, logical thought processes, and creativity to analyze and evaluate situations. Further, critical thinking allows you to use the information gathered to reach a conclusion or answer." (Darril Gibson, "Effective Help Desk Specialist Skills", 2014)

[critical thinking:] "Thinking that is characterized by careful evaluation and judgment; this involves thinking about one’s thinking (metacognition). Critical thinking may involve examining contradictory lines of reasoning and/or using different lines of reasoning to cross-examine alternatives." (Ken Sylvester, "Negotiating in the Leadership Zone", 2015)

[critical thinking:] "Evaluation of products and ideas such as critiquing an e-learning course or preparing an argument for a position." (Ruth C Clark & Richard E Mayer, "e-Learning and the Science of Instruction", 2016)

[critical thinking:] "The capacity of an individual to effectively engage in a process of making decisions or solving problems by analyzing and evaluating evidence, arguments, claims, beliefs, and alternative points of view; synthesizing and making connections between information and arguments; interpreting information; and making inferences using reasoning appropriate to the situation." (Yigal Rosen & Maryam Mosharraf, "Evidence-Centered Concept Map in Computer-Based Assessment of Critical Thinking", 2016) 

"A more detailed, but still uncontroversial comprehensive, definition is that philosophy is rationally critical thinking, of a more or less systematic kind about the general nature of the world (metaphysics or theory of existence), the justification of belief (epistemology or theory of knowledge), and the conduct of life (ethics or theory of value)." (Anthony Quinton)

❄️Systems Thinking: On Synergy (Quotes)

"The constructive process inheres in all forms of synergy, and the cooperation of antithetical forces in nature always results in making, that is, in creating something that did not exist before. But in the organic world this character of structure becomes the leading feature, and we have synthetic products consisting of tissues and organs serving definite purposes, which we call functions." (Lester F Ward, "Pure Sociology", 1903)

"[...] there is a universal principle, operating in every department of nature and at every stage of evolution, which is conservative, creative and constructive. [...] I have at last fixed upon the word synergy, as the term best adapted to express its twofold character of ‘energy’ and ‘mutuality’ or the systematic and organic ‘working together’ of the antithetical forces of nature. [...] Synergy is a synthesis of work, or synthetic work, and this is what is everywhere taking place. It may be said to begin with the primary atomic collision in which mass, motion, time, and space are involved, and to find its simplest expression in the formula for force, which implies a plurality of elements, and signifies an interaction of these elements." (Lester F Ward, "Pure Sociology", 1903)

"This compromise among the contending forces of nature was effected through organization and the formation of chemical systems, which are so many reservoirs of power, this power being represented by what we call the properties of matter. These systems store up energy and expend it in work, but the work is always a collaboration or cooperation of all the competing forces involved. It is synergy." (Lester F Ward, "Pure Sociology", 1903)

"Social structures are the products of social synergy, i.e., of the interaction of different social forces, all of which, in and of themselves, are destructive, but whose combined effect, mutually checking, constraining, and equilibrating one another, is to produce structures. The entire drift is toward economy, conservatism, and the prevention of waste. Social structures are mechanisms for the production of results, and the results cannot be secured without them. They are reservoirs of power." (James Q Dealey & Lester F Ward, "A Text-book of Sociology", 1905)

"The true nature of the universal principle of synergy pervading all nature and creating all the different kinds of structure that we observe to exist, must now be made clearer. Primarily and essentially it is a process of equilibration, i.e., the several forces are first brought into a state of partial equilibrium. It begins in collision, conflict, antagonism, and opposition, and then we have the milder phases of antithesis, competition, and interaction, passing next into a modus vivendi, or compromise, and ending in collaboration and cooperation. […] The entire drift is toward economy, conservatism, and the prevention of waste." (James Q Dealey & Lester F Ward, "A Text-book of Sociology", 1905)

"Synergy is the only word in our language that means behavior of whole systems unpredicted by the separately observed behaviors of any of the system's separate parts or any subassembly of the system's parts." (R Buckminster Fuller, "Operating Manual for Spaceship Earth", 1963)

"Synergy means behavior of whole systems unpredicted by the behavior of their parts taken separately." (R Buckminster Fuller, "Synergetics: Explorations in the Geometry of Thinking", 1975)

"[...] synergy is the consequence of the energy expended in creating order. It is locked up in the viable system created, be it an organism or a social system. It is at the level of the system. It is not discernible at the level of the system. It is not discernible at the level of the system’s components. Whenever the system is dismembered to examine its components, this binding energy dissipates." (J-C Spender, "Organizational Knowledge, Collective Practice and Penrose Rents", 1999)

"There is a multilayering of global networks in the key strategic activities that structure and destructure the planet. When these multilayered networks overlap in some node, when there is a node that belongs to different networks, two major consequences follow. First, economies of synergy between these different networks take place in that node: between financial markets and media businesses; or between academic research and technology development and innovation; between politics and media." (Manuel Castells, "The Rise of the Network Society", 1996)

"With the growing interest in complex adaptive systems, artificial life, swarms and simulated societies, the concept of “collective intelligence” is coming more and more to the fore. The basic idea is that a group of individuals (e. g. people, insects, robots, or software agents) can be smart in a way that none of its members is. Complex, apparently intelligent behavior may emerge from the synergy created by simple interactions between individuals that follow simple rules." (Francis Heylighen, "Collective Intelligence and its Implementation on the Web", 1999)

"Systems thinking means the ability to see the synergy of the whole rather than just the separate elements of a system and to learn to reinforce or change whole system patterns. Many people have been trained to solve problems by breaking a complex system, such as an organization, into discrete parts and working to make each part perform as well as possible. However, the success of each piece does not add up to the success of the whole. to the success of the whole. In fact, sometimes changing one part to make it better actually makes the whole system function less effectively." (Richard L Daft, "The Leadership Experience", 2002)

"Self-organization can be seen as a spontaneous coordination of the interactions between the components of the system, so as to maximize their synergy. This requires the propagation and processing of information, as different components perceive different aspects of the situation, while their shared goal requires this information to be integrated. The resulting process is characterized by distributed cognition: different components participate in different ways to the overall gathering and processing of information, thus collectively solving the problems posed by any perceived deviation between the present situation and the desired situation." (Carlos Gershenson & Francis Heylighen, "How can we think the complex?", 2004)

[synergy:] "Measure describing how one agent or system increases the satisfaction of other agents or systems." (Carlos Gershenson, "Design and Control of Self-organizing Systems", 2007)

"To develop a Control, the designer should find aspect systems, subsystems, or constraints that will prevent the negative interferences between elements (friction) and promote positive interferences (synergy). In other words, the designer should search for ways of minimizing frictions that will result in maximization of the global satisfaction" (Carlos Gershenson, "Design and Control of Self-organizing Systems", 2007)

