On Ideals

"So completely is nature mathematical that some of the more exact natural sciences, in particular astronomy and physics, are in their theoretic phases largely mathematical in character, while other sciences which have hitherto been compelled by the complexity of their phenomena and the inexactitude of their data to remain descriptive and empirical, are developing towards the mathematical ideal, proceeding upon the fundamental assumption that mathematical relations exist between the forces and the phenomena, and that nothing short, of the discovery and formulations of these relations would constitute definitive knowledge of the subject. Progress is measured by the closeness of the approximation to this ideal formulation." (Jacob W A Young, "The Teaching of Mathematics", 1907)

"Much of the skill of the true mathematical physicist and of the mathematical astronomer consists in the power of adapting methods and results carried out on an exact mathematical basis to obtain approximations sufficient for the purposes of physical measurements. It might perhaps be thought that a scheme of Mathematics on a frankly approximative basis would be sufficient for all the practical purposes of application in Physics, Engineering Science, and Astronomy, and no doubt it would be possible to develop, to some extent at least, a species of Mathematics on these lines. Such a system would, however, involve an intolerable awkwardness and prolixity in the statements of results, especially in view of the fact that the degree of approximation necessary for various purposes is very different, and thus that unassigned grades of approximation would have to be provided for. Moreover, the mathematician working on these lines would be cut off from the chief sources of inspiration, the ideals of exactitude and logical rigour, as well as from one of his most indispensable guides to discovery, symmetry, and permanence of mathematical form. The history of the actual movements of mathematical thought through the centuries shows that these ideals are the very life-blood of the science, and warrants the conclusion that a constant striving toward their attainment is an absolutely essential condition of vigorous growth. These ideals have their roots in irresistible impulses and deep-seated needs of the human mind, manifested in its efforts to introduce intelligibility in certain great domains of the world of thought." (Ernest W Hobson, [address] 1910)

"The belief that mathematics, because it is abstract, because it is static and cold and gray, is detached from life, is a mistaken belief. Mathematics, even in its purest and most abstract estate, is not detached from life. It is just the ideal handling of the problems of life, as sculpture may idealize a human figure or as poetry or painting may idealize a figure or a scene. Mathematics is precisely the ideal handling of the problems of life, and the central ideas of the science, the great concepts about which its stately doctrines have been built up, are precisely the chief ideas with which life must always deal and which, as it tumbles and rolls about them through time and space, give it its interests and problems, and its order and rationality. " (Cassius J Keyser, "The Humanization of the Teaching of Mathematics", 1912)

"The ideal of thought is rigor; mathematics is the name that usage employs to designate, not attainment of the ideal, for it cannot be attained, but its devoted pursuit and close approximation." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)

"The rigor of mathematics is not absolute - absolute rigor is an ideal, to be, like other ideals, aspired unto, forever approached, but never quite attained, for such attainment would mean that every possibility of error or indetermination, however slight, had been eliminated from idea, from symbol, and from argumentation." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)

"Absolute certainty is a privilege of uneducated minds and fanatics. - It is, for scientific folk, an unattainable ideal." (Cassius J Keyser, "Mathematical Philosophy: A Study of Fate and Freedom", 1922)

"The sciences bring into play the imagination, the building of images in which the reality, of the past is blended with the ideals for the future, and from the picture there springs the prescience of genius." (William J Mayo, "Contributions of Pure Science to Progressive Medicine", The Journal of the American Medical Association Vol. 84 (20), 1925)

On Idealization: Systems Theory

"It is difficult, however, to learn all these things from situations such as occur in everyday life. What we need is a series of abstract and quite impersonal situations to argue about in which one side is surely right and the other surely wrong. The best source of such situations for our purposes is geometry. Consequently we shall study geometric situations in order to get practice in straight thinking and logical argument, and in order to see how it is possible to arrange all the ideas associated with a given subject in a coherent, logical system that is free from contradictions. That is, we shall regard the proof of each proposition of geometry as an example of correct method in argumentation, and shall come to regard geometry as our ideal of an abstract logical system. Later, when we have acquired some skill in abstract reasoning, we shall try to see how much of this skill we can apply to problems from real life." (George D Birkhoff & Ralph Beately, "Basic Geometry", 1940)

