Mental Models LXVII: On Causal Maps

"Causal maps are representations of individuals (or groups) beliefs about causal relations. They include elements, with only two kinds of properties. The first property is 'relevance'. The second  is the possibility of being in one (of two) 'influence relationships' (positive or negative) with one (of three) strengths (weak. moderate, or strong)." (Kivia Markoczy & Jeff Goldberg, "A method for eliciting and comparing causal maps", 1995)

"Short-term memory can hold 7 ± 2 chunks of information at once. This puts a rather sharp limit on the effective size and complexity of a causal map. Presenting a complex causal map all at once makes it hard to see the loops, understand which are important, or understand how they generate the dynamics. Resist the temptation to put all the loops you and your clients have identified into a single comprehensive diagram." (John D Sterman, "Business Dynamics Systems Thinking and Modeling for a Complex World", 2000)

"The robustness of the misperceptions of feedback and the poor performance they cause are due to two basic and related deficiencies in our mental model. First, our cognitive maps of the causal structure of systems are vastly simplified compared to the complexity of the systems themselves. Second, we are unable to infer correctly the dynamics of all but the simplest causal maps. Both are direct consequences of bounded rationality, that is, the many limitations of attention, memory, recall, information processing capability, and time that constrain human decision making." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"A causal map is an abstract representation of the causal relationships among kinds of objects and events in the world. Such relationships are not, for the most part, directly observable, but they can often be accurately inferred from observations. This includes both observations of patterns of contingency and correlation among events as well as observations of the effects of experimental interventions. We can think of everyday theories and theory-formation processes as cognitive systems that allow us to recover an accurate causal map of the world." (Alison Gopnik & Clark Glymour, "Causal maps and Bayes nets: a cognitive and computational account of theory-formation" [in "The cognitive basis of science"], 2002)

"Causal mapping is a simple and useful technique for addressing situations where thinking - as an individual or as a group - matters. A causal map is a word-and-arrow diagram in which ideas and actions are causally linked with one another through the use of arrows. The arrows indicate how one idea or action leads to another. Causal mapping makes it possible to articulate a large number of ideas and their interconnections in such a way that people can know what to do in an area of concern, how to do it and why, because the arrows indicate the causes and consequences of an idea or action." (John M Bryson et al, "Visible Thinking: Unlocking Causal Mapping For Practical Business Results", 2004)

"Causal mapping is [...]  a technique for linking strategic thinking and acting, helping make sense of complex problems, and communicating to oneself and others what might be done about them. With practice, the use of causal mapping can assist you in moving from 'winging it' when thinking matters to a more concrete and rigorous approach that helps you and others achieve success in an easy and far more reliable way" (John M Bryson et al, "Visible Thinking: Unlocking Causal Mapping For Practical Business Results", 2004)

"Causal mapping makes it possible to articulate a large number of ideas and their interconnections in such a way that we can better understand an area of concern. Causal mapping also helps us know what to do about the issue, what it would take to do those things, and what we would like to get out of having done so. Causal mapping is therefore a particularly powerful technique for making sense of complex problems, linking strategic thinking and acting, and helping to communicate to others what might or should be done. " (John M Bryson et al, "Visible Thinking: Unlocking Causal Mapping For Practical Business Results", 2004)

"When an individual uses causal mapping to help clarify their own thinking, we call this technique cognitive mapping, because it is related to personal thinking or cognition. When a group maps their own ideas, we call it oval mapping, because we often use oval-shaped cards to record individuals’ ideas so that they can be arranged into a group’s map. Cognitive maps and oval maps can be used to create a strategic plan, because the maps include goals, strategies and actions, just like strategic plans." (John M Bryson et al, "Visible Thinking: Unlocking Causal Mapping For Practical Business Results", 2004)

"Causal maps include elements called nodes, which are allowed to have causal relationships of different strengths of positive or negative loading depicted with a number, usually in the range of from 1 (weak) to 3 (strong). The relationships of the nodes are depicted with arcs or links labeled with the assumed polarity and loading factor or strength of causality, Links with positive polarity refer to dependency (when A increases B increases proportionally to the loading factor) and negative to inverse dependency (when A increases, B decreases)." (Hannu Kivijärvi et al, "A Support System for the Strategic Scenario Process", Encyclopedia of Decision Making and Decision Support Technologies, 2008)

"Fifth principle: (a) in finding solutions for systemic problems do not be content with symptomatic solutions but look for systemic-structural levers that can produce the more incisive effect; (b) if there are several systemic levers, choose the most efficient, that which produces the maximum effects with the minimum effort; (c) to activate the chosen structural lever identify the most effective decisional lever (action variable) taking into account the time necessary to produce the desired effect; (d) the choice of structural and decisional levers, as well as the intensity of the actions to modify their values, must follow from a careful construction, interpretation and assessment of the system’s causal map." (Piero Mella, "Systems Thinking: Intelligence in Action", 2012)

"(1) The causal maps are only models of a world of variables and processes; (2) They are models suitable for depicting that world only if they represent a logical image; (3) A logical image is made up of a network of arrows that depict the cause and effect connections among the variables and processes in the world; this network cannot be in contradiction to the world; (4) This depiction of the world relates to the boundaries between the represented and the external systems; the causal maps always depict a portion of a vaster world;" (Piero Mella, "Systems Thinking: Intelligence in Action", 2012)

"In constructing causal maps, whatever technique is adopted, there is always the problem of identifying or defining the system’s boundaries, either if we zoom in or broaden our perspective by zooming out." (Piero Mella, "Systems Thinking: Intelligence in Action", 2012)

"A Causal Map is hierarchical in structure (linking means to ends) and built with a focus on achieving goals. The process of creating the maps is ideally a group process and this in itself will add lots of value to a collective understanding of goals around EDI, what is required to achieve these and some of the potential challenges around this." (Nicola Morrill, "Supporting Your Efforts on Diversity", 2021)

On Maps: Definitions

"Two important characteristics of maps should be noticed. A map is not the territory it represents, but, if correct, it has a similar structure to the territory, which accounts for its usefulness." (Alfred Korzybski, "Science and Sanity: An Introduction to Non-Aristotelian Systems and General Semantics", 1958)

"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)

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

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

"The representational nature of maps, however, is often ignored - what we see when looking at a map is not the word, but an abstract representation that we find convenient to use in place of the world. When we build these abstract representations we are not revealing knowledge as much as are creating it." (Alan MacEachren, "How Maps Work: Representation, Visualization, and Design", 1995)

"Maps are models, and every model represents some aspect of reality or an idea that is of interest. A model is a simplification. It is an interpretation of reality that abstracts the aspects relevant to solving the problem at hand and ignores extraneous detail." (Eric Evans, "Domain-Driven Design: Tackling complexity in the heart of software", 2003)

"A map does not just chart, it unlocks and formulates meaning; it forms bridges between here and there, between disparate ideas that we did not know were previously connected." (Reif Larsen, "The Selected Works of T S Spivet", 2009)

Winifred E Newman - Collected Quotes

"'Computational thinking', an idea borrowed from computer science, quickly radicalized design disciplines and created productive debate around what and how to represent data in the design process. Making complex maps using large datasets is one option enabling designers to bridge the gap between human and machine observation. Maps are efficient, evocative, and indeterminate. They model data well enough for us to infer relations, forecast possibilities, while maintaining data consistency. Additionally, maps are reproducible and transmittable to other designers." (Winifred E Newman, "Data Visualization for Design Thinking: Applied Mapping", 2017)

"Making a map is the physical production including conceptualization and design. Mapping is the mental interpretation of the world and although it must precede the map, it does not necessarily result in a map artifact. Mapping defined in mathematics is the correspondence between each element of a given set with each element of another. Similarly in linguistics emphasis is on the correspondence between associated elements of different types. For designers all drawings are maps - they represent relationships between objects, places and ideas." (Winifred E Newman, "Data Visualization for Design Thinking: Applied Mapping", 2017)

"Mapping is based on our perceptions of what we are experiencing and our conceptions framing how we interpret sensorial input. The map isn’t ‘real’ but the objects in the map are at some point in the world or of the world. Just as we have to disabuse ourselves of the idea maps, as semiotic texts are value neutral, similarly mapping calls into question how we interpret perceptions." (Winifred E Newman, "Data Visualization for Design Thinking: Applied Mapping", 2017)

"[...] mapping is the collective set of practices structuring correspondence between physical phenomena, lived experience, or conceptual frameworks." (Winifred E Newman, "Data Visualization for Design Thinking: Applied Mapping", 2017)