"Synergy is the combined action that occurs when people work together to create new alternatives and solutions. In addition, the greatest opportunity for synergy occurs when people have different viewpoints, because the differences present new opportunities. The essence of synergy is to value and respect differences and take advantage of them to build on strengths and compensate for weaknesses." (Richard L Daft, "The Leadership Experience" 4th Ed., 2008)

"Synergy occurs when organizational parts interact to produce a joint effect that is greater than the sum of the parts acting alone. As a result the organization may attain a special advantage with respect to cost, market power, technology, or employee." (Richard L Daft, "The Leadership Experience" 4th Ed., 2008)

"In short, synergy is the consequence of the energy expended in creating order. It is locked up in the viable system created, be it an organism or a social system. It is at the level of the system. It is not discernible at the level of the system. It is not discernible at the level of the system's components. Whenever the system is dismembered to examine its components, this binding energy dissipates." (J-C Spender, "Organizational Knowledge, Collective Practice and Penrose Rents", 2009)

"Synergy is defined as the surplus gained by working together. A task which couldn’t be fulfilled by one individual, can be completed by the work of different individuals together. To maximize synergy, first, the initial task is divided into different sub-tasks. Different agents perform different tasks, which is called division of labor. An end product of one work is used for another work, which is called workflow. Finally, everything needs to be put together. We call this aggregation. This isn’t as linear as it looks. At every step in the process it can happen that a task is divided into sub tasks or aggregated with other tasks." (Evo Busseniers, "Self-organization versus hierarchical organization", [thesis] 2018)

❄️Systems Thinking: On Paradigm (Quotes)

"All crises begin with the blurring of a paradigm and the consequent loosening of the rules for normal research […] Or finally, the case that will most concern us here, a crisis may end with the emergence of a new candidate for paradigm and with the ensuing battle over its acceptance." (Thomas S Kuhn, "The Structure of Scientific Revolutions", 1962)

"Discovery commences with the awareness of anomaly, i.e., with the recognition that nature has somehow violated the paradigm-induced expectations that govern normal science. It then continues with a more or less extended exploration of the area of anomaly. And it closes only when the paradigm theory has been adjusted so that the anomalous has become the expected. […] Until he has learned to see nature in a different way - the new fact is not quite a scientific fact at all." (Thomas S Kuhn, "The Structure of Scientific Revolutions", 1962)

"Probably, the single most prevalent claim advanced by the proponents of a new paradigm is that they can solve the problems that led the old one to a crisis." (Thomas S Kuhn, "The Structure of Scientific Revolutions", 1962)

"The transition from a paradigm to a new one from which a new tradition of normal science can emerge is far from a cumulative process, one achieved by an articulation or extension of the old paradigm. Rather it is a reconstruction of the field from new fundamentals, a reconstruction that changes some of the field’s most elementary theoretical generalizations as well as many of its paradigm methods and applications." (Thomas S Kuhn, "The Structure of Scientific Revolutions", 1962)

"[…] paradigms, the core of the culture of science, are transmitted and sustained just as is culture generally: scientists accept them and become committed to them as a result of training and socialization, and the commitment is maintained by a developed system of social control." (Barry Barnes, "Thomas Kuhn", 1985)

"For our purposes, a simple way to understand paradigms is to see them as maps. We all know that ‘the map is not the territory’. A map is simply an explanation of certain aspects of the territory. That’s exactly what a paradigm is. It is a theory, an explanation, or model of something else." (Stephen R Covey, "The 7 Habits of Highly Effective People", 1989)

"The word paradigm comes from the Greek. It was originally a scientific term, and is more commonly used today to mean a model, theory, perception, assumption, or frame of reference. In the more general sense, it's the way we 'see' the world - not in terms of our visual sense of sight, but in terms of perceiving, understanding, and interpreting." (Stephen Covey, "The 7 Habits of Highly Effective People", 1989)

“Each of us carries within us a worldview, a set of assumptions about how the world works - what some call a paradigm - that forms the very questions we allow ourselves to ask, and determines our view of future possibilities.” (Frances M Lappé, “Rediscovering America's Values”, 1991)

"Paradigms are powerful because they create the lens through which we see the world." (Stephen Covey, "Daily Reflections for Highly Effective People", 1994)

"The realm of the particularity of each experienced item differs from the formal realm of concepts. [...] The power of paradigmatic thought is to bring order to experience by seeing individual things as belonging to a category." (Donald E Polkinghorne, “Narrative configuration in qualitative analysis", International Journal of Qualitative Studies in Education Vol. 8 (1), 1995)

"The shift of paradigms requires an expansion not only of our perceptions and ways of thinking, but also of our values. […] scientific facts emerge out of an entire constellation of human perceptions, values, and actions-in one word, out of a paradigm-from which they cannot be separated. […] Today the paradigm shift in science, at its deepest level, implies a shift from physics to the life sciences." (Fritjof Capra, "The Web of Life", 1996)

"All scientific theories, even those in the physical sciences, are developed in a particular cultural context. Although the context may help to explain the persistence of a theory in the face of apparently falsifying evidence, the fact that a theory arises from a particular context is not sufficient to condemn it. Theories and paradigms must be accepted, modified or rejected on the basis of evidence." (Richard P Bentall,  "Madness Explained: Psychosis and Human Nature", 2003)

"Paradigms are the most general-rather like a philosophical or ideological framework. Theories are more specific, based on the paradigm and designed to describe what happens in one of the many realms of events encompassed by the paradigm. Models are even more specific providing the mechanisms by which events occur in a particular part of the theory's realm. Of all three, models are most affected by empirical data - models come and go, theories only give way when evidence is overwhelmingly against them and paradigms stay put until a radically better idea comes along." (Lee R Beach, "The Psychology of Decision Making: People in Organizations", 2005)

"A paradigm is a shared mindset that represents a fundamental way of thinking about, perceiving, and understanding the world." (Richard L Daft, "The Leadership Experience" 4th Ed., 2008)

“The crucial concept that brings all of this together is one that is perhaps as rich and suggestive as that of a paradigm: the concept of a model." (Otávio Bueno, [in" Springer Handbook of Model-Based Science", Ed. by Lorenzo Magnani & Tommaso Bertolotti, 2017])

🏷️Knowledge Representation: On Problem Solving (Quotes)

"Reflexion is careful and laborious thought, and watchful attention directed to the agreeable effect of one's plan. Invention, on the other hand, is the solving of intricate problems and the discovery of new principles by means of brilliancy and versatility." (Marcus Vitruvius Pollio, "De architectura" ["On Architecture], cca. 15BC)

"This diagrammatic method has, however, serious inconveniences as a method for solving logical problems. It does not show how the data are exhibited by cancelling certain constituents, nor does it show how to combine the remaining constituents so as to obtain the consequences sought. In short, it serves only to exhibit one single step in the argument, namely the equation of the problem; it dispenses neither with the previous steps, i.e., 'throwing of the problem into an equation' and the transformation of the premises, nor with the subsequent steps, i.e., the combinations that lead to the various consequences. Hence it is of very little use, inasmuch as the constituents can be represented by algebraic symbols quite as well as by plane regions, and are much easier to deal with in this form." (Louis Couturat, "The Algebra of Logic", 1914)

