Mental Models VI

“The intellectual capacity thinks the forms in the phantasmata (mental images) […] And for the following reason, as without having perceptual awareness no one could either learn or understand anything, so when one engages in intellectual activity one must at that time do so by means of a phantasma. For, phantasmata are just as perceptual states (aisthemata) are [in actual external perception] but without matter.” (Aristotle, “De Anima” III, cca. 350 BC)

“Nor is it enough to say that the intelligible notions formed by the active intellect subsist somehow in the phantasmata (mental image), which are certainly intrinsic to us; for as we have already observed in treating the passive intellect, objects only become actually intelligible when abstracted from phantasmata; so that merely by way of the phantasmata, we cannot attribute the work of the active intellect to ourselves” (St. Thomas Aquinas, “De Anima” III, cca. 1268) [On Aristotle's phantasmata]

“From all this I am beginning to have a rather better understanding of what I am. But it still appears - and I cannot stop thinking this - that the corporeal things of which images are formed in my thought, and which the senses investigate, are known with much more distinctness than this puzzling 'I' which cannot be pictured in the imagination.” (René Descartes, “Meditations” II, 1641)

“For after the object is removed, or the eye shut, we still retain an image of the thing seen, though more obscure than when we see it. And this is it the Latins call imagination, from the image made in seeing, and apply the same, though improperly, to all the other senses. But the Greeks call it fancy, which signifies appearance, and is as proper to one sense as to another. IMAGINATION, therefore, is nothing but decaying sense; and is found in men and many other living creatures, as well sleeping as waking.” (Thomas Hobbes, “Leviathan: The Matter, Form and Power of a Commonwealth  Ecclesiastical and Civil”, 1651)

“It is also only by virtue of the continual action of God upon us that we have in our soul the ideas of all things; that is to say, since every effect expresses its cause, the essence of our soul is a certain expression, imitation or image of the divine essence, thought, and will and of all the ideas which are comprised in God.“ (Gottfried W Leibniz, “Discourse on Metaphysics”, 1686)

“As in a block of marble all possible figures are potentially contained in it, and can be drawn out of it by the movement or by the action of the chisel, so in the same way all intelligible figures are potentially in intelligible extension and are discovered in it according to the different ways in which this extension is represented to the mind, as a consequence of the general laws which God has established according to which he continuously acts in us. “ (Nicolas Malebranche , “Dialogues On Metaphysics And Religion”, 1688)

“But to form the idea of an object, and to form an idea simply is the same thing; the reference of the idea to an object being an extraneous denomination, of which in itself it bears no mark or character. Now as it is impossible to form an idea of an object, that is possessed of quantity and quality, and yet is possessed of no precise degree of either; it follows, that there is an equal impossibility of forming an idea, that is not limited and confined in both these particulars. Abstract ideas are therefore in themselves individual, however they may become general in their representation. The image in the mind is only that of a particular object, though the application of it in our reasoning be the same, as if it were universal.” (David Hume, “Treatise of Human Nature”, 1738)

“While all that we have is a relation of phenomena, a mental image, as such, in juxtaposition with or soldered to a sensation, we can not as yet have assertion or denial, a truth or a falsehood. We have mere reality, which is, but does not stand for anything, and which exists, but by no possibility could be true. […] the image is not a symbol or idea. It is itself a fact, or else the facts eject it. The real, as it appears to us in perception, connects the ideal suggestion with itself, or simply expels it from the world of reality. […] you possess explicit symbols all of which are universal and on the other side you have a mind which consists of mere individual impressions and images, grouped by the laws of a mechanical attraction.” (Francis H Bradley, "Principles of Logic", 1883)

“Memory-images, it is true, cannot be directly aroused through external sense impressions, but follow them after a longer or shorter interval. Still, it is obvious that their attributes, and especially their relation to the primary ideas through direct impressions, can be most accurately be learned, not by waiting for their chance arrival, but by using such memory-ideas as may be aroused in a systematic, experimental way, through immediately preceding impressions.” (Wilhelm M Wundt, “Outlines of Psychology”, 1897)