"Maps are parenthetical - maps frame what you want to hold apart from the real in the world. Maps do this by creating conceptual representations of the milieu using symbols and relations between symbols. [...] Maps, any map and every map, begin with a frame. This is the literal and conceptual demarcation between what is in the map and what is not. Making a map begins with an observation which is both a thought about thinking and the object of thought itself. The undifferentiated world cannot be apprehended, therefore; all maps have a frame whether a concept or a cosmography." (Winifred E Newman, "Data Visualization for Design Thinking: Applied Mapping", 2017)

"The presence of structure distinguishes maps from diagrams. They are similar in kind but not degree. Diagrams share some of the structure but don’t quantify or qualify spatio-temporal relationships in the same way as maps. They are ‘simplified figures to convey essential meaning,’ whereas maps tend toward robust meaning relative to the subject. Symbols in diagrams have multiple possible significations until we specify or point to their meaning through context using an index. Diagrams are indexical, i.e. they point to something, but they aren’t indexed: they don’t order or organize within a larger context nor do they have a spatio-temporal dimension like maps." (Winifred E Newman, "Data Visualization for Design Thinking: Applied Mapping", 2017)

"The utility of mapping as a form of data visualization isn’t in accuracy or precision, but rather the map’s capacity to help us make and organize hypothesis about the world of ideas and things. hypothesis-making through the map isn’t strictly inductive or deductive, although it can use the thought process of either, but it is often based on general observations." (Winifred E Newman, "Data Visualization for Design Thinking: Applied Mapping", 2017)

"Using maps as communication tools masks their complexity as a mode of thinking. Maps act like language: we attribute the signs or marks in the map to a natural extension of thought. But post-structuralism exposed maps (like language) as artificial signs whose meaning is tethered to time, place, culture, gesture, smell - in short, a plethora of cognitive and phenomenal attributes of our communication ecology." (Winifred E Newman, "Data Visualization for Design Thinking: Applied Mapping", 2017)

"Visual representations of data engage our cognitive perceptions, graphic skills, and rational capacity to synthesize complex ideas. For designers, maps represent observations about places, ideas or relationships. Being able to frame, represent, and order data through maps empowers the process of thinking, including thinking visually. In the twenty-first century, synthesis of complex data is a hallmark of critical thinking in all areas of knowledge. organizing information means creating links between chains of data, links between quantitative and qualitative data, and links between ideas through data. Visually organizing data means managing these links at the intersection of visual perception, representation, and the imagination." (Winifred E Newman, "Data Visualization for Design Thinking: Applied Mapping", 2017)

Mark S Monmonier - Collected Quotes

"A good map tells a multitude of little white lies; it suppresses truth to help the user see what needs to be seen. Reality is three-dimensional, rich in detail, and far too factual to allow a complete yet uncluttered two-dimensional graphic scale model. Indeed, a map that did not generalize would be useless. But the value of a map depends on how well its generalized geometry and generalized content reflect a chosen aspect of reality." (Mark S Monmonier, "How to Lie with Maps" 2nd Ed., 1996)

"Enhancement adds detail to give map symbols a more realistic appearance. Lines representing streams, for instance, might be given typical meander loops, whereas shorelines might be made to look more coast-like. Enhanced map symbols are more readily interpreted as well as more aesthetic." (Mark S Monmonier, "How to Lie with Maps" 2nd Ed., 1996)

"Graphic scales are not only the most helpful means of communicating map scale but also the safest. An alternative to blind trust in the user's sense of distance and skill in mental arithmetic, the simple bar scale typically portrays a series of conveniently rounded distances appropriate to the map's function and the area covered." (Mark S Monmonier, "How to Lie with Maps" 2nd Ed., 1996)

"Graphic symbols complement map scale and projection by making visible the features, places, and other locational information represented on the map. By describing and differentiating features and places, map symbols serve as a graphic code for storing and retrieving data in a two-dimensional geographic framework." (Mark S Monmonier, "How to Lie with Maps" 2nd Ed., 1996)

"Map projections distort five geographic relationships: areas, angles, gross shapes, distances, and directions. Although some projections preserve local angles but not areas, others preserve areas but not local angles. All distort large shapes noticeably (but some distort continental shapes more than others), and all distort at least some distances and some directions." (Mark S Monmonier, "How to Lie with Maps" 2nd Ed., 1996)

"Maps have three basic attributes: scale, projection, and symbolization. Each element is a source of distortion. As a group, they describe the essence of the map's possibilities and limitations. No one can use maps or make maps safely and effectively without understanding map scales, map projections, and map symbols." (Mark S Monmonier, "How to Lie with Maps" 2nd Ed., 1996)

"Not only is it easy to lie with maps, it's essential. To portray meaningful relationships for a complex, three-dimensional world on a flat sheet of paper or a video screen, a map must distort reality. As a scale model, the map must use symbols that almost always are proportionally much bigger or thicker than the features they represent. To avoid hiding critical information in a fog of detail, the map must offer a selective, incomplete view of reality. There's no escape from the cartographic paradox: to present a useful and truthful picture, an accurate map must tell white lies." (Mark S Monmonier, "How to Lie with Maps" 2nd Ed., 1996)

"Smoothing, which also diminishes detail and angularity, might displace some points and add others to the list. A prime objective of smoothing is to avoid a series of abruptly joined straight line segments." (Mark S Monmonier, "How to Lie with Maps" 2nd Ed., 1996)

"Some maps fail because of the mapmaker's ignorance or oversight. The range of blunders affecting maps includes graphic scales that invite users to estimate distances from world maps, maps based on incompatible sources, misspelled place-names, and graytone symbols changed by poor printing by poor planning. By definition a blunder is not a lie, but the informed map user must be aware of cartographic fallibility; and even of a bit of mischief." (Mark S Monmonier, "How to Lie with Maps" 2nd Ed., 1996)

Menno-Jan Kraak - Collected Quotes

"Before a map can be drawn, a cartographer has to consider constraints that will influence its design. These include the purpose of the map, user characteristics, the use environment, and data characteristics. The purpose of a map, which can be manifold, relates to the questions the map will have to answer, or it use requirements. […] Above all, however, data characteristics will influence a mapmaker’s choice of symbology, because qualitative and quantitative data cannot be expressed in the same way." (Menno-Jan Kraak, "Mapping Time: Illustrated by Minard’s map of Napoleon’s Russian Campaign of 1812", 2014)

"Designs that play the role of presentation usually inform about spatial patterns and relations. This reflects the traditional cartographic approach. The cartographer begins with a known set of data and must select an appropriate visualization technique that will produce a high-quality explanation of facts. In other words, the visualization process ends with the best possible map. Map design, then, is important." (Menno-Jan Kraak, "Mapping Time: Illustrated by Minard’s map of Napoleon’s Russian Campaign of 1812", 2014)

"Graphic representations such as maps and diagrams should be able to answer all kinds of spatiotemporal questions, whatever the given situation. Not all graphics are suitable; even if they are, the data behind the graphics should be archived in a well-organized database." (Menno-Jan Kraak, "Mapping Time: Illustrated by Minard’s map of Napoleon’s Russian Campaign of 1812", 2014)

"In contrast to flow maps, origin-destination maps’ paths are highly structured, and do not use arrowheads to indicate direction. Both types of maps illustrate the volume of flow by varying the thickness of the path line’s shaft, some by gradually trimming the thickness of the shaft, others by splitting the shaft into sections and giving each section its own uniform thickness." (Menno-Jan Kraak, "Mapping Time: Illustrated by Minard’s map of Napoleon’s Russian Campaign of 1812", 2014)

"[…] information is lost or gained during the communication process. Information could be lost whenever a cartographer chooses to suppress information and/or users fail to understand all of it. It can be gained whenever the cartographer clarifies the original set of data and/or users combine the map information with their prior knowledge." (Menno-Jan Kraak, "Mapping Time: Illustrated by Minard’s map of Napoleon’s Russian Campaign of 1812", 2014)

"One cannot always be fully acquainted with the data, however. In such cases, exploration helps to judge the data’s usefulness. The exploration process, which emphasizes discovery, often facilitates an interactive, undirected search for structures and trends. This may result in conclusions that lead to alternative hypotheses." (Menno-Jan Kraak, "Mapping Time: Illustrated by Minard’s map of Napoleon’s Russian Campaign of 1812", 2014)

"We use maps to help us understand the world around us in the most effective and efficient way. Maps can summarize, clarify, explain, and emphasize aspects of our environment. Maps can play many roles. They support navigation and decision making, they of f er insight into spatial patterns and relationships among mapped phenomena, and […] they can tell stories. Maps do this well because they symbolize and abstract the reality they represent." (Menno-Jan Kraak, "Mapping Time: Illustrated by Minard’s map of Napoleon’s Russian Campaign of 1812", 2014)