"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)

"[The] function of thinking is not just solving an actual problem but discovering, envisaging, going into deeper questions. Often, in great discovery the most important thing is that a certain question is found." (Max Wertheimer, "Productive Thinking", 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, 1945)

"I believe, that the decisive idea which brings the solution of a problem is rather often connected with a well-turned word or sentence. The word or the sentence enlightens the situation, gives things, as you say, a physiognomy. It can precede by little the decisive idea or follow on it immediately; perhaps, it arises at the same time as the decisive idea. […]  The right word, the subtly appropriate word, helps us to recall the mathematical idea, perhaps less completely and less objectively than a diagram or a mathematical notation, but in an analogous way. […] It may contribute to fix it in the mind." (George Pólya [in a letter to Jaque Hadamard, "The Psychology of Invention in the Mathematical Field", 1949])

"Solving problems is the specific achievement of intelligence." (George Pólya, 1957)

"Systems engineering embraces every scientific and technical concept known, including economics, management, operations, maintenance, etc. It is the job of integrating an entire problem or problem to arrive at one overall answer, and the breaking down of this answer into defined units which are selected to function compatibly to achieve the specified objectives." (Instrumentation Technology, 1957)

"Solving problems can be regarded as the most characteristically human activity." (George Pólya, "Mathematical Discovery", 1962)

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

"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)

"Every problem-solving effort must begin with creating a representation for the problem - a problem space in which the search for the solution can take place. Of course, for most of the problems we encounter in our daily personal or professional lives, we simply retrieve from memory a representation that we have already stored and used on previous occasions. Sometimes, we have to adapt the representation a bit to the new situation, but that is usually a rather simple matter." (Herbert A Simon, "The Sciences of the Artificial", 1968)

"An expert problem solver must be endowed with two incompatible qualities, a restless imagination and a patient pertinacity.” (Howard W Eves, “In Mathematical Circles”, 1969)

"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)

"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)

“Solving problems can be regarded as the most characteristically human activity.” (George Polya, 1981)

"The problem solver needs to stand back and examine problem contexts in the light of different 'Ws' (Weltanschauungen). Perhaps he can then decide which 'W' seems to capture the essence of the particular problem context he is faced with. This whole process needs formalizing if it is to be carried out successfully. The problem solver needs to be aware of different paradigms in the social sciences, and he must be prepared to view the problem context through each of these paradigms." (Michael C Jackson, "Towards a System of Systems Methodologies", 1984)

“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)

"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)

"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)

“A mental model is a knowledge structure that incorporates both declarative knowledge (e.g., device models) and procedural knowledge (e.g., procedures for determining distributions of voltages within a circuit), and a control structure that determines how the procedural and declarative knowledge are used in solving problems (e.g., mentally simulating the behavior of a circuit).” (Barbara Y White & John R Frederiksen, “Causal Model Progressions as a Foundation for Intelligent Learning Environments”, Artificial Intelligence 42, 1990)

"An important symptom of an emerging understanding is the capacity to represent a problem in a number of different ways and to approach its solution from varied vantage points; a single, rigid representation is unlikely to suffice." (Howard Gardner, "The Unschooled Mind", 1991)

“[By understanding] I mean simply a sufficient grasp of concepts, principles, or skills so that one can bring them to bear on new problems and situations, deciding in which ways one’s present competencies can suffice and in which ways one may require new skills or knowledge.” (Howard Gardner, “The Unschooled Mind”, 1991)

"We consider the notion of ‘system’ as an organising concept, before going on to look in detail at various systemic metaphors that may be used as a basis for structuring thinking about organisations and problem situations." (Michael C Jackson, "Creative Problem Solving: Total Systems Intervention", 1991)

“But our ways of learning about the world are strongly influenced by the social preconceptions and biased modes of thinking that each scientist must apply to any problem. The stereotype of a fully rational and objective ‘scientific method’, with individual scientists as logical (and interchangeable) robots, is self-serving mythology.” (Stephen Jay Gould, “This View of Life: In the Mind of the Beholder”, “Natural History”, Vol. 103, No. 2, 1994)

“The term mental model refers to knowledge structures utilized in the solving of problems. Mental models are causal and thus may be functionally defined in the sense that they allow a problem solver to engage in description, explanation, and prediction. Mental models may also be defined in a structural sense as consisting of objects, states that those objects exist in, and processes that are responsible for those objects’ changing states.” (Robert Hafner & Jim Stewart, “Revising Explanatory Models to Accommodate Anomalous Genetic Phenomena: Problem Solving in the ‘Context of Discovery’”, Science Education 79 (2), 1995)

"The purpose of a conceptual model is to provide a vocabulary of terms and concepts that can be used to describe problems and/or solutions of design. It is not the purpose of a model to address specific problems, and even less to propose solutions for them. Drawing an analogy with linguistics, a conceptual model is analogous to a language, while design patterns are analogous to rhetorical figures, which are predefined templates of language usages, suited particularly to specific problems." (Peter P Chen [Ed.], "Advances in Conceptual Modeling", 1999)

"The three basic mechanisms of averaging, feedback and division of labor give us a first idea of a how a CMM [Collective Mental Map] can be developed in the most efficient way, that is, how a given number of individuals can achieve a maximum of collective problem-solving competence. A collective mental map is developed basically by superposing a number of individual mental maps. There must be sufficient diversity among these individual maps to cover an as large as possible domain, yet sufficient redundancy so that the overlap between maps is large enough to make the resulting graph fully connected, and so that each preference in the map is the superposition of a number of individual preferences that is large enough to cancel out individual fluctuations. The best way to quickly expand and improve the map and fill in gaps is to use a positive feedback that encourages individuals to use high preference paths discovered by others, yet is not so strong that it discourages the exploration of new paths." (Francis Heylighen, "Collective Intelligence and its Implementation on the Web", 1999)

"What it means for a mental model to be a structural analog is that it embodies a representation of the spatial and temporal relations among, and the causal structures connecting the events and entities depicted and whatever other information that is relevant to the problem-solving talks. […] The essential points are that a mental model can be nonlinguistic in form and the mental mechanisms are such that they can satisfy the model-building and simulative constraints necessary for the activity of mental modeling." (Nancy J Nersessian, "Model-based reasoning in conceptual change", 1999)

"A model is an imitation of reality and a mathematical model is a particular form of representation. We should never forget this and get so distracted by the model that we forget the real application which is driving the modelling. In the process of model building we are translating our real world problem into an equivalent mathematical problem which we solve and then attempt to interpret. We do this to gain insight into the original real world situation or to use the model for control, optimization or possibly safety studies." (Ian T Cameron & Katalin Hangos, "Process Modelling and Model Analysis", 2001)