"A material model is the representation of a complex system by a system which is assumed simpler and which is also assumed to have some properties similar to those selected for study in the original complex system. A formal model is a symbolic assertion in logical terms of an idealised relatively simple situation sharing the structural properties of the original factual system." (Arturo Rosenblueth & Norbert Wiener, "The Role of Models in Science", Philosophy of Science Vol. 12 (4), 1945)

"To call a model an idealization is to suggest that the model is a simplification of what occurs in reality, usually a simplification which omits some relevant features, such as the extended mass of the planets or, in the example of the circuit model, the resistance in the bypass capacitor. Sometimes the omitted factors make only an insignificant contribution to the effect under study. But that does not seem to be essential to idealizations, especially to the idealizations that in the end are applied by engineers to study real things. In calling something an idealization it seems not so important that the contributions from omitted factors be small, but that they be ones for which we know how to correct. If the idealization is to be of use, when the time comes to apply it to a real system we had better know how to add back the contributions of the factors that have been left out. In that case the use of idealizations does not seem to counter realism: either the omitted factors do not matter much, or in principle we know how to treat them." (Nancy Cartwright, "How the Laws of Physics Lie", 1983)

"[…] if a system is sufficiently complicated, the time it takes to return near a state already visited is huge (think of the hundred fleas on the checkerboard). Therefore if you look at the system for a moderate amount of time, eternal return is irrelevant, and you had better choose another idealization." (David Ruelle, "Chance and Chaos", 1991)

"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 first important remark is that nature gives us mathematical hints. […] A second important remark is that mathematical physics deals with idealized systems. […] The third important remark is that nature may hint at a theorem but does not state clearly under which conditions is true." (David Ruelle, "The Mathematician's Brain", 2007)

"Abstract formulations of simply stated concrete ideas are often the result of efforts to create idealized models of complex systems. The models are 'idealized' in the sense that they retain only the most fundamental properties of the original systems. The vocabulary is chosen to be as inclusive as possible so that research into the model reveals facts about a wide variety of similar systems. Unfortunately, it is often the case that over time the connection between a model and the systems on which it was based is lost, and the interested reader is faced with something that looks as if it were created to be deliberately complicated - deliberately confusing - but the original intention was just the opposite. Often, the model was devised to be simpler and more transparent than any of the systems on which it was based." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"Unanticipated forms of collective behavior emerge that are not obvious from the properties of the individuals themselves. All the models are extremely simplified, of course, but that's the point. If even their idealized behavior can surprise us, we may find clues about what to expect in the real thing. […] the collective dynamics of a crowd can be exquisitely sensitive to its composition, which may be one reason why mobs are so unpredictable, which may be one reason why mobs are so unpredictable." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)

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

On Idealization: Science

"The belief that mathematics, because it is abstract, because it is static and cold and gray, is detached from life, is a mistaken belief. Mathematics, even in its purest and most abstract estate, is not detached from life. It is just the ideal handling of the problems of life, as sculpture may idealize a human figure or as poetry or painting may idealize a figure or a scene. Mathematics is precisely the ideal handling of the problems of life, and the central ideas of the science, the great concepts about which its stately doctrines have been built up, are precisely the chief ideas with which life must always deal and which, as it tumbles and rolls about them through time and space, give it its interests and problems, and its order and rationality. " (Cassius J Keyser, "The Humanization of the Teaching of Mathematics", 1912)

"Science is reduction. Mathematics is its ideal, its form par excellence, for it is in mathematics that assimilation, identification, is most perfectly realized. The universe, scientifically explained, would be a certain formula, one and eternal, regarded as the equivalent of the entire diversity and movement of things." (Émile Boutroux, "Natural law in Science and Philosophy", 1914)