String Theory III

"String theory promises to take a further step beyond that taken by Einstein's picture of force subsumed within curved space and time geometry. Indeed, string theory contains Einstein's theory of gravitation within itself. Loops of string behave like the exchange particles of the gravitational forces, or 'gravitons' as they are called in the point-particle picture of things. But it has been argued that it must be possible to extract even the geometry of space and time from the characteristics of the strings and their topological properties. At present, it is not known how to do this and we merely content ourselves with understanding how strings behave when they sit in a background universe of space and time." (John D. Barrow, "Theories of Everything: The Quest for Ultimate Explanation", 1991)

"A five-dimensional space is not a strange deformation of ordinary space, one that only mathematicians can see, but a place where numbers are collected in ordered sets. When string theorists talk of the eleven dimensions required by their latest theory, they are not encouraging one another to search for eight otherwise familiar spatial dimensions that have somehow become lost. They are saying only that for their purposes, eleven numbers are needed to specify points. Where they are is no one’s business." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005) 

"One could also question whether we are looking for a single overarching mathematical structure or a combination of different complementary points of view. Does a fundamental theory of Nature have a global definition, or do we have to work with a series of local definitions, like the charts and maps of a manifold, that describe physics in various 'duality frames'. At present string theory is very much formulated in the last kind of way." (Robbert Dijkgraaf, "Mathematical Structures", 2005)

"Quantum physics, in particular particle and string theory, has proven to be a remarkable fruitful source of inspiration for new topological invariants of knots and manifolds. With hindsight this should perhaps not come as a complete surprise. Roughly one can say that quantum theory takes a geometric object (a manifold, a knot, a map) and associates to it a (complex) number, that represents the probability amplitude for a certain physical process represented by the object." (Robbert Dijkgraaf, "Mathematical Structures", 2005)

"String theory was not invented to describe gravity; instead it originated in an attempt to describe the strong interactions, wherein mesons can be thought of as open strings with quarks at their ends. The fact that the theory automatically described closed strings as well, and that closed strings invariably produced gravitons and gravity, and that the resulting quantum theory of gravity was finite and consistent is one of the most appealing aspects of the theory." (David Gross, "Einstein and the Search for Unification", 2005)

"Like many a maturing beauty, string theory has gotten rich in relationships, complicated, hard to handle and widely influential. Its tentacles have reached so deeply into so many areas in theoretical physics, it’s become almost unrecognizable, even to string theorists." (K C Cole, "The Strange Second Life of String Theory", Quanta Magazine", 2016) [source] 

"String theory today looks almost fractal. The more closely people explore any one corner, the more structure they find. Some dig deep into particular crevices; others zoom out to try to make sense of grander patterns. The upshot is that string theory today includes much that no longer seems stringy. Those tiny loops of string whose harmonics were thought to breathe form into every particle and force known to nature (including elusive gravity) hardly even appear anymore on chalkboards at conferences." (K C Cole, "The Strange Second Life of String Theory", Quanta Magazine", 2016) [source]

Alan M MacEachren - Collected Quotes

"Cartography is about representation. This statement may seem obvious, but it has been overlooked in our search for organizing principles for the field. Rather than restricting research in cartography to maps that present well-defined messages (and suggesting a single, map-engineering approach to improving the transmission of these messages, as the communication approach did), attention to maps as spatial representation expands the field." (Alan M MacEachren, "How Maps Work: Representation, Visualization, and Design", 1995)

"Exploring maps as representation forges important links between cartography and a variety of cognate fields concerned with this topic in its various facets (including geographical information systems [GIs] and remote sensing, as well as art, cognitive science, sociology, cognitive and environmental psychology, semiotics, and even the history and philosophy of science)." (Alan M MacEachren, "How Maps Work: Representation, Visualization, and Design", 1995)

"It may be that the human brain not only perceives but stores the essentials of a visual scene using the same geometrical, quasi-symbolic, minimalist vocabulary found in maps." (Alan M MacEachren, "How Maps Work: Representation, Visualization, and Design", 1995)

"Maps, due to their melding of scientific and artistic approaches, always involve complex interaction between the denotative and the connotative meanings of signs they contain." (Alan M MacEachren, "How Maps Work: Representation, Visualization, and Design", 1995)

"The fact that map is a fuzzy and radial, rather than a precisely defined, category is important because what a viewer interprets a display to be will influence her expectations about the display and how she interacts with it." (Alan M MacEachren, "How Maps Work: Representation, Visualization, and Design", 1995)

"The representational nature of maps, however, is often ignored - what we see when looking at a map is not the word, but an abstract representation that we find convenient to use in place of the world. When we build these abstract representations we are not revealing knowledge as much as are creating it." (Alan M MacEachren, "How Maps Work: Representation, Visualization, and Design", 1995)

"To make maps that work, we must depict categories using methods that match the structures of human mental categorization." (Alan M MacEachren, "How Maps Work: Representation, Visualization, and Design", 1995)

"Understanding how maps work and why maps work (or do not work) as representations in their own right and as prompts to further representations, and what it means for a map to work, are critical issues as we embark on a visual information age." (Alan M MacEachren, "How Maps Work: Representation, Visualization, and Design", 1995)

"When visualization tools act as a catalyst to early visual thinking about a relatively unexplored problem, neither the semantics nor the pragmatics of map signs is a dominant factor. On the other hand, syntactics (or how the sign-vehicles, through variation in the visual variables used to construct them, relate logically to one another) are of critical importance." (Alan M MacEachren, "How Maps Work: Representation, Visualization, and Design", 1995)

"Cartography as a discipline has a significant stake in the evolving role of maps within systems for scientific visualization, within spatial decision support systems, within hypermedia information access systems, and within virtual reality environments." (Alan M MacEachren, "Exploratory cartographic visualization: advancing the agenda", 1997)

"The nature of maps and of their use in science and society is in the midst of remarkable change - change that is stimulated by a combination of new scientific and societal needs for geo-referenced information and rapidly evolving technologies that can provide that information in innovative ways. A key issue at the heart of this change is the concept of ‘visualization’." (Alan M MacEachren, "Exploratory cartographic visualization: advancing the agenda", 1997)

"Maps have been a successful form of representation for centuries by making the world understandable through systematic abstraction that retains the iconicity of space depicting space. Advances in methods and technologies are blurring the lines among maps and other forms of visual representation and pushing the bounds of 'map' as a concept toward both more realistic and more abstract depiction. As a result, there are a variety of unanswered questions about the attributes and implications of 'maps'." (Alan M MacEachren, "Research Challenges in Geovisualization", 2001)

On Imagination (1800-1849)

"The philosopher who is really useful to the cause of science, is he who, uniting to a fertile imagination, a rigid severity in investigation and observation, is at once tormented by the desire of ascertaining the cause of the phenomena, and by the fear of deceiving himself in that which he assigns." (Pierre-Simon Laplace, "System of the World" Vol. 2, 1809)

"When the eye or the imagination is struck with an uncommon work, the next transition of an active mind is to the means by which it was performed." (Samuel Johnson, 1810)

"The imagination […] that reconciling and mediatory power, which incorporating the reason in images of the sense and organizing (as it were) the flux of the senses by the permanence and self-circling energies of the reason, gives birth to a system of symbols, harmonious in themselves, and consubstantial with the truths of which they are the conductors." (Samuel T Coleridge, "The Statesman's Manual", 1816)

"It seems to be like taking the pieces of a dissected map out of its box. We first look at one part, and then at another, then join and dove-tail them; and when the successive acts of attention have been completed, there is a retrogressive effort of mind to behold it as a whole. The poet should paint to the imagination, not to the fancy; and I know no happier case to exemplify the distinction between these two faculties." (Samuel T Coleridge," Biographia Literaria", 1817)

"Whilst chemical pursuits exalt the understanding, they do not depress the imagination or weaken genuine feeling; whilst they give the mind habits of accuracy, by obliging it to attend to facts, they likewise extend its analogies; and, though conversant with the minute forms of things, they have for their ultimate end the great and magnificent objects of nature." (Sir Humphry Davy, "Consolations in Travel, or the Last Days of a Philosopher", 1830)

"No occupation is more worthy of an intelligent and enlightened mind, than the study of Nature and natural objects; and whether we labour to investigate the structure and function of the human system, whether we direct our attention to the classification and habits of the animal kingdom, or prosecute our researches in the more pleasing and varied field of vegetable life, we shall constantly find some new object to attract our attention, some fresh beauties to excite our imagination, and some previously undiscovered source of gratification and delight." (Sir Joseph Paxton, "A Practical Treatise on the Cultivation of the Dahlia", 1838)

"But a thousand unconnected observations have no more value, as a demonstrative proof, than a single one. If we do not succeed in discovering causes by our researches, we have no right to create them by the imagination; we must not allow mere fancy to proceed beyond the bounds of our knowledge."(Justus von Liebig, "The Lancet", 1844)