"Good numeric representation is a key to effective thinking that is not limited to understanding risks. Natural languages show the traces of various attempts at finding a proper representation of numbers. [...] The key role of representation in thinking is often downplayed because of an ideal of rationality that dictates that whenever two statements are mathematically or logically the same, representing them in different forms should not matter. Evidence that it does matter is regarded as a sign of human irrationality. This view ignores the fact that finding a good representation is an indispensable part of problem solving and that playing with different representations is a tool of creative thinking." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

"The key role of representation in thinking is often downplayed because of an ideal of rationality that dictates that whenever two statements are mathematically or logically the same, representing them in different forms should not matter. Evidence that it does matter is regarded as a sign of human irrationality. This view ignores the fact that finding a good representation is an indispensable part of problem solving and that playing with different representations is a tool of creative thinking." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

"Mathematical modeling is as much ‘art’ as ‘science’: it requires the practitioner to (i) identify a so-called ‘real world’ problem (whatever the context may be); (ii) formulate it in mathematical terms (the ‘word problem’ so beloved of undergraduates); (iii) solve the problem thus formulated (if possible; perhaps approximate solutions will suffice, especially if the complete problem is intractable); and (iv) interpret the solution in the context of the original problem." (John A Adam, "Mathematics in Nature", 2003)

"What is a mathematical model? One basic answer is that it is the formulation in mathematical terms of the assumptions and their consequences believed to underlie a particular ‘real world’ problem. The aim of mathematical modeling is the practical application of mathematics to help unravel the underlying mechanisms involved in, for example, economic, physical, biological, or other systems and processes." (John A Adam, "Mathematics in Nature", 2003)

"Alternative models are neither right nor wrong, just more or less useful in allowing us to operate in the world and discover more and better options for solving problems." (Andrew Weil," The Natural Mind: A Revolutionary Approach to the Drug Problem", 2004)

“A conceptual model is a mental image of a system, its components, its interactions. It lays the foundation for more elaborate models, such as physical or numerical models. A conceptual model provides a framework in which to think about the workings of a system or about problem solving in general. An ensuing operational model can be no better than its underlying conceptualization.” (Henry N Pollack, “Uncertain Science … Uncertain World”, 2005)

"In specific cases, we think by applying mental rules, which are similar to rules in computer programs. In most of the cases, however, we reason by constructing, inspecting, and manipulating mental models. These models and the processes that manipulate them are the basis of our competence to reason. In general, it is believed that humans have the competence to perform such inferences error-free. Errors do occur, however, because reasoning performance is limited by capacities of the cognitive system, misunderstanding of the premises, ambiguity of problems, and motivational factors. Moreover, background knowledge can significantly influence our reasoning performance. This influence can either be facilitation or an impedance of the reasoning process." (Carsten Held et al, "Mental Models and the Mind", 2006)

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

"Mental models are formed over time through a deep enculturation process, so it follows that any attempt to align mental models must focus heavily on collective sense making. Alignment only happens through a process of socialisation; people working together, solving problems together, making sense of the world together." (Robina Chatham & Brian Sutton, "Changing the IT Leader’s Mindset", 2010)

"Mathematical modeling is the application of mathematics to describe real-world problems and investigating important questions that arise from it." (Sandip Banerjee, "Mathematical Modeling: Models, Analysis and Applications", 2014)

"Mental imagery is often useful in problem solving. Verbal descriptions of problems can become confusing, and a mental image can clear away excessive detail to bring out important aspects of the problem. Imagery is most useful with problems that hinge on some spatial relationship. However, if the problem requires an unusual solution, mental imagery alone can be misleading, since it is difficult to change one’s understanding of a mental image. In many cases, it helps to draw a concrete picture since a picture can be turned around, played with, and reinterpreted, yielding new solutions in a way that a mental image cannot." (James Schindler, "Followership", 2014)

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

🧬Cybernetics: On Man vs. Machine (Quotes)

"Moreover, it must be confessed that perception and that which depends upon it are inexplicable on mechanical grounds, that is to say, by means of figures and motions. And supposing there were a machine, so constructed as to think, feel, and have perception, it might be conceived as increased in size, while keeping the same proportions, so that one might go into it as into a mill. That being so, we should, on examining its interior, find only parts which work one upon another, and never anything by which to explain a perception. Thus it is in a simple substance, and not in a compound or in a machine, that perception must be sought for." (Gottfried W Leibniz,  "Monadology", 1714)

"In imagining that there is a machine whose construction would enable it to think, to sense, and to have perception, one could conceive it enlarged while retaining the same proportions, so that one could enter into it, just like into a windmill. Supposing this, one should, when visiting within it, find only parts pushing one another, and never anything by which to explain perception. (Gottfried W Leibniz, "Monadology", 1714) [alternative translation]

"Man is so complicated a machine that it is impossible to get a clear idea of the machine beforehand, and hence impossible to define it. For this reason, all the investigations have been vain, which the greatest philosophers have made à priori, that is to say, in so far as they use, as it were, the wings of the spirit. Thus it is only à posteriori or by trying to disentangle the soul from the organs of the body, so to speak, that one can reach the highest probability concerning man's own nature, even though one can not discover with certainty what his nature is." (Julien Offray de La Mettrie, "Man a Machine", 1747) 

"The machines that are first invented to perform any particular movement are always the most complex, and succeeding artists generally discover that, with fewer wheels the same effects may be more easily produced." (Adam Smith, "An Inquiry into the Nature and Causes of the Wealth of Nations", 1776)

"As nature has uncovered from under this hard shell the seed for which she most tenderly cares - the propensity and vocation to free thinking - this gradually works back upon the character of the people, who thereby gradually become capable of managing freedom; finally, it affects the principles of government, which finds it to its advantage to treat men, who are now more than machines, in accordance with their dignity." (Immanuel Kant, "An Answer to the Question: What Is Enlightenment?", 1784)

"Under the Providence of God, our means of education are the grand machinery by which the 'raw material' of human nature can be worked up into inventors and discoverers, into skilled artisans and scientific farmers, into scholars and jurists, into the founders of benevolent institutions, and the great expounders of ethical and theological science." (Horace Mann, "Annual Reports of the Secretary of the Board of Education of Massachusetts for the years 1845-1848", 1848)

"Human nature is not a machine to be built after a model, and set to do exactly the work prescribed for it, but a tree, which requires to grow and develop itself on all sides, according to the tendency of the inward forces which make it a living thing." (John S Mill, "On Liberty Source: On Liberty", 1859)

"The machine unmakes the man. Now that the machine is so perfect, the engineer is nobody." (Ralph W Emerson, "Society and Solitude", 1870)

