" […] as a general rule, that in selecting a particular case for constructing a model the first prerequisite is regularity. By selecting a symmetrical form for the model, not only is the execution simplified, but what is of more importance, the model will be of such a character as to impress itself readily on the mind." (Felix Klein, 1893)
"To use an old analogy - and here we can hardly go except upon analogy - while the building of Nature is growing spontaneously from within, the model of it, which we seek to construct in our descriptive science, can only be constructed by means of scaffolding from without, a scaffolding of hypotheses. While in the real building all is continuous, in our model there are detached parts which must be connected with the rest by temporary ladders and passages, or which must be supported till we can see how to fill in the understructure. To give the hypotheses equal validity with facts is to confuse the temporary scaffolding with the building itself." (John H Poynting, 1899)
"[…] we can only study Nature through our senses - that is […] we can only study the model of Nature that our senses enable our minds to construct; we cannot decide whether that model, consistent though it be, represents truly the real structure of Nature; whether, indeed, there be any Nature as an ultimate reality behind its phenomena." (William C Dampier, "The Recent Development of Physical Science", 1904)
"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 sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work" (John Von Neumann, "Method in the Physical Sciences", 1955)
"General Systems Theory is a name which has come into use to describe a level of theoretical model-building which lies somewhere between the highly generalized constructions of pure mathematics and the specific theories of the specialized disciplines. Mathematics attempts to organize highly general relationships into a coherent system, a system however which does not have any necessary connections with the 'real' world around us. It studies all thinkable relationships abstracted from any concrete situation or body of empirical knowledge." (Kenneth E Boulding, "General Systems Theory - The Skeleton of Science", Management Science Vol. 2 (3), 1956)
"[a pictorial representation] is not a faithful record of a visual experience, but the faithful construction of a relational model […] Such a model can be constructed to any required degree of accuracy . What is decisive here is clearly the word 'required'. The form of a representation cannot be divorced from its purpose and the requirements of the society in which the given visual language gains currency." (Ernst H Gombrich," Art and illusion", 1960)
"In fact, the construction of mathematical models for various fragments of the real world, which is the most essential business of the applied mathematician, is nothing but an exercise in axiomatics." (Marshall Stone, cca 1960)
"[…] 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)
"[...] sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work - that is, correctly to describe phenomena from a reasonably wide area. Furthermore, it must satisfy certain aesthetic criteria - that is, in relation to how much it describes, it must be rather simple." (John von Neumann, "Method in the physical sciences", 1961)
"Cybernetics is concerned primarily with the construction of theories and models in science, without making a hard and fast distinction between the physical and the biological sciences. The theories and models occur both in symbols and in hardware, and by 'hardware’ we shall mean a machine or computer built in terms of physical or chemical, or indeed any handleable parts. Most usually we shall think of hardware as meaning electronic parts such as valves and relays. Cybernetics insists, also, on a further and rather special condition that distinguishes it from ordinary scientific theorizing: it demands a certain standard of effectiveness. In this respect it has acquired some of the same motive power that has driven research on modern logic, and this is especially true in the construction and application of artificial languages and the use of operational definitions. Always the search is for precision and effectiveness, and we must now discuss the question of effectiveness in some detail. It should be noted that when we talk in these terms we are giving pride of place to the theory of automata at the expense, at least to some extent, of feedback and information theory." (Frank H George, "The Brain As A Computer", 1962)
"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." (Northrop Frye, "The Educated Imagination", 1964)
"The usefulness of the models in constructing a testable theory of the process is severely limited by the quickly increasing number of parameters which must be estimated in order to compare the predictions of the models with empirical results" (Anatol Rapoport, "Prisoner's Dilemma: A study in conflict and cooperation", 1965)
"Knowing reality means constructing systems of transformations that correspond, more or less adequately, to reality. They are more or less isomorphic to transformations of reality. The transformational structures of which knowledge consists are not copies of the transformations in reality; they are simply possible isomorphic models among which experience can enable us to choose. Knowledge, then, is a system of transformations that become progressively adequate." (Jean Piaget, "Genetic Epistemology", 1968)
"Models are not intended to either reflect or construct a single objective reality. Rather, their purpose is to simulate some aspect of a possible reality. In NLP, for instance, it is not important whether or not a model is 'true' , but rather that it is 'useful'. In fact, all models can be perceived as symbolic or metaphoric, as opposed to reflective of reality. Whether the description being used is metaphorical or literal, the usefulness of a model depends on the degree to which it allows us to move effectively to the next step in the sequence of transformations connecting deeper structures and surface structures. Instead of 'constructing' reality, models establish a set of functions that serve as a tool or a bridge between deep structures and surface structures. It is this bridge that forms our 'understanding' of reality and allows us to generate new experiences and expressions of reality." (Richard Bandler & John Grinder, "The Structure of Magic", 1975)
"In physics it is usual to give alternative theoretical treatments of the same phenomenon. We construct different models for different purposes, with different equations to describe them. Which is the right model, which the 'true' set of equations? The question is a mistake. One model brings out some aspects of the phenomenon; a different model brings out others. Some equations give a rougher estimate for a quantity of interest, but are easier to solve. No single model serves all purposes best." (Nancy Cartwright, "How the Laws of Physics Lie", 1983)
"Physics is like that. It is important that the models we construct allow us to draw the right conclusions about the behaviour of the phenomena and their causes. But it is not essential that the models accurately describe everything that actually happens; and in general it will not be possible for them to do so, and for much the same reasons. The requirements of the theory constrain what can be literally represented. This does not mean that the right lessons cannot be drawn. Adjustments are made where literal correctness does not matter very much in order to get the correct effects where we want them; and very often, as in the staging example, one distortion is put right by another. That is why it often seems misleading to say that a particular aspect of a model is false to reality: given the other constraints that is just the way to restore the representation." (Nancy Cartwright, "How the Laws of Physics Lie", 1983)