"The nose of a mob is its imagination. By this, at any time, it can be quietly led." (Edgar A Poe, "The Works of Edgar Allan Poe", 1849)

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On Nonlinearity I

"In complex systems cause and effect are often not closely related in either time or space. The structure of a complex system is not a simple feedback loop where one system state dominates the behavior. The complex system has a multiplicity of interacting feedback loops. Its internal rates of flow are controlled by nonlinear relationships. The complex system is of high order, meaning that there are many system states (or levels). It usually contains positive-feedback loops describing growth processes as well as negative, goal-seeking loops. In the complex system the cause of a difficulty may lie far back in time from the symptoms, or in a completely different and remote part of the system. In fact, causes are usually found, not in prior events, but in the structure and policies of the system." (Jay Wright Forrester, "Urban dynamics", 1969)

"Self-organization can be defined as the spontaneous creation of a globally coherent pattern out of local interactions. Because of its distributed character, this organization tends to be robust, resisting perturbations. The dynamics of a self-organizing system is typically non-linear, because of circular or feedback relations between the components. Positive feedback leads to an explosive growth, which ends when all components have been absorbed into the new configuration, leaving the system in a stable, negative feedback state. Non-linear systems have in general several stable states, and this number tends to increase (bifurcate) as an increasing input of energy pushes the system farther from its thermodynamic equilibrium. " (Francis Heylighen, "The Science Of Self-Organization And Adaptivity", 1970)

"[The] system may evolve through a whole succession of transitions leading to a hierarchy of more and more complex and organized states. Such transitions can arise in nonlinear systems that are maintained far from equilibrium: that is, beyond a certain critical threshold the steady-state regime become unstable and the system evolves into a new configuration." (Ilya Prigogine, Gregoire Micolis & Agnes Babloyantz, "Thermodynamics of Evolution", Physics Today 25 (11), 1972)

"An artificial neural network is an information-processing system that has certain performance characteristics in common with biological neural networks. Artificial neural networks have been developed as generalizations of mathematical models of human cognition or neural biology, based on the assumptions that: 1. Information processing occurs at many simple elements called neurons. 2. Signals are passed between neurons over connection links. 3. Each connection link has an associated weight, which, in a typical neural net, multiplies the signal transmitted. 4. Each neuron applies an activation function (usually nonlinear) to its net input (sum of weighted input signals) to determine its output signal." (Laurene Fausett, "Fundamentals of Neural Networks", 1994)

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

"[…] nonlinear interactions almost always make the behavior of the aggregate more complicated than would be predicted by summing or averaging."  (John H Holland," Hidden Order: How Adaptation Builds Complexity", 1995)

“[…] self-organization is the spontaneous emergence of new structures and new forms of behavior in open systems far from equilibrium, characterized by internal feedback loops and described mathematically by nonlinear equations.” (Fritjof  Capra, “The web of life: a new scientific understanding of living  systems”, 1996)

"Bounded rationality simultaneously constrains the complexity of our cognitive maps and our ability to use them to anticipate the system dynamics. Mental models in which the world is seen as a sequence of events and in which feedback, nonlinearity, time delays, and multiple consequences are lacking lead to poor performance when these elements of dynamic complexity are present." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"Even if our cognitive maps of causal structure were perfect, learning, especially double-loop learning, would still be difficult. To use a mental model to design a new strategy or organization we must make inferences about the consequences of decision rules that have never been tried and for which we have no data. To do so requires intuitive solution of high-order nonlinear differential equations, a task far exceeding human cognitive capabilities in all but the simplest systems."  (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"All forms of complex causation, and especially nonlinear transformations, admittedly stack the deck against prediction. Linear describes an outcome produced by one or more variables where the effect is additive. Any other interaction is nonlinear. This would include outcomes that involve step functions or phase transitions. The hard sciences routinely describe nonlinear phenomena. Making predictions about them becomes increasingly problematic when multiple variables are involved that have complex interactions. Some simple nonlinear systems can quickly become unpredictable when small variations in their inputs are introduced." (Richard N Lebow, "Forbidden Fruit: Counterfactuals and International Relations", 2010)

On Symbols (2000-2009)

"Precision is greatly enhanced by the human capacity to symbolize. Symbols can be devised to stand for mathematical ideas, entities, operations, and relations. Symbols also permit precise and repeatable calculation." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"The motion of the mind is conveyed along a cloud of meaning. There is this paradox that we get to meaning only when we strip the meaning from symbols." (David Berlinski, "The Advent of the Algorithm: The Idea that Rules the World", 2000)

"A symbol is a mental representation regarding the internal reality referring to its object by a convention and produced by the conscious interpretation of a sign. In contrast to signals, symbols may be used every time if the receiver has the corresponding representation. Symbols also relate to feelings and thus give access not only to information but also to the communicator’s motivational and emotional state. The use of symbols makes it possible for the organism using it to evoke in the receiver the same response it evokes in himself. To communicate with symbols is to use a language." (Lars Skyttner, "General Systems Theory: Ideas and Applications", 2001)

"In the definition of meaning, it is assumed that both the source and receiver have previously coded (and stored) signals of the same or similar referents, such that the messages may have meaning and relate to behaviour. That is, the used symbols must have the same signification for both sender and receiver. If not, the receiver will create a different mental picture than intended by the transmitter. Meaning is generated by individuals in a process of social interaction with a more or less common environment. It is a relation subsisting within a field of experience and appears as an emergent property of a symbolic representation when used in culturally accepted interaction. The relation between the symbolic representation and its meaning is random. Of this, however, the mathematical theory has nothing to say. If human links in the chain of communication are missing, of course no questions of meaning will arise." (Lars Skyttner, "General Systems Theory: Ideas and Applications", 2001)

"A person thinking in the nonverbal mode is actually thinking with the meaning of the language in the form of mental pictures of the concepts and ideas it contains. Nonverbal thought doesn't require literacy. An illiterate person can communicate without knowing what the symbols look like. [...] Literacy, then, is established as the person learns how the symbols look and becomes able to recognize them as representing certain things or concepts." (Ronald D Davis & Eldon M Braun, "The Gift of Learning", 2003)

"Science does not speak of the world in the language of words alone, and in many cases it simply cannot do so. The natural language of science is a synergistic integration of words, diagrams, pictures, graphs, maps, equations, tables, charts, and other forms of visual and mathematical expression. […] [Science thus consists of] the languages of visual representation, the languages of mathematical symbolism, and the languages of experimental operations." (Jay Lemke, "Teaching all the languages of science: Words, symbols, images and actions", 2003)

"I often told the fanatics of realism that there is no such thing as realism in art: it only exists in the mind of the observer. Art is a symbol, a thing conjuring up reality in our mental image. That is why I don't see any contradiction between abstract and figurative art either." (Antoni Tàpies, "Tàpies, Werke auf Papier 1943 – 2003", 2004)

"A symbol is an object, act, or event that conveys meaning to others. Symbols can be considered a rich, non-verbal language that vibrantly conveys the organization’s important values concerning how people relate to one another and interact with the environment" (Richard L Daft & Dorothy Marcic, "Understanding Management" 5th Ed., 2006)

"But because of the way in which depictions represent, there is a correspondence between parts and spatial relations of the representation and those of the object; this structural mapping, which confers a type of resemblance, underlies the way images convey specific content. In this respect images are like pictures. Unlike words and symbols, depictions are not arbitrarily paired with what they represent." (Stephen Kosslyn et al," The Case for Mental Imagery", 2006)

"Imagination has the creative task of making symbols, joining things together in such a way that they throw new light on each other and on everything around them. The imagination is a discovering faculty, a faculty for seeing relationships, for seeing meanings that are special and even quite new." (Thomas Merton, "Angelic Mistakes: The Art of Thomas Merton", 2006)

"[...] the scientific models of concrete things are symbolic rather than iconic: they are systems of propositions, not pictures. Besides, such models are seldom if ever completely accurate, if only because they involve more or less brutal simplifications, such as pretending that a metallic surface is smooth, a crystal has no impurities, a biopopulation has a single predator, or a market is in equilibrium.  These are all fictions. However, they are stylizations rather than wild fantasies. Hence, introducing and using them to account for real existents does not commit us to fictionism, just as defending the role of experience need not make us empiricists, nor is admitting the role of intuition enough to qualify as intuitionist." (Mario Bunge, "Chasing Reality: Strife over Realism", 2006)

"But notice, a subatomic particle is itself a holon [hole/parts]. And so is a cell. And so is a symbol, and an image, and a concept. What all of those entities are, before they are anything else, is a holon. So the world is not composed of atoms or symbols or cells or concepts. It is composed of holons." (Ken Wilber, "A Brief History of Everything", 2007)