"It is because the body is a machine that education is possible. Education is the formation of habits, a superinducing of an artificial organisation upon the natural organisation of the body: so that acts, which at first required a conscious effort, eventually became unconscious and mechanical." (Thomas H Huxley, "Descartes’ Discourse on Method", 1904)

"A machine is a machine because it cannot think." (Gilbert K Chesterton, 1916)

"We are automata entirely controlled by the forces of the medium being tossed about like corks on the surface of the water, but mistaking the resultant of the impulses from the outside for free will. The movements and other actions we perform are always life preservative and tho seemingly quite independent from one another, we are connected by invisible links." (Nikola Tesla, "My Inventions", 1919)

"Thanks to the psycho-physical reversibility, we can materialize the act of creation. Undoubtedly, the inventive machine has not yet been created, but we can see its creation soon." (Stefan Odobleja, "Consonant Psychology", 1938)

"[…] there is something wonderful in the idea that man’s brain is the greatest machine of all, imitating within its tiny network events happening in the most distant stars, […] On our model theory neural or other mechanisms can imitate or parallel the behaviour and interaction of physical objects and so supply us with information on physical processes which are not directly observable to us." (Kenneth Craik, "The Nature of Explanation", 1943)

"Every time one combines and records facts in accordance with established logical processes, the creative aspect of thinking is concerned only with the selection of the data and the process to be employed, and the manipulation thereafter is repetitive in nature and hence a fit matter to be relegated to the machines." (Vannevar Bush, "As We May Think", 1945) 

"In other words then, if a machine is expected to be infallible, it cannot also be intelligent. There are several theorems which say almost exactly that. But these theorems say nothing about how much intelligence may be displayed if a machine makes no pretense at infallibility." (Alan M Turing, 1946)

"Let it be remarked [...] that an important difference between the way in which we use the brain and the machine is that the machine is intended for many successive runs, either with no reference to each other, or with a minimal, limited reference, and that it can be cleared between such runs; while the brain, in the course of nature, never even approximately clears out its past records. Thus the brain, under normal circumstances, is not the complete analogue of the computing machine but rather the analogue of a single run on such a machine." (Norbert Wiener, "Cybernetics: Or Control and Communication in the Animal and the Machine", 1948)

"The machine is only a tool after all, which can help humanity progress faster by taking some of the burdens of calculations and interpretations off its back. The task of the human brain remains what it has always been; that of discovering new data to be analyzed, and of devising new concepts to be tested." (Isaac Asimov, "I, Robot", 1950)

"The act of discovery escapes logical analysis; there are no logical rules in terms of which a 'discovery machine' could be constructed that would take over the creative function of the genius. But it is not the logician’s task to account for scientific discoveries; all he can do is to analyze the relation between given facts and a theory presented to him with the claim that it explains these facts. In other words, logic is concerned with the context of justification." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

"The brain has been compared to a digital computer because the neuron, like a switch or valve, either does or does not complete a circuit. But at that point the similarity ends. The switch in the digital computer is constant in its effect, and its effect is large in proportion to the total output of the machine. The effect produced by the neuron varies with its recovery from [the] refractory phase and with its metabolic state. The number of neurons involved in any action runs into millions so that the influence of any one is negligible. [...] Any cell in the system can be dispensed with. [...] The brain is an analogical machine, not digital. Analysis of the integrative activities will probably have to be in statistical terms. (Karl S Lashley, "The problem of serial order in behavior", 1951)

"Machines might give us more time to think but will never do our thinking for us." (Thomas Watson Jr., 1957)

"Although it sounds implausible, it might turn out that above a certain level of complexity, a machine ceased to be predictable, even in principle, and started doing things on its own account, or, to use a very revealing phrase, it might begin to have a mind of its own." (John R Lucas, "Minds, Machines and Gödel", 1959)

"The study of thinking machines teaches us more about the brain than we can learn by introspective methods. Western man is externalizing himself in the form of gadgets." (William S Burroughs, "Naked Lunch Benway Naked Lunch", 1959)

"Let an ultraintelligent machine be defined as a machine that can far surpass all the intellectual activities of any man however clever. Since the design of machines is one of these intellectual activities, an ultraintelligent machine could design even better machines; there would then unquestionably be an 'intelligence explosion:, and the intelligence of man would be left far behind. Thus the first ultraintelligent machine is the last invention that man need ever make." (Irving J Good, "Speculations Concerning the First Ultraintelligent Machine", Advances in Computers Vol. 6, 1965) 

"These machines have no common sense; they have not yet learned to "think," and they do exactly as they are told, no more and no less. This fact is the hardest concept to grasp when one first tries to use a computer." (Donald Knuth, "The Art of Computer Programming, Volume 1: Fundamental Algorithms", 1968)

"Questions are the engines of intellect, the cerebral machines which convert energy to motion, and curiosity to controlled inquiry." (David H Fischer, "Historians’ Fallacies", 1970)

"A smart machine will first consider which is more worth its while: to perform the given task or, instead, to figure some way out of it." (Stanisław Lem, "The Futurological Congress The Futurological Congress", 1971) 

"If cognitive processes can be realized in a general machine then it is possible to execute mental operations in artifacts that are not necessarily subject to the embarrassing spatio-temporal limitations and structural frailties of a biological processor." (Gordon Pask, "Conversation, Cognition and Learnin", 1975)

"A higher-level formal language is an abstract machine." (Joseph Weizenbaum, "Computer power and human reason: From judgment to calculation", 1976)

"Man is not a machine, [...] although man most certainly processes information, he does not necessarily process it in the way computers do. Computers and men are not species of the same genus. [...] No other organism, and certainly no computer, can be made to confront genuine human problems in human terms. [...] However much intelligence computers may attain, now or in the future, theirs must always be an intelligence alien to genuine human problems and concerns." (Joesph Weizenbaum, Computer Power and Human Reason: From Judgment to Calculation, 1976)

"There are now machines in the world that think, that learn and create. Moreover, their ability to do these things is going to increase rapidly until - in the visible future - the range of problems they can handle will be coextensive with the range to which the human mind has been applied." (Allen Newell & Herbert A Simon, "Human problem solving", 1976)

"The idea of making machines that think has an unfailing fascination, not only for science fiction readers, but for all who can see it is a possible way of gaining some understanding of the working of our own minds. Thinking, however, is not an easily defined phenomenon, although it is often considered to be the process of solving problems." (Edward Ihnatowicz, "The Relevance of Manipulation to the Process of Perception", 1977) 

"We can divide those who uphold the doctrine that men are machines, or a similar doctrine, into two categories: those who deny the existence of mental events, or personal experiences, or of consciousness; [...] and those who admit the existence of mental events, but assert that they are 'epiphenomena' - that everything can be explained without them, since the material world is causally closed." (Karl Popper & John Eccles, "The self and its brain", 1977)