"Scientific models have all these connotations. They are representations of states, objects, and events. They are idealized in the sense that they are less complicated than reality and hence easier to use for research purposes. These models are easier to manipulate and 'carry' than the real thing. The simplicity of models, compared with reality, lies in the fact that only the relevant properties of reality are represented." (Russell L Ackoff, "Scientific Method: optimizing applied research decisions", 1962) 

"We realize, however, that all scientific laws merely represent abstractions and idealizations expressing certain aspects of reality. Every science means a schematized picture of reality, in the sense that a certain conceptual construct is unequivocally related to certain features of order in reality […]" (Ludwig von Bertalanffy, "General System Theory", 1968)

"The idea of approximation to the truth is, in my view, one of the most important ideas in the theory of science. [...] The idea of approximation to the truth - like the idea of truth as are gulative principle - presupposes a realistic view of the world. It does not presuppose that reality is as our scientific theories describe it; but it does presuppose that there is a reality and that we and our theories - which are ideas we have ourselves created and are therefore always idealizations - can draw closer and closer to an adequate description of reality, if we employ the four-stage method of trial and error." (Karl R Popper, "The Logic and Evolution of Scientific Theory", [in "All Life is Problem Solving", 1999] 1972)

"[…] 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)

"When scientists need to explain difficult points of theory, illustration by hypothetical example - rather than by total abstraction - works well (perhaps indispensably) as a rhetorical device. Such cases do not function as speculations in the pejorative sense - as silly stories that provide insight into complex mechanisms - but rather as idealized illustrations to exemplify a difficult point of theory." (Stephen Jay Gould, "Leonardo's Mountain of Clams and the Diet of Worms", 1998)

"Through modeling, scientists manipulate symbols with meanings to represent an environment with structure. Such manipulations take place to fulfill a human need, solve a problem, or create a product. When constructing a model, one works in the cognitive space of ideas. Models are used to encapsulate, highlight, replicate or represent patterns of events and the structures of things. Of course, no model provides an exact duplication of the subject matter being modeled. Details are hidden, features are skewed, and certain properties are emphasized. Models are abstract and idealized. As an abstraction, a model omits some features of the subject matter, while retaining only significant properties. As an idealization, a model depicts a subject's properties in a more perfect form." (Daniel Rothbart [Ed.], "Modeling: Gateway to the Unknown", 2004)

On Idealization: Truth

"There is a logic of language and a logic of mathematics. The former is supple and lifelike, it follows our experience. The latter is abstract and rigid, more ideal. The latter is perfectly necessary, perfectly reliable: the former is only sometimes reliable and hardly ever systematic. But the logic of mathematics achieves necessity at the expense of living truth, it is less real than the other, although more certain. It achieves certainty by a flight from the concrete into abstraction." (Thomas Merton, "The Secular Journal of Thomas Merton", 1959)

"The idea of approximation to the truth is, in my view, one of the most important ideas in the theory of science. [...] The idea of approximation to the truth - like the idea of truth as are gulative principle - presupposes a realistic view of the world. It does not presuppose that reality is as our scientific theories describe it; but it does presuppose that there is a reality and that we and our theories - which are ideas we have ourselves created and are therefore always idealizations - can draw closer and closer to an adequate description of reality, if we employ the four-stage method of trial and error." (Karl R Popper, "The Logic and Evolution of Scientific Theory", [in "All Life is Problem Solving", 1999] 1972)

"The introduction and gradual acceptance of concepts that have no immediate counterparts in the real world certainly forced the recognition that mathematics is a human, somewhat arbitrary creation, rather than an idealization of the realities in nature, derived solely from nature. But accompanying this recognition and indeed propelling its acceptance was a more profound discovery - mathematics is not a body of truths about nature." (Morris Kline, "Mathematical Thought from Ancient to Modern Times" Vol. III, 1972)

"The assumption of rationality has a favored position in economics. It is accorded all the methodological privileges of a self-evident truth, a reasonable idealization, a tautology, and a null hypothesis. Each of these interpretations either puts the hypothesis of rational action beyond question or places the burden of proof squarely on any alternative analysis of belief and choice. The advantage of the rational model is compounded because no other theory of judgment and decision can ever match it in scope, power, and simplicity." (Amos Tversky & Daniel Kahneman, "Rational Choice and the Framing of Decisions", The Journal of Business Vol. 59 (4), 1986)