"Language use is a curious behavior, but once the transition to language is made, literature is a likely consequence, since it is linked to the dynamic of the linguistic symbol through the functioning of the imagination." (Russell Berman, "Fiction Sets You Free: Literature, Liberty and Western Culture", 2007)

"Images and pictures […] have played a key role in shaping our scientific picture of the world. […] Carefully constructed families of pictures can act as a calculus all their own. Like any successful systems of symbols, with an appropriate grammar they enlarge the number of things that we can do without consciously thinking." (John D Barrow, "Cosmic Imagery: Key Images in the History of Science", 2008)

"How are we to explain the contrast between the matter-of-fact way in which √-1 and other imaginary numbers are accepted today and the great difficulty they posed for learned mathematicians when they first appeared on the scene? One possibility is that mathematical intuitions have evolved over the centuries and people are generally more willing to see mathematics as a matter of manipulating symbols according to rules and are less insistent on interpreting all symbols as representative of one or another aspect of physical reality. Another, less self-congratulatory possibility is that most of us are content to follow the computational rules we are taught and do not give a lot of thought to rationales." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2009)

"Mathematicians are sometimes described as living in an ideal world of beauty and harmony. Instead, our world is torn apart by inconsistencies, plagued by non sequiturs and, worst of all, made desolate and empty by missing links between words, and between symbols and their referents; we spend our lives patching and repairing it. Only when the last crack disappears are we rewarded by brief moments of harmony and joy." (Alexandre V Borovik, "Mathematics under the Microscope: Notes on Cognitive Aspects of Mathematical Practice", 2009)

"Mathematical ideas like number can only be really 'seen' with the 'eyes of the mind' because that is how one 'sees' ideas. Think of a sheet of music which is important and useful but it is nowhere near as interesting, beautiful or powerful as the music it represents. One can appreciate music without reading the sheet of music. Similarly, mathematical notation and symbols on a blackboard are just like the sheet of music; they are important and useful but they are nowhere near as interesting, beautiful or powerful as the actual mathematics (ideas) they represent." (Fiacre O Cairbre, "The Importance of Being Beautiful in Mathematics", IMTA Newsletter 109, 2009)

On Networks (1960-1969)

"Any pattern of activity in a network, regarded as consistent by some observer, is a system, Certain groups of observers, who share a common body of knowledge, and subscribe to a particular discipline, like 'physics' or 'biology' (in terms of which they pose hypotheses about the network), will pick out substantially the same systems. On the other hand, observers belonging to different groups will not agree about the activity which is a system." (Gordon Pask, The Natural History of Networks, 1960)

"Clearly, if the state of the system is coupled to parameters of an environment and the state of the environment is made to modify parameters of the system, a learning process will occur. Such an arrangement will be called a Finite Learning Machine, since it has a definite capacity. It is, of course, an active learning mechanism which trades with its surroundings. Indeed it is the limit case of a self-organizing system which will appear in the network if the currency supply is generalized." (Gordon Pask, "The Natural History of Networks", 1960)

"I am using the term 'network' in a general sense, to imply any set of interconnected and measurably active physical entities. Naturally occurring networks, of interest because they have a, self-organizing character, are, for example, a marsh, a colony of microorganisms, a research team, and a man." (Gordon Pask, "The Natural History of Networks", 1960)

"A NETWORK is a collection of connected lines, each of which indicates the movement of some quantity between two locations. Generally, entrance to a network is via a source (the starting point) and exit from a network is via a sink (the finishing point); the lines which form the network are called links (or arcs), and the points at which two or more links meet are called nodes." (Cecil W Lowe, "Critical Path Analysis by Bar Chart", 1966)

"In a network, one can plot the figures on a plane which has no meaning, and then look for the arrangement which produces the minimum number of intersections, or the simplest figure. After this transformation, the graphic will yield maximum efficiency, based on the discovery of a meaningful order expressed by the plane." (Jacques Bertin, "Semiology of graphics", 1967)

"To analyse graphic representation precisely, it is helpful to distinguish it from musical, verbal and mathematical notations, all of which are perceived in a linear or temporal sequence. The graphic image also differs from figurative representation essentially polysemic, and from the animated image, governed by the laws of cinematographic time. Within the boundaries of graphics fall the fields of networks, diagrams and maps. The domain of graphic imagery ranges from the depiction of atomic structures to the representation of galaxies and extends into the spheres of topography and cartography."  (Jacques Bertin, "Semiology of graphics", 1967)

On Diagrams (2000-2019)

"Concept maps have long provided visual languages widely used in many different disciplines and application domains. Abstractly, they are sorted graphs visually represented as nodes having a type, name and content, some of which are linked by arcs. Concretely, they are structured diagrams having discipline- and domain-specific interpretations for their user communities, and, sometimes, formally defining computer data structures. Concept maps have been used for a wide range of purposes and it would be useful to make such usage available over the World Wide Web." (Brian R Gaines, "WebMap: Concept Mapping on the Web", 2001) 

"Data is transformed into graphics to understand. A map, a diagram are documents to be interrogated. But understanding means integrating all of the data. In order to do this it’s necessary to reduce it to a small number of elementary data. This is the objective of the 'data treatment' be it graphic or mathematic." (Jacques Bertin, [interview] 2003)

"Science does not speak of the world in the language of words alone, and in many cases it simply cannot do so. The natural language of science is a synergistic integration of words, diagrams, pictures, graphs, maps, equations, tables, charts, and other forms of visual and mathematical expression. […] [Science thus consists of] the languages of visual representation, the languages of mathematical symbolism, and the languages of experimental operations." (Jay Lemke, "Teaching all the languages of science: Words, symbols, images and actions", 2003)

"A causal map is a word-and-arrow diagram in which ideas and actions are causally linked with one another through the use of arrows. The arrows indicate how one idea or action leads to another. Causal mapping makesit possible to articulate a large number of ideas and their interconnections in such a way that people can know what to do in an area of concern, how to do it and why, because the arrows indicate the causes and consequences of an idea or action. Causal mapping is therefore a technique for linking strategic thinking and acting, helping make sense of complex problems, and communicating to oneself and others what might be done about them." (John M Bryson et al, "Visible Thinking: Unlocking Causal Mapping For Practical Business Results", 2004)

"Graphical design notations have been with us for a while [...] their primary value is in communication and understanding. A good diagram can often help communicate ideas about a design, particularly when you want to avoid a lot of details. Diagrams can also help you understand either a software system or a business process. As part of a team trying to figure out something, diagrams both help understanding and communicate that understanding throughout a team. Although they aren't, at least yet, a replacement for textual programming languages, they are a helpful assistant." (Martin Fowler," UML Distilled: A Brief Guide to the Standard Object Modeling", 2004)

"In mathematics a good diagram is an invaluable aid to clear reasoning, whereas a bad one can seriously mislead. But it is not often that a diagram is good enough to suggest fresh avenues of inquiry altogether." (Anthony W F Edwards, "Cogwheels of the mind: The story of Venn diagrams", 2004)

"Good diagrams clarify. Very good diagrams force the ideas upon the viewer. The best diagrams compellingly embody the ideas themselves." (R W Oldford & W H Cherry, "Picturing Probability: the poverty of Venn diagrams, the richness of Eikosograms", 2006)

"A diagram is a graphic shorthand. Though it is an ideogram, it is not necessarily an abstraction. It is a representation of something in that it is not the thing itself. In this sense, it cannot help but be embodied. It can never be free of value or meaning, even when it attempts to express relationships of formation and their processes. At the same time, a diagram is neither a structure nor an abstraction of structure." (Peter Eisenman, "Written Into the Void: Selected Writings", 1990-2004, 2007)

"A modeling language is usually based on some kind of computational model, such as a state machine, data flow, or data structure. The choice of this model, or a combination of many, depends on the modeling target. Most of us make this choice implicitly without further thinking: some systems call for capturing dynamics and thus we apply for example state machines, whereas other systems may be better specified by focusing on their static structures using feature diagrams or component diagrams. For these reasons a variety of modeling languages are available." (Steven Kelly & Juha-Pekka Tolvanen, "Domain-specific Modeling", 2008)

"What was clearly useful was the use of diagrams to prove certain results either in algebraic topology, homological algebra or algebraic geometry. It is clear that doing category theory, or simply applying category theory, implies manipulating diagrams: constructing the relevant diagrams, chasing arrows by going via various paths in diagrams and showing they are equal, etc. This practice suggests that diagram manipulation, or more generally diagrams, constitutes the natural syntax of category theory and the category-theoretic way of thinking. Thus, if one could develop a formal language based on diagrams and diagrams manipulation, one would have a natural syntactical framework for category theory. However, moving from the informal language of categories which includes diagrams and diagrammatic manipulations to a formal language based on diagrams and diagrammatic manipulations is not entirely obvious." (Jean-Pierre Marquis, "From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory", 2009)