“In perception, a knowledge forms itself slowly; in the [mental] image the knowledge is immediate. We see now that the image is a synthetic act which unites a concrete, nonimagined, knowledge to elements which are more actually representative. The image teaches nothing: it is organized exactly like the objects which do produce knowledge, but it is complete at the very moment of its appearance. […] Thus, the object presents itself in the image as having to be apprehended in a multiplicity of synthetic acts. Due to this fact, and because its content retains a sensible opacity, like a phantom, because it does not involve either essences or generating laws but only an irrational quality, it gives the impression of being an object of observation: from this point of view the image appears to be more like a perception than a concept.” (Jean-Paul Sartre, “The Psychology of Imagination”, 1940)

See also:
Mental Models I, II, III, IV, V, VII, VIII

Mental Models V

"He [Simonides] inferred that persons desiring to train this faculty [of memory] must select places and form mental images of the things they wish to remember and store those images in the places, so that the order of the places will preserve the order of the things, and the images of the things will denote the things themselves, and we shall employ the places and images respectively as a wax writing-tablet and the letters written on it." (Marcus Tullius Cicero [cited in Yates 1966], “De Oratore”, 55 BC)

“For the Mind feels those things that it conceives in understanding no less than those it has in the memory. For the eyes of the mind, by which it sees and observes things, are demonstrations [descriptions] themselves.” (Baruch Spinoza, “Ethics, Demonstrated in Geometrical Order”, 1677)

“One says of a person who has travelled much, that he has seen the world. to the knowledge of the world than just seeing it. Whoever wants to must draw up a plan beforehand and must not just regard the world senses.” (Immanuel Kant “Physische Geographie” [Physical Geography], 1802)

“Acting is the physical representation of a mental picture and the projection of an emotional concept.” (Laurette Taylor, “The Quality You Need Most”, Green Book Magazine, 1914)

“The conception of lines of force was introduced by Faraday to form a mental picture of the processes going on in the electric field. To him these lines were not mere mathematical abstractions. He ascribed to them properties that gave them a real physical significance.” (Hendrik van der Bijl, “The Thermionic Vacuum Tube and Its Applications”, 1920)

“A geometrical-physical theory as such is incapable of being directly pictured, being merely a system of concepts. But these concepts serve the purpose of bringing a multiplicity of real or imaginary sensory experiences into connection in the mind. To ‘visualise’ a theory, or bring it home to one's mind, therefore means to give a representation to that abundance of experiences for which the theory supplies the schematic arrangement” (Albert Einstein, “Geometry and Experience”, 1921)

“[…] learning consists not in stimulus-response connections but in the building up in the nervous system of sets which function like cognitive maps […] such cognitive maps may be usefully characterized as varying from a narrow strip variety to a broader comprehensive variety.” (Edward C Tolman, "Cognitive maps in rats and men", 1948)

"We never have any understanding of any subject matter except in terms of our own mental constructs of ‘things’ and ‘happenings’ of that subject matter.” (Douglas T Ross, "Structured analysis (SA): A language for communicating ideas", IEEE Transactions on Software Engineering Vol. 3 No. 1, 1977)

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

See also:
Mental Models I, II, III, IV, VI, VII, VIII

Mental Models IV (Limitations I)

“However, and conversely, our models fall far short of representing the world fully. That is why we make mistakes and why we are regularly surprised. In our heads, we can keep track of only a few variables at one time. We often draw illogical conclusions from accurate assumptions, or logical conclusions from inaccurate assumptions. Most of us, for instance, are surprised by the amount of growth an exponential process can generate. Few of us can intuit how to damp oscillations in a complex system.” (Donella H Meadows, “Limits to Growth”, 1972)