"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)
"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)
"The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing?" (Stephen Hawking, "A Brief History of Time", 1988)
"We build mental models that represent significant aspects of our physical and social world, and we manipulate elements of those models when we think, plan, and try to explain events of that world. The ability to construct and manipulate valid models of reality provides humans with our distinctive adaptive advantage; it must be considered one of the crowning achievements of the human intellect." (Gordon H Bower & Daniel G Morrow, 1990)
"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 model is something one tries to construct when one has to describe a complicated situation. A model is therefore an approximate description of reality and invariably involves many simplifying assumptions. […] models are convenient idealisations." (Ganeschan Venkataraman, "Chandrasekhar and His Limit", 1992)
"Pedantry and sectarianism aside, the aim of theoretical physics is to construct mathematical models such as to enable us, from the use of knowledge gathered in a few observations, to predict by logical processes the outcomes in many other circumstances. Any logically sound theory satisfying this condition is a good theory, whether or not it be derived from ‘ultimate’ or ‘fundamental’ truth." (Clifford Truesdell & Walter Noll, "The Non-Linear Field Theories of Mechanics" 2nd Ed., 1992)
"[…] 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)
"The science of statistics may be described as exploring, analyzing and summarizing data; designing or choosing appropriate ways of collecting data and extracting information from them; and communicating that information. Statistics also involves constructing and testing models for describing chance phenomena. These models can be used as a basis for making inferences and drawing conclusions and, finally, perhaps for making decisions." (Fergus Daly et al, "Elements of Statistics", 1995)
"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)
"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)
"The claim that scientific models are metaphors is tied to the fact that often an analogy is exploited to construct a model about a phenomenon. [...] Scientific models appear to be, contrary to past research traditions, as central in scientific practice for describing and communicating aspects of the empirical world as metaphors are in ordinary language." (Daniela M Bailer-Jones," Models, Metaphors and Analogies", 2002)
"Fuzzy models should make good predictions even when they are asked to predict on regions that were not excited during the construction of the model. The generalization capabilities can be controlled by an appropriate initialization of the consequences (prior knowledge) and the use of the recursive least squares to improve the prior choices. The prior knowledge can be obtained from the data." (Jairo Espinosa et al, "Fuzzy Logic, Identification and Predictive Control", 2005)
"Although fiction is not fact, paradoxically we need some fictions, particularly mathematical ideas and highly idealized models, to describe, explain, and predict facts. This is not because the universe is mathematical, but because our brains invent or use refined and law-abiding fictions, not only for intellectual pleasure but also to construct conceptual models of reality." (Mario Bunge, "Chasing Reality: Strife over Realism", 2006)
"Prom the processing view, the model theory distinguishes between three different operations. In the construction phase, reasoners construct the mental model that reflects the information from the premises. In the inspection phase, this model is inspected to find new information that is not explicitly given in the premises. In most variants of the model theory, the inspection process is conceptualized as a spatial focus that scans the model to find new information not given in the premises.. In the variation phase, reasoners try to construct alternative models from the premises that refute the putative conclusion. If no such model is found, the putative conclusion is considered true." (Carsten Held et al, "Mental Models and the Mind", 2006)
"In order to understand how mathematics is applied to understanding of the real world it is convenient to subdivide it into the following three modes of functioning: model, theory, metaphor. A mathematical model describes a certain range of phenomena qualitatively or quantitatively. […] A (mathematical) metaphor, when it aspires to be a cognitive tool, postulates that some complex range of phenomena might be compared to a mathematical construction." (Yuri I Manin," Mathematics as Metaphor: Selected Essays of Yuri I. Manin", 2007)
"Just as physicists have created models of the atom based on observed data and intuitive synthesis of the patterns in their data, so must designers create models of users based on observed behaviors and intuitive synthesis of the patterns in the data. Only after we formalize such patterns can we hope to systematically construct patterns of interaction that smoothly match the behavior patterns, mental models, and goals of users. Personas provide this formalization." (Alan Cooper et al, "About Face 3: The Essentials of Interaction Design", 2007)
"Although complexity of the physical system is both intimidating and unavoidable in typical networks, for the purposes of control design it is frequently possible to construct models of reduced complexity that lead to effective control solutions for the physical system of interest. These idealized models also serve to enhance intuition regarding network behavior." (Sean Meyn, "Control Techniques for Complex Networks", 2008)
"It is impossible to construct a model that provides an entirely accurate picture of network behavior. Statistical models are almost always based on idealized assumptions, such as independent and identically distributed (i.i.d.) interarrival times, and it is often difficult to capture features such as machine breakdowns, disconnected links, scheduled repairs, or uncertainty in processing rates." (Sean Meyn, "Control Techniques for Complex Networks", 2008)
"There are actually two sides to the success of mathematics in explaining the world around us (a success that Wigner dubbed ‘the unreasonable effectiveness of mathematics’), one more astonishing than the other. First, there is an aspect one might call ‘active’. When physicists wander through nature’s labyrinth, they light their way by mathematics - the tools they use and develop, the models they construct, and the explanations they conjure are all mathematical in nature. This, on the face of it, is a miracle in itself. […] But there is also a ‘passive’ side to the mysterious effectiveness of mathematics, and it is so surprising that the 'active' aspect pales by comparison. Concepts and relations explored by mathematicians only for pure reasons - with absolutely no application in mind—turn out decades (or sometimes centuries) later to be the unexpected solutions to problems grounded in physical reality!" (Mario Livio, "Is God a Mathematician?", 2011)
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