"Diagrams are information graphics that are made up primarily of geometric shapes, such as rectangles, circles, diamonds, or triangles, that are typically (but not always) interconnected by lines or arrows. One of the major purposes of a diagram is to show how things, people, ideas, activities, etc. interrelate and interconnect. Unlike quantitative charts and graphs, diagrams are used to show interrelationships in a qualitative way." (Robbie T Nakatsu, "Diagrammatic Reasoning in AI", 2010)

"What advantages do diagrams have over verbal descriptions in promoting system understanding? First, by providing a diagram, massive amounts of information can be presented more efficiently. A diagram can strip down informational complexity to its core - in this sense, it can result in a parsimonious, minimalist description of a system. Second, a diagram can help us see patterns in information and data that may appear disordered otherwise. For example, a diagram can help us see mechanisms of cause and effect or can illustrate sequence and flow in a complex system. Third, a diagram can result in a less ambiguous description than a verbal description because it forces one to come up with a more structured description." (Robbie T Nakatsu, "Diagrammatic Reasoning in AI", 2010)

"[…] a conceptual model is a diagram connecting variables and constructs based on theory and logic that displays the hypotheses to be tested." (Mary Wolfinbarger Celsi et al, "Essentials of Business Research Methods", 2011)

"Diagrams furnish only approximate information. They do not add anything to the meaning of the data and, therefore, are not of much use to a statistician or research worker for further mathematical treatment or statistical analysis. On the other hand, graphs are more obvious, precise and accurate than the diagrams and are quite helpful to the statistician for the study of slopes, rates of change and estimation, (interpolation and extrapolation), wherever possible." (S C Gupta & Indra Gupta, "Business Statistics", 2013)

"Systems archetypes thus provide a good starting theory from which we can develop further insights into the nature of a particular system. The diagram that results from working with an archetype should not be viewed as the 'truth', however, but rather a good working model of what we know at any point in time." (Daniel H Kim, "Systems Archetypes as Dynamic Theories", The Systems Thinker Vol. 24 (1), 2013)

"As mathematics gets more abstract, diagrams become more and more prominent as the ways that things fit together abstractly become both more subtle and more important. Moreover, the diagram often sums up the situation more succinctly than the explanation in words, [..]" (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)

"Models are formal structures represented in mathematics and diagrams that help us to understand the world. Mastery of models improves your ability to reason, explain, design, communicate, act, predict, and explore." (Scott E Page, "The Model Thinker", 2018) 

Jacques Bertin - Collected Quotes

"A graphic should not only show the leaves, it should show the branches as well as the entire tree." (Jacques Bertin, "The Semiology of Graphics", 1967)

"Graphic representation constitutes one of the basic sign-systems conceived by the human mind for the purposes of storing, understanding, and communicating essential information. As a "language" for the eye, graphics benefits from the ubiquitous properties of visual perception. As a monosemic system, it forms the rational part of the world of images. […] Graphics owes its special significance to its double function as a storage mechanism and a research instrument."  (Jacques Bertin, "The Semiology of graphics" ["Semiologie Graphique"], 1967)

"The aim of the graphic is to make the relationship among previously defined sets appear." (Jacques Bertin, "The Semiology of graphics" ["Semiologie Graphique"], 1967)

"The great difference between the graphic representation of yesterday, which was poorly dissociated from the figurative image, and the graphics of tomorrow, is the disappearance of the congential fixity of the image. […] When one can superimpose, juxtapose, transpose, and permute graphic images in ways that lead to groupings and classings, the graphic image passes from the dead image, the 'illustration,' to the living image, the widely accessible research instrument it is now becoming. The graphic is no longer only the 'representation' of a final simplification, it is a point of departure for the discovery of these simplifications and the means for their justification. The graphic has become, by its manageability, an instrument for information processing." (Jacques Bertin, "The Semiology of graphics" ["Semiologie Graphique"], 1967)

"The plane is the mainstay of all graphic representation. It is so familiar that its properties seem self-evident, but the most familiar things are often the most poorly understood. The plane is homogeneous and has two dimensions. The visual consequences of these properties must be fully explored." (Jacques Bertin, "The Semiology of graphics" ["Semiologie Graphique"], 1967)

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

"There are as many types of questions as components in the information." (Jacques Bertin, "The Semiology of graphics" ["Semiologie Graphique"], 1967)

"To analyse graphic representation precisely, it is helpful to distinguish it from musical, verbal and mathematical notations, all of which are perceived in a linear or temporal sequence. The graphic image also differs from figurative representation essentially polysemic, and from the animated image, governed by the laws of cinematographic time. Within the boundaries of graphics fall the fields of networks, diagrams and maps. The domain of graphic imagery ranges from the depiction of atomic structures to the representation of galaxies and extends into the spheres of topography and cartography." (Jacques Bertin, "The Semiology of graphics" ["Semiologie Graphique"], 1967)

"As with any graphic, networks are used in order to discover pertinent troups of to inform others of the groups and structures dis(Jacques Bertin, "The Semiology of graphics" ["Semiologie Graphique"], 1967)overed. It is a good means of displaying structures, However, it ceases to be a means of discovery when the elements are numerous. The figure rapidly becomes complex, illegible and untransformable." (Jacques Bertin, "Graphics and graphic information processing", 1977)

"Computers are able to multiply useless images without taking into account that, by definition, every graphic corresponds to a table. This table allows you to think about three basic questions that go from the particular to the general level. When this last one receives an answer, you have answers for all of them. Understanding means accessing the general level and discovering significant grouping (patterns). Consequently, the function of a graphic is answering the three following questions:

Which are the X,Y, Z components of the data table? (What it’s all about?)

What are the groups in X, in Y that Z builds? (What the information at the general level is?

What are the exceptions?

These questions can be applied to every kind of problem. They measure the usefulness of whatever construction or graphical invention allowing you to avoid useless graphics." (Jacques Bertin, [interview] 2003)

"Data is transformed into graphics to understand. A map, a diagram are documents to be interrogated. But understanding means integrating all of the data. In order to do this it’s necessary to reduce it to a small number of elementary data. This is the objective of the 'data treatment' be it graphic or mathematic." (Jacques Bertin, [interview] 2003)

"The use of computers shouldn't ignore the objectives of graphics, that are: (1) Treating data to get information. (2) Communicating, when necessary, the information obtained." (Jacques Bertin, [interview] 2003)

"Graphics is the visual means of resolving logical problems." (Jacques Bertin, "Graphics and Graphic Information Processing", 2011)

Mental Models XLVI

"Things which do not now exist in the mind itself, can only be perceived, remembered, or imagined, by means of the ideas or images in the mind, which are the immediate objects of perception, remembrance, and imagination." (Thomas Reid, "An Inquiry into the Human Mind on the Principles", 1764)

"There are conceptions which may be called fancy pictures. They are commonly called creatures of fancy, or of imagination. They are not the copies of any original that exists, but are originals themselves […]. They were conceived by their creators, and may be conceived by others, but they never existed. We do not ascribe the qualities of true or false to them, because they are not accompanied with any belief, nor do they imply any affirmation or negation." (Thomas Reid,"Essays on the Intellectual Powers of Man", 1785)

"For our purpose it is not necessary that they [images] should be in conformity with the things in any other respect whatever. As a matter of fact, we do not know, nor have we any means of knowing, whether our conceptions of things are in conformity with them in any other than this one fundamental respect."  (Heinrich Hertz, "The Principles of Mechanics Presented in a New Form", 1894)

"The problem of the transformation of images is of great importance in the theory of economic development. […] The problem here is that of the initiation and imitation of superior processes. Both these phenomena require transformation of the image; a new process always starts as a new image, as a new idea. The process itself is merely a form of transcription of the new image." (Kenneth E Boulding, "The Image: Knowledge in life and society", 1956)

"Discourses are ways of referring to or constructing knowledge about a particular topic of practice: a cluster (or formation) of ideas, images and practices, which provide ways of talking about, forms of knowledge and conduct associated with, a particular topic, social activity or institutional site in society. These discursive formations, as they are known, define what is and is not appropriate in our formulation of, and our practices in relation to, a particular subject or site of social activity." (Stuart Hall, "Representation: Cultural Representations and Signifying Practices", 1997)