"It is essential to realize that a computer is not a mere 'number cruncher', or supercalculating arithmetic machine, although this is how computers are commonly regarded by people having no familiarity with artificial intelligence. Computers do not crunch numbers; they manipulate symbols. [...] Digital computers originally developed with mathematical problems in mind, are in fact general purpose symbol manipulating machines." (Margaret A Boden, "Minds and mechanisms", 1981)

"Computers and robots replace humans in the exercise of mental functions in the same way as mechanical power replaced them in the performance of physical tasks. As time goes on, more and more complex mental functions will be performed by machines. Any worker who now performs his task by following specific instructions can, in principle, be replaced by a machine. This means that the role of humans as the most important factor of production is bound to diminish - in the same way that the role of horses in agricultural production was first diminished and then eliminated by the introduction of tractors."  (Wassily Leontief, National perspective: The definition of problem and opportunity, 1983)

"It's difficult to be rigorous about whether a machine really 'knows', 'thinks', etc., because we're hard put to define these things. We understand human mental processes only slightly better than a fish understands swimming." (John McCarthy, "The Little Thoughts of Thinking Machines", Psychology Today, 1983)

"The digital-computer field defined computers as machines that manipulated numbers. The great thing was, adherents said, that everything could be encoded into numbers, even instructions. In contrast, scientists in AI [artificial intelligence] saw computers as machines that manipulated symbols. The great thing was, they said, that everything could be encoded into symbols, even numbers." (Allen Newell, "Intellectual Issues in the History of Artificial Intelligence", 1983) 

"[...] two programs can be thought of as strongly equivalent or as different realizations of the same algorithm or the same cognitive process if they can be represented by the same program in some theoretically specified virtual machine. A simple way of stating this is to say that we individuate cognitive processes in terms of their expression in the canonical language of this virtual machine. The formal structure of the virtual machine - or what I call its functional architecture - thus represents the theoretical definition of, for example, the right level of specificity (or level of aggregation) at which to view mental processes, the sort of functional resources the brain makes available - what operations are primitive, how memory is organized and accessed, what sequences are allowed, what limitations exist on the passing of arguments and on the capacities of various buffers, and so on." (Zenon W Pylyshyn, "Computation and cognition: Towards a foundation for cognitive science", 1984)

"Under pressure from the computer, the question of mind in relation to machine is becoming a central cultural preoccupation." (Sherry Turkle, "The Second Self: Computers and the Human Spirit", 1984)

"The hardest problems we have to face do not come from philosophical questions about whether brains are machines or not. There is not the slightest reason to doubt that brains are anything other than machines with enormous numbers of parts that work in perfect accord with physical laws. As far as anyone can tell, our minds are merely complex processes. The serious problems come from our having had so little experience with machines of such complexity that we are not yet prepared to think effectively about them." (Marvin Minsky, 1986)

"What makes people smarter than machines? They certainly are not quicker or more precise. Yet people are far better at perceiving objects in natural scenes and noting their relations, at understanding language and retrieving contextually appropriate information from memory, at making plans and carrying out contextually appropriate actions, and at a wide range of other natural cognitive tasks. People are also far better at learning to do these things more accurately and fluently through processing experience." (James L McClelland et al, "The appeal of parallel distributed processing", 1986)

"A popular myth says that the invention of the computer diminishes our sense of ourselves, because it shows that rational thought is not special to human beings, but can be carried on by a mere machine. It is a short stop from there to the conclusion that intelligence is mechanical, which many people find to be an affront to all that is most precious and singular about their humanness." (Jeremy Campbell, "The improbable machine", 1989)

"Fuzziness, then, is a concomitant of complexity. This implies that as the complexity of a task, or of a system for performing that task, exceeds a certain threshold, the system must necessarily become fuzzy in nature. Thus, with the rapid increase in the complexity of the information processing tasks which the computers are called upon to perform, we are reaching a point where computers will have to be designed for processing of information in fuzzy form. In fact, it is the capability to manipulate fuzzy concepts that distinguishes human intelligence from the machine intelligence of current generation computers. Without such capability we cannot build machines that can summarize written text, translate well from one natural language to another, or perform many other tasks that humans can do with ease because of their ability to manipulate fuzzy concepts." (Lotfi A Zadeh, "The Birth and Evolution of Fuzzy Logic", 1989)

"Looking at ourselves from the computer viewpoint, we cannot avoid seeing that natural language is our most important 'programming language'. This means that a vast portion of our knowledge and activity is, for us, best communicated and understood in our natural language. [...] One could say that natural language was our first great original artifact and, since, as we increasingly realize, languages are machines, so natural language, with our brains to run it, was our primal invention of the universal computer. One could say this except for the sneaking suspicion that language isn’t something we invented but something we became, not something we constructed but something in which we created, and recreated, ourselves. (Justin Leiber, "Invitation to cognitive science", 1991)

"On the other hand, those who design and build computers know exactly how the machines are working down in the hidden depths of their semiconductors. Computers can be taken apart, scrutinized, and put back together. Their activities can be tracked, analyzed, measured, and thus clearly understood - which is far from possible with the brain. This gives rise to the tempting assumption on the part of the builders and designers that computers can tell us something about brains, indeed, that the computer can serve as a model of the mind, which then comes to be seen as some manner of information processing machine, and possibly not as good at the job as the machine. (Theodore Roszak, "The Cult of Information", 1994)

"Once a computer achieves human intelligence it will necessarily roar past it." (Ray Kurzweil, "The Age of Spiritual Machines: When Computers Exceed Human Intelligence", 1999) 

"As machines slip from human control they will do more than become conscious. They will become spiritual beings, whose inner life is no more limited by conscious thought than ours. Not only will they think and have emotions. They will develop the errors and illusions that go with self-awareness." (John Gray, “Straw Dogs: Thoughts on Humans and Other Animals”, 2002)

"Machine consciousness refers to attempts by those who design and analyse informational machines to apply their methods to various ways of understanding consciousness and to examine the possible role of consciousness in informational machines." (Igor Aleksander, "Machine Consciousness", 2008)

"Now think about the prospect of competition from computers instead of competition from human workers. On the supply side, computers are far more different from people than any two people are different from each other: men and machines are good at fundamentally different things. People have intentionality - we form plans and make decisions in complicated situations. We’re less good at making sense of enormous amounts of data. Computers are exactly the opposite: they excel at efficient data processing, but they struggle to make basic judgments that would be simple for any human." (Peter Thiel & Blake Masters, "Zero to One: Notes on Startups, or How to Build the Future", 2014)

"Either mathematics is too big for the human mind, or the human mind is more than a machine." (Kurt Gödel)

"Every living being is an engine geared to the wheelwork of the universe. Though seemingly affected only by its immediate surrounding, the sphere of external influence extends to infinite distance." (Nikola Tesla)

"He’s fed in enough data for a dozen forecasts — let the electronic brains do the rest. While the THINK machines grind out prophecies, he can relax and contemplate the cosmos." (Lydia Strong)