"The problem with mental models lie not in whether they are right or wrong - by definition, all models are simplifications. The problem with mental models arise when they become implicit - when they exist below the level of our awareness. “[…] models, if unexamined, limit an organization's range of actions to what is familiar and comfortable. [...] Each person's mental model focuses on different parts of the system. Each emphasizes different cause-effect chains. This makes it virtually impossible for a shared picture of the system as a whole to emerge in normal conversation." (Peter Senge, “The Fifth Discipline”, 1990)

“Mental models are the images, assumptions, and stories which we carry in our minds of ourselves, other people, institutions, and every aspect of the world. Like a pane of glass framing and subtly distorting our vision, mental models determine what we see. Human beings cannot navigate through the complex environments of our world without cognitive ‘mental maps’; and all of these mental maps, by definition, are flawed in some way.” (Peter M Senge et al, “The Fifth Discipline Fieldbook: Strategies and Tools for Building a Learning Organization”, 1994)

“What are the models? Well, the first rule is that you’ve got to have multiple models - because if you just have one or two that you’re using, the nature of human psychology is such that you’ll torture reality so that it fits your models, or at least you’ll think it does.” (Charles Munger, 1994)

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

“[…] our mental models fail to take into account the complications of the real world - at least those ways that one can see from a systems perspective. It is a warning list. Here is where hidden snags lie. You can’t navigate well in an interconnected, feedback-dominated world unless you take your eyes off short-term events and look for long-term behavior and structure; unless you are aware of false boundaries and bounded rationality; unless you take into account limiting factors, nonlinearities and delays. You are likely to mistreat, misdesign, or misread systems if you don’t respect their properties of resilience, self-organization, and hierarchy.” (Donella H Meadows, “Thinking in Systems: A Primer”, 2008)

“The discrepancy between our mental models and the real world may be a major problem of our times; especially in view of the difficulty of collecting, analyzing, and making sense of the unbelievable amount of data to which we have access today.” (Ugo Bardi, “The Limits to Growth Revisited”, 2011)

Mental Models III

"Theories usually result from the precipitate reasoning of an impatient mind which would like to be rid of phenomena and replace them with images, concepts, indeed often with mere words." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1833) 

“Everything we think we know about the world is a model. Every word and every language is a model. All maps and statistics, books and databases, equations and computer programs are models. So are the ways I picture the world in my head - my mental models. None of these is or ever will be the real world. […] Our models usually have a strong congruence with the world. That is why we are such a successful species in the biosphere. Especially complex and sophisticated are the mental models we develop from direct, intimate experience of nature, people, and organizations immediately around us.” (Donella Meadows, “Limits to Growth”, 1972)

“Concepts are inventions of the human mind used to construct a model of the world. They package reality into discrete units for further processing, they support powerful mechanisms for doing logic, and they are indispensable for precise, extended chains of reasoning. […] A mental model is a cognitive construct that describes a person's understanding of a particular content domain in the world.” (John Sown, “Conceptual Structures: Information Processing in Mind and Machine”, 1984)

“We construct mental models that provide us with situations in which we can interact with mental objects that represent objects, properties and relations and that behave in ways that simulate the objects, properties and relations that our models represent. […] The concepts and principles that a person understands, in this sense, are embedded in the kinds of objects that he or she includes in mental models and in the ways in which those objects behave, including how they combine and separate to form other objects.” (James G Greeno, “Number sense as situated knowing in a conceptual domain”, Journal for Research on Mathematics Education Vol. 22 No. 3, 1991)

“A mental model is conceived […] as a knowledge structure possessing slots that can be filled not only with empirically gained information but also with ‘default assumptions’ resulting from prior experience. These default assumptions can be substituted by updated information so that inferences based on the model can be corrected without abandoning the model as a whole. Information is assimilated to the slots of a mental model in the form of ‘frames’ which are understood here as ‘chunks’ of knowledge with a well-defined meaning anchored in a given body of shared knowledge.” (Jürgen Renn, “Before the Riemann Tensor: The Emergence of Einstein’s Double Strategy”, “The Universe of General Relativity” Ed. by A.J. Kox & Jean Eisenstaedt, 2005)