"A collective mental map functions first of all as a shared memory. Various discoveries by members of the collective are registered and stored in this memory, so that the information will remain available for as long as necessary. The storage capacity of this memory is in general much larger than the capacities of the memories of the individual participants. This is because the shared memory can potentially be inscribed over the whole of the physical surroundings, instead of being limited to a single, spatially localized nervous system. Thus, a collective mental map differs from cultural knowledge, such as the knowledge of a language or a religion, which is shared among different individuals in a cultural group but is limited by the amount of knowledge a single individual can bear in mind." (Francis Heylighen, "Collective Intelligence and its Implementation on the Web", 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)

"Because feeling does not have a form; it has to be treated like an inner sensation which can only be understood in terms of the images it triggers. These images do not, however, represent the feeling as such, for they are independently existing representations or fantasies that are merely associated at the moment of pleasure or pain." (Angelika Rauch, "The Hieroglyph of Tradition: Freud, Benjamin, Gadamer, Novalis, Kant", 2000)

"[...] information feedback about the real world not only alters our decisions within the context of existing frames and decision rules but also feeds back to alter our mental models. As our mental models change we change the structure of our systems, creating different decision rules and new strategies. The same information, processed and interpreted by a different decision rule, now yields a different decision. Altering the structure of our systems then alters their patterns of behavior. The development of systems thinking is a double-loop learning process in which we replace a reductionist, narrow, short-run, static view of the world with a holistic, broad, long-term, dynamic view and then redesign our policies and institutions accordingly." (John D Sterman, "Business dynamics: Systems thinking and modeling for a complex world", 2000)

"To avoid policy resistance and find high leverage policies requires us to expand the boundaries of our mental models so that we become aware of and understand the implications of the feedbacks created by the decisions we make. That is, we must learn about the structure and dynamics of the increasingly complex systems in which we are embedded." (John D Sterman, "Business dynamics: Systems thinking and modeling for a complex world", 2000)

On Mental Models XLV

"A symbol, therefore, may have no effect and indeed ordinarily will have no effect on the image of the immediate future around one.   It does produce an effect, however, of what might be called the image of the image, on the image of the future, on the image of the past, on the image of the potential or even of the image of the possible." (Kenneth E Boulding, "The Image: Knowledge in life and society", 1956)

"I have suggested that one of the basic theorems of the theory of the image is that it is the image which in fact determines what might be called the current behavior of any organism […] The image acts as a field. The behavior consists in gravitating toward the most highly valued part of the field." (Kenneth E Boulding, "The Image: Knowledge in life and society", 1956)

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

"As perceivers we select from all the stimuli falling on our senses only those which interest us, and our interests are governed by a pattern-making tendency, sometimes called a schema. In a chaos of shifting impressions each of us constructs a stable world in which objects have recognisable shapes, are located in depth and have permanence." (Mary Douglas, "Purity and Danger", 1966)

"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)

"A vision is a clear mental picture of a desired future outcome. If you have ever put together a large 1,000-piece jigsaw puzzle, the chances are you used the picture on the top of the puzzle box to guide the placement of the pieces. That picture on the top of the box is the end result or the vision of what you are trying to turn into a reality. It is much more difficult - if not impossible - to put the jigsaw puzzle together without ever looking at the picture." (Jane Flaherty & Peter B Stark, "The Manager's Pocket Guide to Leadership Skills", 1999)

"Deep change in mental models, or double-loop learning, arises when evidence not only alters our decisions within the context of existing frames, but also feeds back to alter our mental models. As our mental models change, we change the structure of our systems, creating different decision rules and new strategies. The same information, interpreted by a different model, now yields a different decision. Systems thinking is an iterative learning process in which we replace a reductionist, narrow, short-run, static view of the world with a holistic, broad, long-term, dynamic view, reinventing our policies and institutions accordingly." (John D Sterman, "Learning in and about complex systems", Systems Thinking Vol. 3 2003)

"Eliciting and mapping the participant's mental models, while necessary, is far from sufficient [...] the result of the elicitation and mapping process is never more than a set of causal attributions, initial hypotheses about the structure of a system, which must then be tested. Simulation is the only practical way to test these models. The complexity of the cognitive maps produced in an elicitation workshop vastly exceeds our capacity to understand their implications. Qualitative maps are simply too ambiguous and too difficult to simulate mentally to provide much useful information on the adequacy of the model structure or guidance about the future development of the system or the effects of policies." (John D Sterman, "Learning in and about complex systems", Systems Thinking Vol. 3 2003)

"To form a mental picture of the event, the knowledge developer attempts to integrate his or her perception of the situation with the expert’s perception. That mental picture is then recorded. What happens is a continuous shuttle process; the knowledge developer mentally moves back and forth from the initial impression of the event to the later evaluation of the event. What is finally recorded is the evaluation made during this retrospective period. Because a time lapse can make details of a situation less clear, the information is not always valid." (Elias M Awad, "Knowledge Management", 2003)

"Images are generally resistant to change and ignore messages that do not conform to their internal settings. Sometimes, however, they do react and can alter in an incremental or even revolutionary manner. Humans can talk about and share their images and, in the symbolic universe they create, reflect upon what is and what might be." (Michael C Jackson, "Critical Systems Thinking and the Management of Complexity", 2019)

Mental Models XLII

"Nay farther, even with relation to that succession, we cou'd only admit of those perceptions, which are immediately present to our consciousness, nor cou'd those lively images, with which the memory presents us, be ever receiv'd as true pictures of past perceptions. The memory, senses, and understanding are, therefore, all of them founded on the imagination, or the vivacity of our ideas."(David Hume, "A Treatise of Human Nature A Treatise of Human Nature", 1739) 

"Every presentation of philosophy, whether oral or written, is to be taken and can only be taken in the sense of a means. Every system is only an expression or image of reason, and hence only an object of reason, an object which reason - a living power that procreates itself in new thinking beings - distinguishes from itself and posits as an object of criticism. Every system that is not recognized and appropriated as just a means, limits and warps the mind for it sets up the indirect and formal thought in the place of the direct, original and material thought." (Ludwig Feuerbach, "Towards a Critique of Hegel’s Philosophy", 1839) 

"The various languages placed side by side show that with words it is never a question of truth, never a question of adequate expression; otherwise, there would not be so many languages. The ‘thing in itself’ (which is precisely what the pure truth, apart from any of its consequences, would be) is likewise something quite incomprehensible to the creator of language and something not in the least worth striving for. This creator only designates the relations of things to men, and for expressing these relations he lays hold of the boldest metaphors. To begin with, a nerve stimulus is transferred into an image: first metaphor. The image, in turn, is imitated in a sound: second metaphor. And each time there is a complete overleaping of one sphere, right into the middle of an entirely new and different one." (Friedrich Nietzsche, "On Truth and Lie in an Extra-Moral Sense", 1873) 

"Mental states of every kind - sensations, feelings, images - which were at one time present in consciousness and then have disappeared from it - have not with their disappearance absolutely ceased to exist. Although the inwardly - turned look may no longer be able to find them, nevertheless they have not been utterly destroyed and annulled, but in a certain manner they, continue to exist, stored up to speak, in the memory." (Hermann Ebbinghaus, "Memory: A contribution to experimental psychology", 1885) 

"To see is one thing; to picture or visualise is another. A person can see things, only when his eyes are open, and when his surroundings are illuminated; but he can have pictures in his mind’s eye, when his eyes are shut and when the world is dark." (Gilbert Ryle, "The Concept of Mind", 1949) 

"Nature is more subtle, more deeply intertwined and more strangely integrated than any of our pictures of her - than any of our errors. It is not merely that our pictures are not full enough; each of our pictures in the end turns out to be so basically mistaken that the marvel is that it worked at all." (Jacob Bronowski, "Science and Human Values", 1956)

"Perhaps our ultimate understanding of scientific topics is measured in terms of our ability to generate metaphoric pictures of what is going on. Maybe understanding is coming up with metaphoric pictures." (Per Bak, "How Nature Works: the science of self-organized criticality", 1996) 

"Science does not speak of the world in the language of words alone, and in many cases it simply cannot do so. The natural language of science is a synergistic integration of words, diagrams, pictures, graphs, maps, equations, tables, charts, and other forms of visual and mathematical expression. [… Science thus consists of] the languages of visual representation, the languages of mathematical symbolism, and the languages of experimental operations." (Jay Lemke, "Teaching all the languages of science: Words, symbols, images and actions", 2003)

"Images and pictures […] have played a key role in shaping our scientific picture of the world. […] Carefully constructed families of pictures can act as a calculus all their own. Like any successful systems of symbols, with an appropriate grammar they enlarge the number of things that we can do without consciously thinking." (John D Barrow, "Cosmic Imagery: Key Images in the History of Science", 2008)

On Abstraction (1990-1999)

"All of engineering involves some creativity to cover the parts not known, and almost all of science includes some practical engineering to translate the abstractions into practice." (Richard W Hamming, "The Art of Probability for Scientists and Engineers", 1991)