"Human nature is not a machine to be built after a model, and set to do exactly the work prescribed for it, but a tree, which requires to grow and develop itself on all sides, according to the tendency of the inward forces which make it a living thing." (John Stuart Mill)

"Science is opposed to theological dogmas because science is founded on fact. To me, the universe is simply a great machine which never came into being and never will end. The human being is no exception to the natural order. Man, like the universe, is a machine." (Nikola Tesla)

"The whole thinking process is rather mysterious to us, but I believe that the attempt to make a thinking machine will help us greatly in finding out how we think ourselves." (Alan M Turing)

🏷️Knowledge Representation: On Mathematical Models (Quotes)

"Experience teaches that one will be led to new discoveries almost exclusively by means of special mechanical models." (Ludwig Boltzmann, "Lectures on Gas Theory", 1896)

"The atomic theory plays a part in physics similar to that of certain auxiliary concepts in mathematics: it is a mathematical model for facilitating the mental reproduction of facts." (Ernst Mach, "The Science of Mechanics" 5th Ed, 1942)

"If the system exhibits a structure which can be represented by a mathematical equivalent, called a mathematical model, and if the objective can be also so quantified, then some computational method may be evolved for choosing the best schedule of actions among alternatives. Such use of mathematical models is termed mathematical programming." (George Dantzig, "Linear Programming and Extensions", 1959)

"In fact, the construction of mathematical models for various fragments of the real world, which is the most essential business of the applied mathematician, is nothing but an exercise in axiomatics." (Marshall Stone, cca 1960)

"[...] sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work - that is, correctly to describe phenomena from a reasonably wide area. Furthermore, it must satisfy certain aesthetic criteria - that is, in relation to how much it describes, it must be rather simple." (John von Neumann, "Method in the physical sciences", 1961)

"Mathematical statistics provides an exceptionally clear example of the relationship between mathematics and the external world. The external world provides the experimentally measured distribution curve; mathematics provides the equation (the mathematical model) that corresponds to the empirical curve. The statistician may be guided by a thought experiment in finding the corresponding equation." (Marshall J Walker, "The Nature of Scientific Thought", 1963)

"The most valuable use of such [mathematical] models usually lies less in turning out the answer in an uncertain world than in shedding light on how much difference an alteration in the assumptions and/or variables used would make in the answer yielded by the models." (Edward G. Bennion, "New Decision-Making Tools for Managers", 1963)

"The use of [a] model or any of the mathematical techniques of the operations researcher does not imply management by computer." (James C Hetrick, "New Decision-Making Tools for Managers", 1963)

"A mathematical model is neither an hypothesis nor a theory. Unlike the scientific hypothesis, a model is not verifiable directly by experiment. For all models are both true and false. Almost any plausible proposed relation among aspects of nature is likely to be true in the sense that it occurs (although rarely and slightly). Yet all models leave out a lot and are in that sense false, incomplete, inadequate. The validation of a model is not that it is ' 'true" but that it generates good testable hypotheses relevant to important problems. A model may be discarded in favor of a more powerful one, but it usually is simply outgrown when the live issues are not any longer those for which it was designed." (Richard Levins, "The Strategy of Model Building in Population Biology", American Scientist 54(4), 1966)

"Mathematical models are more precise and less ambiguous than quantitative models and are therefore of greater value in obtaining specific answers to certain managerial questions." (Henry L Tosi & Stephen J Carrol, "Management", 1976)

"A mathematical model is any complete and consistent set of mathematical equations which are designed to correspond to some other entity, its prototype. The prototype may be a physical, biological, social, psychological or conceptual entity, perhaps even another mathematical model."  (Rutherford Aris, "Mathematical Modelling", 1978)

"Mathematical model making is an art. If the model is too small, a great deal of analysis and numerical solution can be done, but the results, in general, can be meaningless. If the model is too large, neither analysis nor numerical solution can be carried out, the interpretation of the results is in any case very difficult, and there is great difficulty in obtaining the numerical values of the parameters needed for numerical results." (Richard E Bellman, "Eye of the Hurricane: An Autobiography", 1984)

"Theoretical scientists, inching away from the safe and known, skirting the point of no return, confront nature with a free invention of the intellect. They strip the discovery down and wire it into place in the form of mathematical models or other abstractions that define the perceived relation exactly. The now-naked idea is scrutinized with as much coldness and outward lack of pity as the naturally warm human heart can muster. They try to put it to use, devising experiments or field observations to test its claims. By the rules of scientific procedure it is then either discarded or temporarily sustained. Either way, the central theory encompassing it grows. If the abstractions survive they generate new knowledge from which further exploratory trips of the mind can be planned. Through the repeated alternation between flights of the imagination and the accretion of hard data, a mutual agreement on the workings of the world is written, in the form of natural law." (Edward O Wilson, "Biophilia", 1984)

"All the [...] computer models and fancy forms will quickly lose their appeal (if they had any to begin with) unless top management pays a lot of attention to the results they generate." (Bernard C Reimann, "The Planning Forum", 1987)

"Symmetries abound in nature, in technology, and - especially - in the simplified mathematical models we study so assiduously. Symmetries complicate things and simplify them. They complicate them by introducing exceptional types of behavior, increasing the number of variables involved, and making vanish things that usually do not vanish. They simplify them by introducing exceptional types of behavior, increasing the number of variables involved, and making vanish things that usually do not vanish. They violate all the hypotheses of our favorite theorems, yet lead to natural generalizations of those theorems. It is now standard to study the 'generic' behavior of dynamical systems. Symmetry is not generic. The answer is to work within the world of symmetric systems and to examine a suitably restricted idea of genericity." (Ian Stewart, "Bifurcation with symmetry", 1988)

"The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing?" (Stephen Hawking, "A Brief History of Time", 1988)

"Mathematical modeling is about rules - the rules of reality. What distinguishes a mathematical model from, say, a poem, a song, a portrait or any other kind of ‘model’, is that the mathematical model is an image or picture of reality painted with logical symbols instead of with words, sounds or watercolors." (John L Casti, "Reality Rules, The Fundamentals", 1992)

"Pedantry and sectarianism aside, the aim of theoretical physics is to construct mathematical models such as to enable us, from the use of knowledge gathered in a few observations, to predict by logical processes the outcomes in many other circumstances. Any logically sound theory satisfying this condition is a good theory, whether or not it be derived from ‘ultimate’ or ‘fundamental’ truth." (Clifford Truesdell & Walter Noll, "The Non-Linear Field Theories of Mechanics" 2nd Ed., 1992)