“Mental models can be literal representations of the external world (as they often are with visual imagery) or arbitrary representations (as they are with propositional, mathematical, or verbal models). In either case, they are explanatory or descriptive representations of the external world.” (Gregory J Feist, “The Psychology of Science and the Origins of the Scientific Mind”, 2006)

“Mental models reflect the beliefs, values, and assumptions that we personally hold, and they underlie our reasons for doing things the way we do.” (Kambiz E Maani & Robert Y Cavana, “Systems Methodology”, The Systems Thinker Vol. 18 No. 8, 2007)

“All models (whether mental or those turned into computer maps/models) are developed using a particular lens of what we value - what we think is important to understand, or what performance we wish to develop or improve. Although organizations can build forum models focusing on the performance measure du jour, they would be well advised to use a systemic or integral framework for what to include.” (Peggy Holman et al, “The Change Handbook”, 2007)

"[...] a model is a tool for taking decisions and any decision taken is the result of a process of reasoning that takes place within the limits of the human mind. So, models have eventually to be understood in such a way that at least some layer of the process of simulation is comprehensible by the human mind. Otherwise, we may find ourselves acting on the basis of models that we don’t understand, or no model at all.” (Ugo Bardi, “The Limits to Growth Revisited”, 2011)

See also:
Mental Models I, II, IV, V, VI, VII, VIII

Mathematical Models II

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

"It is characteristic of modern physics to represent all processes in terms of mathematical equations. But the close connection between the two sciences must not blur their essential difference." (Hans Reichenbach, "The Theory of Relativity and A Priori Knowledge", 1920)

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

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

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

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

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

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

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

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

See also:
Mathematical Models I
Models in Physics
On Models
Good Models
Models vs. Facts

On Models: Are All Models Wrong?

"[…] no models are [true] = not even the Newtonian laws. When you construct a model you leave out all the details which you, with the knowledge at your disposal, consider inessential. […] Models should not be true, but it is important that they are applicable, and whether they are applicable for any given purpose must of course be investigated. This also means that a model is never accepted finally, only on trial." (Georg Rasch, "Probabilistic Models for Some Intelligence and Attainment Tests", 1960)

"Celestial navigation is based on the premise that the Earth is the center of the universe. The premise is wrong, but the navigation works. An incorrect model can be a useful tool." (R A J Phillips, "A Day in the Life of Kelvin Throop", Analog Science Fiction and Science Fact, Vol. 73 No. 5, 1964)

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

"Since all models are wrong the scientist cannot obtain a ‘correct’ one by excessive elaboration. On the contrary following William of Occam he should seek an economical description of natural phenomena. Just as the ability to devise simple but evocative models is the signature of the great scientist so overelaboration and overparameterization is often the mark of mediocrity." (George Box, "Science and Statistics", Journal of the American Statistical Association 71, 1976)

"A model of the universe does not require faith, but a telescope. If it is wrong, it is wrong." (Paul C W Davies, "Space and Time in the Modern Universe", 1977)

"Competent scientists do not believe their own models or theories, but rather treat them as convenient fictions. […] The issue to a scientist is not whether a model is true, but rather whether there is another whose predictive power is enough better to justify movement from today's fiction to a new one." (Steve Vardeman," Comment", Journal of the American Statistical Association 82, 1987)

"The fact that [the model] is an approximation does not necessarily detract from its usefulness because models are approximations. All models are wrong, but some are useful." (George Box, 1987)

"Statistical models for data are never true. The question whether a model is true is irrelevant. A more appropriate question is whether we obtain the correct scientific conclusion if we pretend that the process under study behaves according to a particular statistical model." (Scott Zeger, "Statistical reasoning in epidemiology", American Journal of Epidemiology, 1991)

"[…] it does not seem helpful just to say that all models are wrong. The very word model implies simplification and idealization. The idea that complex physical, biological or sociological systems can be exactly described by a few formulae is patently absurd. The construction of idealized representations that capture important stable aspects of such systems is, however, a vital part of general scientific analysis and statistical models, especially substantive ones, do not seem essentially different from other kinds of model." (Sir David Cox, "Comment on ‘Model uncertainty, data mining and statistical inference’", Journal of the Royal Statistical Society, Series A 158, 1995)