"That is, the physicist likes to learn from particular illustrations of a general abstract concept. The mathematician, on the other hand, often eschews the particular in pursuit of the most abstract and general formulation possible. Although the mathematician may think from, or through, particular concrete examples in coming to appreciate the likely truth of very general statements, he will hide all those intuitive steps when he comes to present the conclusions of his thinking to outsiders. It presents the results of research as a hierarchy of definitions, theorems and proofs after the manner of Euclid; this minimizes unnecessary words but very effectively disguises the natural train of thought that led to the original results." (John D Barrow, "New Theories of Everything", 1991)

"Great mathematics seldom comes from idle speculation about abstract spaces and symbols. More often than not it is motivated by definite questions arising in the worlds of nature and humans." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"The word theory, as used in the natural sciences, doesn’t mean an idea tentatively held for purposes of argument - that we call a hypothesis. Rather, a theory is a set of logically consistent abstract principles that explain a body of concrete facts. It is the logical connections among the principles and the facts that characterize a theory as truth. No one element of a theory [...] can be changed without creating a logical contradiction that invalidates the entire system. Thus, although it may not be possible to substantiate directly a particular principle in the theory, the principle is validated by the consistency of the entire logical structure." (Alan Cromer, "Uncommon Sense: The Heretical Nature of Science", 1993)

"A mental model is not normally based on formal definitions but rather on concrete properties that have been drawn from life experience. Mental models are typically analogs, and they comprise specific contents, but this does not necessarily restrict their power to deal with abstract concepts, as we will see. The important thing about mental models, especially in the context of mathematics, is the relations they represent. […]  The essence of understanding a concept is to have a mental representation or mental model that faithfully reflects the structure of that concept. (Lyn D. English & Graeme S. Halford, "Mathematics Education: Models and Processes", 1995)

"Music and math together satisfied a sort of abstract 'appetite', a desire that was partly intellectual, partly aesthetic, partly emotional, partly, even, physical." (Edward Rothstein, "Emblems of Mind: The Inner Life of Music and Mathematics", 1995)

"The larger, more detailed and complex the model - the less abstract the abstraction – the smaller the number of people capable of understanding it and the longer it takes for its weaknesses and limitations to be found out." (John Adams, "Risk", 1995)

"The representational nature of maps, however, is often ignored - what we see when looking at a map is not the word, but an abstract representation that we find convenient to use in place of the world. When we build these abstract representations we are not revealing knowledge as much as are creating it." (Alan M MacEachren, "How Maps Work: Representation, Visualization, and Design", 1995)

"Abstract concepts are largely metaphorical." (George Lakoff, "Philosophy in the Flesh: The Embodied Mind and Its Challenge to Western Thought", 1999)

"The abstractions of science are stereotypes, as two-dimensional and as potentially misleading as everyday stereotypes. And yet they are as necessary to the process of understanding as filtering is to the process of perception." (K C Cole, "First You Build a Cloud and Other Reflections on Physics as a Way of Life", 1999)

On Abstraction (1980-1989)

"Mathematical reality is in itself mysterious: how can it be highly abstract and yet applicable to the physical world? How can mathematical theorems be necessary truths about an unchanging realm of abstract entities and at the same time so useful in dealing with the contingent, variable and inexact happenings evident to the senses?" (Salomon Bochner, "The Role of Mathematics in the Rise of Science", 1981)

"Today abstraction is no longer that of the map, the double, the mirror, or the concept. Simulation is no longer that of a territory, a referential being or substance. It is the generation by models of a real without origin or reality: A hyperreal. The territory no longer precedes the map, nor does it survive it. It is nevertheless the map that precedes the territory - precession of simulacra - that engenders the territory." (Baudrillard Jean, "Simulacra and Simulation", 1981)

"[…] a mathematician's ultimate concern is that his or her inventions be logical, not realistic. This is not to say, however, that mathematical inventions do not correspond to real things. They do, in most, and possibly all, cases. The coincidence between mathematical ideas and natural reality is so extensive and well documented, in fact, that it requires an explanation. Keep in mind that the coincidence is not the outcome of mathematicians trying to be realistic - quite to the contrary, their ideas are often very abstract and do not initially appear to have any correspondence to the real world. Typically, however, mathematical ideas are eventually successfully applied to describe real phenomena […]"(Michael Guillen,"Bridges to Infinity: The Human Side of Mathematics", 1983)

"Language is the most formless means of expression. Its capacity to describe concepts without physical or visual references carries us into an advanced state of abstraction." (Ian Wilson, "Conceptual Art", 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)

"A central problem in teaching mathematics is to communicate a reasonable sense of taste - meaning often when to, or not to, generalize, abstract, or extend something you have just done." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"There is no agreed upon definition of mathematics, but there is widespread agreement that the essence of mathematics is extension, generalization, and abstraction [… which] often bring increased confidence in the results of a specific application, as well as new viewpoints."  (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"In mathematics itself abstract algebra plays a dual role: that of a unifying link between disparate parts of mathematics  and that of a research subject with a highly active life of its own." (Israel N Herstein, "Abstract Algebra", 1986)

"We can hardly have a serious discussion of a science without abstraction. What makes science materialist is that the process of abstraction is explicit and recognized as historically contingent within the science. Abstraction becomes destructive when the abstract is reified and when the historical process of abstraction is forgotten, so that the abstract descriptions are taken for descriptions of the actual objects. The level of abstraction appropriate in a given science at a given time is a historical issue." (Richard Levins & Richard C Lewontin, "The Dialectical Biologist", 1985)

"A mental model is a data structure, in a computational system, that represents a part of the real world or of a fictitious world. It is assumed that there can be mental models of abstract realms, such as that of mathematics, but little more will be said about them. A model-theoretic semanticist is free to think of the entities in his model as actual items in the world.[...] Mental model is an appropriate term for the mental representations that underlie everyday reasoning about the world. To understand the everyday world is to have a theory of how it works." (Alan Granham, "Mental Models as Representations of Discourse and Text", 1987)

"Metaphor [is] a pervasive mode of understanding by which we project patterns from one domain of experience in order to structure another domain of a different kind. So conceived metaphor is not merely a linguistic mode of expression; rather, it is one of the chief cognitive structures by which we are able to have coherent, ordered experiences that we can reason about and make sense of. Through metaphor, we make use of patterns that obtain in our physical experience to organise our more abstract understanding. " (Mark Johnson, "The Body in the Mind", 1987)

"The essence of modeling, as we see it, is that one begins with a nontrivial word problem about the world around us. We then grapple with the not always obvious problem of how it can be posed as a mathematical question. Emphasis is on the evolution of a roughly conceived idea into a more abstract but manageable form in which inessentials have been eliminated. One of the lessons learned is that there is no best model, only better ones."  (Edward Beltrami, "Mathematics for Dynamic Modeling", 1987)

"Probabilities are summaries of knowledge that is left behind when information is transferred to a higher level of abstraction." (Judea Pearl, Probabilistic Reasoning in Intelligent Systems: Network of Plausible, Inference, 1988)

"Western culture’s world-view appears to be dominated by material objects. […] One of the ways mathematics has gained its power is through the activity of objectivising the abstractions from reality. Through its symbols (letters, numerals, figures) mathematics has taught people how to deal with abstract entities, as if they were objects." (Alan J Bishop, "Mathematics education in its cultural context", Educational Studies in Mathematics 19, 1988)

"[…] a model is the picture of the real - a short form of the whole. Hence, a model is an abstraction or simplification of a system. It is a technique by which aspects of reality can be 'artificially' represented or 'simulated' and at the same time simplified to facilitate comprehension." (Laxmi K Patnaik, "Model Building in Political Science", The Indian Journal of Political Science, Vol. 50, No. 2, 1989)

"As a practical matter, mathematics is a science of pattern and order. Its domain is not molecules or cells, but numbers, chance, form, algorithms, and change. As a science of abstract objects, mathematics relies on logic rather than observation as its standard of truth, yet employs observation, simulation, and even experimentation as a means of discovering truth. "(National Research Council, "Everybody Counts", 1989)

"Modeling in its broadest sense is the cost-effective use of something in place of something else for some [cognitive] purpose. It allows us to use something that is simpler, safer, or cheaper than reality instead of reality for some purpose. A model represents reality for the given purpose; the model is an abstraction of reality in the sense that it cannot represent all aspects of reality. This allows us to deal with the world in a simplified manner, avoiding the complexity, danger and irreversibility of reality." (Jeff Rothenberg, "The Nature of Modeling. In: Artificial Intelligence, Simulation, and Modeling", 1989)

On Paradigms I

"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)

"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)

"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 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)

"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)

“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])