"Nature behaves in ways that look mathematical, but nature is not the same as mathematics. Every mathematical model makes simplifying assumptions; its conclusions are only as valid as those assumptions. The assumption of perfect symmetry is excellent as a technique for deducing the conditions under which symmetry-breaking is going to occur, the general form of the result, and the range of possible behaviour. To deduce exactly which effect is selected from this range in a practical situation, we have to know which imperfections are present" (Ian Stewart & Martin Golubitsky, "Fearful Symmetry", 1992)

"Mathematical models are continually invoking ideas of infinitely smooth surfaces, weightless strings, weightless beams, perfectly spherical balls, projectiles flying through airless space, gases which are perfectly compressible and liquids which are perfectly incompressible, and so on. The purpose of such simplifications is, in theory, to understand the world better despite the oversimplification, which you hope either will not matter or will be corrected when you construct a second (better) model." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"A model is an imitation of reality and a mathematical model is a particular form of representation. We should never forget this and get so distracted by the model that we forget the real application which is driving the modelling. In the process of model building we are translating our real world problem into an equivalent mathematical problem which we solve and then attempt to interpret. We do this to gain insight into the original real world situation or to use the model for control, optimization or possibly safety studies." (Ian T Cameron & Katalin Hangos, "Process Modelling and Model Analysis", 2001)

"Formulation of a mathematical model is the first step in the process of analyzing the behaviour of any real system. However, to produce a useful model, one must first adopt a set of simplifying assumptions which have to be relevant in relation to the physical features of the system to be modelled and to the specific information one is interested in. Thus, the aim of modelling is to produce an idealized description of reality, which is both expressible in a tractable mathematical form and sufficiently close to reality as far as the physical mechanisms of interest are concerned." (Francois Axisa, "Discrete Systems" Vol. I, 2001)

"[…] interval mathematics and fuzzy logic together can provide a promising alternative to mathematical modeling for many physical systems that are too vague or too complicated to be described by simple and crisp mathematical formulas or equations. When interval mathematics and fuzzy logic are employed, the interval of confidence and the fuzzy membership functions are used as approximation measures, leading to the so-called fuzzy systems modeling." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)

"Modeling, in a general sense, refers to the establishment of a description of a system (a plant, a process, etc.) in mathematical terms, which characterizes the input-output behavior of the underlying system. To describe a physical system […] we have to use a mathematical formula or equation that can represent the system both qualitatively and quantitatively. Such a formulation is a mathematical representation, called a mathematical model, of the physical system." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)

"What is a mathematical model? One basic answer is that it is the formulation in mathematical terms of the assumptions and their consequences believed to underlie a particular ‘real world’ problem. The aim of mathematical modeling is the practical application of mathematics to help unravel the underlying mechanisms involved in, for example, economic, physical, biological, or other systems and processes." (John A Adam, "Mathematics in Nature", 2003)

"Mathematical modeling is as much ‘art’ as ‘science’: it requires the practitioner to (i) identify a so-called ‘real world’ problem (whatever the context may be); (ii) formulate it in mathematical terms (the ‘word problem’ so beloved of undergraduates); (iii) solve the problem thus formulated (if possible; perhaps approximate solutions will suffice, especially if the complete problem is intractable); and (iv) interpret the solution in the context of the original problem." (John A Adam, "Mathematics in Nature", 2003)

"An important aspect of the global theory of dynamical systems is the stability of the orbit structure as a whole. The motivation for the corresponding theory comes from applied mathematics. Mathematical models always contain simplifying assumptions. Dominant features are modeled; supposed small disturbing forces are ignored. Thus, it is natural to ask if the qualitative structure of the set of solutions - the phase portrait - of a model would remain the same if small perturbations were included in the model. The corresponding mathematical theory is called structural stability." (Carmen Chicone

, "Stability Theory of Ordinary Differential Equations" [Mathematics of Complexity and Dynamical Systems, 2012])

"Models do not and need not match reality in all of its aspects and details to be adequate. A mathematical model is usually developed for a specific class of target systems, and its validity is determined relative to its intended applications. A model is considered valid within

its intended domain of applicability provided that its predictions in that domain fall within an acceptable range of error, specified prior to the model’s development or identification." (Zoltan Domotor, "Mathematical Models in Philosophy of Science" [Mathematics of Complexity and Dynamical Systems, 2012])

"Simplified description of a real world system in mathematical terms, e. g., by means of differential equations or other suitable mathematical structures." (Benedetto Piccoli, Andrea Tosin, "Vehicular Traffic: A Review of Continuum Mathematical Models" [Mathematics of Complexity and Dynamical Systems, 2012])

"Stated loosely, models are simplified, idealized and approximate representations of the structure, mechanism and behavior of real-world systems. From the standpoint of set-theoretic model theory, a mathematical model of a target system is specified by a nonempty set - called the model’s domain, endowed with some operations and relations, delineated by suitable axioms and intended empirical interpretation." (Zoltan Domotor, "Mathematical Models in Philosophy of Science" [Mathematics of Complexity and Dynamical Systems, 2012])

"The standard view among most theoretical physicists, engineers and economists is that mathematical models are syntactic (linguistic) items, identified with particular systems of equations or relational statements. From this perspective, the process of solving a designated system of (algebraic, difference, differential, stochastic, etc.) equations of the target system, and interpreting the particular solutions directly in the context of predictions and explanations are primary, while the mathematical structures of associated state and orbit spaces, and quantity algebras – although conceptually important, are secondary." (Zoltan Domotor, "Mathematical Models in Philosophy of Science" [Mathematics of Complexity and Dynamical Systems, 2012])

"Mathematical modeling is the application of mathematics to describe real-world problems and investigating important questions that arise from it." (Sandip Banerjee, "Mathematical Modeling: Models, Analysis and Applications", 2014)

"A mathematical model is a mathematical description (often by means of a function or an equation) of a real-world phenomenon such as the size of a population, the demand for a product, the speed of a falling object, the concentration of a product in a chemical reaction, the life expectancy of a person at birth, or the cost of emission reductions. The purpose of the model is to understand the phenomenon and perhaps to make predictions about future behavior. [...] A mathematical model is never a completely accurate representation of a physical situation - it is an idealization." (James Stewart, "Calculus: Early Transcedentals" 8th Ed., 2016)

"Different models serve different purposes. Setting up a model involves focusing on features of the phenomenon that are compatible with the methodology being proposed, and neglecting features that are not compatible with it. A mathematical model in applied science explicitly refrains from attempting to be a complete picture of the phenomenon being modeled." (Reuben Hersh, ”Mathematics as an Empirical Phenomenon, Subject to Modeling”, 2017)

"Mathematical modeling is the modern version of both applied mathematics and theoretical physics. In earlier times, one proposed not a model but a theory. By talking today of a model rather than a theory, one acknowledges that the way one studies the phenomenon is not unique; it could also be studied other ways. One's model need not claim to be unique or final. It merits consideration if it provides an insight that isn't better provided by some other model." (Reuben Hersh, "Mathematics as an Empirical Phenomenon, Subject to Modeling", 2017)