"I do not know that my view is more correct; I do not even think that ‘right’ and ‘wrong’ are good categories for assessing complex mental models of external reality - for models in science are judged [as] useful or detrimental, not as true or false." (Stephen Jay Gould, "Dinosaur in a Haystack: Reflections in Natural History", 1995)

"No matter how beautiful the whole model may be, no matter how naturally it all seems to hang together now, if it disagrees with experiment, then it is wrong." (John Gribbin, "Almost Everyone’s Guide to Science", 1999)

"A model is a simplification or approximation of reality and hence will not reflect all of reality. […] Box noted that ‘all models are wrong, but some are useful’. While a model can never be ‘truth’, a model might be ranked from very useful, to useful, to somewhat useful to, finally, essentially useless." (Kenneth P Burnham & David R Anderson, "Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach" 2nd Ed., 2005)

"You might say that there’s no reason to bother with model checking since all models are false anyway. I do believe that all models are false, but for me the purpose of model checking is not to accept or reject a model, but to reveal aspects of the data that are not captured by the fitted model." (Andrew Gelman, "Some thoughts on the sociology of statistics", 2007)

"First, we affirm that all models are wrong, some of them are useful. Since a model is an abstraction of reality, and that too only from a particular perspective, they are fundamentally wrong because they are not reality. That gives no license to models that are wrongly built - after all, two wrongs don’t make a right. So usefulness, or purpose, is what determines a model’s role, given that it is correctly formed. Models therefore have teleological value even though they are ontologically erroneous." (John Boardman & Brian Sauser, "Systems Thinking: Coping with 21st Century Problems", 2008)

"In general, when building statistical models, we must not forget that the aim is to understand something about the real world. Or predict, choose an action, make a decision, summarize evidence, and so on, but always about the real world, not an abstract mathematical world: our models are not the reality - a point well made by George Box in his oft-cited remark that "all models are wrong, but some are useful". (David Hand, "Wonderful examples, but let's not close our eyes", Statistical Science 29, 2014)

"A model is a metaphor, a description of a system that helps us to reason more clearly. Like all metaphors, models are approximations, and will never account for every last detail. A useful mantra here is: all models are wrong, but some models are useful." (James G Scott, "Statistical Modeling: A Gentle Introduction", 2017)

Models with a Twist

"Every theory of the course of events in nature is necessarily based on some process of simplification and is to some extent, therefore, a fairy tale." (Sir Napier Shaw, "Manual of Meteorology", 1932)

"The best model of a cat is a cat. Preferably the same cat." (Arturo Rosenblueth, "Philosophy of Science", 1945) [also attributed to Norbert Wiener]

"A theory has only the alternative of being right or wrong. A model has a third possibility: it may be right, but irrelevant." (Manfred Eigen, 'The Origin of Biological Information' [in “The Physicists's Conception of Nature”, Ed. by Jagdish Mehra, 1973)

"Models - in contrast to those who sat for Renoir - improve with age." (Erwin Chargaff, "Heraclitean Fire", 1978)

"A model is a work of fiction." (Nancy Cartwright, 1983) 

"Old models never die; they just fade away." (Robert M Solow, "How Did Economics Get That Way and What Way Did It Get?", Daedalus, Vol. 126, No. 1, 1997) 

"There are no surprising facts, only models that are surprised by facts; and if a model is surprised by the facts, it is no credit to that model." (Eliezer S Yudkowsky, "Quantum Explanations", 2008)

"In science, as in life, it is extremely dangerous to fall in love with beautiful models." (Vijay Pande)

"It is a paradox in mathematics and physics that we have no good model for the teaching of models." (Hartley Rogers Jr)

"Classical models tell us more than we at first can know." (Karl Popper)

"What is a model? A model is like an Austrian timetable. Austrian trains are always late. A Prussian visitor asks the Austrian conductor why they bother to print timetables. The conductor replies ‘If we did not, howwould we know how late the trains are?’" (Victor F Weisskopf)

Models in Physics

“Physics is the attempt at the conceptual construction of a model of the real world and its lawful structure.” (Albert Einstein, [letter to Moritz Schlick] 1931)

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

“All great discoveries in experimental physics have been due to the intuition of men who made free use of models, which were for them not products of the imagination, but representatives of real things.” (Max Born, “Physical Reality”, Philosophical Quarterly, Vol. 3, 1953)

"Pedantry and sectarianism aside, the aim of theoretical physics is to construct mathematical models such as to enable us, from the use of knowledge gathered in a few observations, to predict by logical processes the outcomes in many other circumstances. Any logically sound theory satisfying this condition is a good theory, whether or not it be derived from 'ultimate' or 'fundamental' truth. It is as ridiculous to deride continuum physics because it is not obtained from nuclear physics as it would be to reproach it with lack of foundation in the Bible." (Clifford Truesdell & Walter Noll, "The Non-Linear Field Theories of Mechanics", 1965)

“The pre-eminence of astronomy rests on the peculiarity that it can be treated mathematically; and the progress of physics, and most recently biology, has hinged equally on finding formulations of their laws that can be displayed as mathematical models.” (Jacob Bronowski, “The Ascent of Man”, 1973)

“Here is one way to look at physics: the physicists are men looking for new interpretations of the Book of Nature. After each pedestrian period of normal science, they dream up a new model, a new picture, a new vocabulary, and then they announce that the true meaning of the Book has been discovered.” (Richard Rorty, “Philosophy as a Kind of Writing”, 1978)

“[…] the more you see how strangely Nature behaves, the harder it is to make a model that explains how even the simplest phenomena actually work. So theoretical physics has given up on that.” (Richard P Feynman, “QED: The Strange Theory of Light and Matter”, 1985)

“Physicists are all too apt to look for the wrong sorts of generalizations, to concoct theoretical models that are too neat, too powerful, and too clean. Not surprisingly, these seldom fit well with data. To produce a really good biological theory, one must try to see through the clutter produced by evolution to the basic mechanisms. What seems to physicists to be a hopelessly complicated process may have been what nature found simplest, because nature could build on what was already there.” (Francis H C Crick, “What Mad Pursuit?: A Personal View of Scientific Discovery”, 1988)

“Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe? The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing?” (Stephen W Hawking, “A Brief History of Time: From the Big Bang to Black Holes”, 1988)

“Physicists’ models are like maps: never final, never complete until they grow as large and complex as the reality they represent.” (James Gleick, “Genius”, 1992)

On Metaphors II

“The drive toward the formation of metaphors is the fundamental human drive, which one cannot for a single instant dispense with in thought, for one would thereby dispense with man himself.” (Friedrich Nietzsche, “On Truth and Lies in a Nonmoral Sense”, 1873)

“A metaphor holds a truth and an untruth, felt as inextricably bound up with each other. If one takes it as it is and gives it some sensual form, in the shape of reality, one gets dreams and art; but between these two and real, full-scale life there is a glass partition. If one analyzes it for its rational content and separates the unverifiable from the verifiable, one gets truth and knowledge but kills the feeling.” (Robert Musil, “Man Without Qualities”, 1943) 

"[...] one cannot describe reality; only give metaphors that indicate it. All human modes of description (photographic, mathematical, and literary) are metaphorical. Even the most precise scientific description of an object or movement is a tissue of metaphors." (John Fowles, “'Notes on an Unfinished Novel”, 1969) 

“A metaphor is a word used in an unfamiliar context to give us a new insight; a good metaphor moves us to see our ordinary world in an extraordinary way.” (Sallie McFague, “Speaking in Parables”, 1975)

"The essence of metaphor is understanding and experiencing one kind of thing in terms of another. […] Metaphor is pervasive in everyday life, not just in language but in thought and action. Our ordinary conceptual system, in terms of which we both think and act, is fundamentally metaphorical in nature.” (George Lakoff and Mark Johnson, Metaphors We Live By, 1980)

“Metaphors can have profound significance because, as images or figures, they allow the mind to grasp or discover unsuspected ideal and material relationships between objects.” (Giuseppe Del Re, “Cosmic Dance”, 1999)

“[metaphors] are always open to more than one interpretation. But far from being a defect this essential openness is the reason why a number of those metaphors have had a very long life and have been able to survive great changes both in science and in the social background against which they first appeared.” (Olaf Pedersen, “The Book of Nature”, 1992)

“A metaphor is not an ornament. It is an organ of perception. Through metaphors, we see the world as one thing or another.” (Neil Postman, “The End of Education: Redefining the Value of School”, 1996)

“Metaphor is evidence of the human ability to visualize the universe as a coherent organism. Proof of our capacity, not just to see one thing in another but to change the very nature of things. When a metaphor is accepted as fact, it enters groupthink, taking on an existence in the real world. [...] Metaphor is the default form of thought, providing many angles from which to literally 'see' the world." (Marcel Danesi, "Poetic Logic: The Role of Metaphor in Thought, Language, and Culture", 2004)

"Metaphor is a primary cognitive tool by which we make sense of the world." (Terry Marks-Tarlow, "Psyche's Veil: Psychotherapy, Fractals and Complexity", 2008)

Mental Models II

"’Mental models’ are deeply ingrained assumptions, generalizations, or even pictures or images that influence how we understand the world and how we take action. Very often, we are not consciously aware of our mental models or the effects they have on our behavior. […] Mental models focus on the openness needed to unearth shortcomings in our present ways of seeing the world. [...] Mental models are deeply held internal images of how the world works, images that limit us to familiar ways of thinking and acting. Very often, we are not consciously aware of our mental models or the effects they have on our behavior.” (Peter Senge, “The Fifth Discipline”, 1990)

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

"We all depend on models to interpret our everyday experiences. We interpret what we see in terms of mental models constructed on past experience and education. They are constructs that we use to understand the pattern of our experiences." (David Bartholomew, “What is Statistics?”, 1995)

“I do not know that my view is more correct; I do not even think that ‘right’ and ‘wrong’ are good categories for assessing complex mental models of external reality - for models in science are judged [as] useful or detrimental, not as true or false.” (Stephen Jay Gould, “Dinosaur in a Haystack: Reflections in Natural History”, 1995)

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

“Our mental model of the way the world works must shift from images of a clockwork, machinelike universe that is fixed and determined, to the model of a universe that is open, dynamic, interconnected, and full of living qualities.” (Joseph Jaworski, “Synchronicity: The Inner Path of Leadership”, 1996)

“Science begins with the world we have to live in, accepting its data and trying to explain its laws. From there, it moves toward the imagination: it becomes a mental construct, a model of a possible way of interpreting experience. The further it goes in this direction, the more it tends to speak the language of mathematics, which is really one of the languages of the imagination, along with literature and music.” (Northrop Frye, “The Educated Imagination”, 2002)

“Each generation builds a mental picture that reflects their own understanding of this world. They construct mental tools that penetrate more and more deeply into it, so that they can explore aspects of it that were previously hidden.” (Alain Connes, “The Princeton Companion to Mathematics”, Ed. by Timothy Gowers et al, 2008)

"When a particular image appears in the mind's eye often enough it begins to connect apparently unrelated ideas leading to models and theories. […] Patterns experienced again and again become intuitions. […] Intuitive judgments are made by our use of imagery; intuition is the result of mental model building. […] The mental model used and the form of the intuition is dependent upon the question being answered." (Roger Frantz, “Two Minds”, 2005)

“We all have mental models: the lens through which we see the world that drive our responses to everything we experience. Being aware of your mental models is key to being objective.” (Elizabeth Thornton, “Learn to Be an Objective Leader without Losing Everything”, 2015)

See also:
Mental Models I, III, IV, V, VI, VII, VIII