"The reason why a 'crude', experimental approach is not adequate for determining mathematical truth lies in the nature of what mathematics is and is intended to be. Though its roots lie in the physical world, mathematics is a precise and idealized discipline. The 'points', 'lines', 'planes', and other ideal constructs of mathematics have no exact counterpart in reality. What the mathematician does is to take a totally abstract, idealized view of the world, and reason with his abstractions in an entirely precise and rigorous fashion." (Keith Devlin, "Mathematics: The New Golden Age", 1998)

On Idealization: Representation

"We may be thinking out a chain of reasoning in abstract Geometry, but if we draw a figure, as we usually must do in order to fix our ideas and prevent our attention from wandering owing to the difficulty of keeping a long chain of syllogisms in our minds, it is excusable if we are apt to forget that we are not in reality reasoning about the objects in the figure, but about objects which ore their idealizations, and of which the objects in the figure are only an imperfect representation. Even if we only visualize, we see the images of more or less gross physical objects, in which various qualities irrelevant for our specific purpose are not entirely absent, and which are at best only approximate images of those objects about which we are reasoning." (Ernest W Hobson, "Squaring the Circle", 1913)

"We realize, however, that all scientific laws merely represent abstractions and idealizations expressing certain aspects of reality. Every science means a schematized picture of reality, in the sense that a certain conceptual construct is unequivocally related to certain features of order in reality […]" (Ludwig von Bertalanffy, "General System Theory", 1968)

"For the mathematician, the physical way of thinking is merely the starting point in a process of abstraction or idealization. Instead of being a dot on a piece of paper or a particle of dust suspended in space, a point becomes, in the mathematician's ideal way of thinking, a set of numbers or coordinates. In applied mathematics we must go much further with this process because the physical problems under consideration are more complex. We first view a phenomenon in the physical way, of course, but we must then go through a process of idealization to arrive at a more abstract representation of the phenomenon which will be amenable to mathematical analysis." (Peter Lancaster, "Mathematics: Models of the Real World", 1976)

"[…] 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)

"Through modeling, scientists manipulate symbols with meanings to represent an environment with structure. Such manipulations take place to fulfill a human need, solve a problem, or create a product. When constructing a model, one works in the cognitive space of ideas. Models are used to encapsulate, highlight, replicate or represent patterns of events and the structures of things. Of course, no model provides an exact duplication of the subject matter being modeled. Details are hidden, features are skewed, and certain properties are emphasized. Models are abstract and idealized. As an abstraction, a model omits some features of the subject matter, while retaining only significant properties. As an idealization, a model depicts a subject's properties in a more perfect form." (Daniel Rothbart [Ed.], "Modeling: Gateway to the Unknown", 2004)

"The intent of these representations is to capture the relevant characteristics of reality, which may overlap but are not identical in each case. The engineer has to ascertain what those are, and then incorporate appropriate assumptions and simplifications accordingly. Two common strategies are abstraction, which involves neglecting certain aspects of reality in order to gain a better understanding of the remaining aspects; and idealization, which involves replacing a complicated and/or complex aspect of reality with a simplified version." (Jon A Schmidt, "Representation and Reality", Structure [magazine], 2015)

“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)

On Idealization: Mathematics

"The belief that mathematics, because it is abstract, because it is static and cold and gray, is detached from life, is a mistaken belief. Mathematics, even in its purest and most abstract estate, is not detached from life. It is just the ideal handling of the problems of life, as sculpture may idealize a human figure or as poetry or painting may idealize a figure or a scene. Mathematics is precisely the ideal handling of the problems of life, and the central ideas of the science, the great concepts about which its stately doctrines have been built up, are precisely the chief ideas with which life must always deal and which, as it tumbles and rolls about them through time and space, give it its interests and problems, and its order and rationality. " (Cassius J Keyser, "The Humanization of the Teaching of Mathematics", 1912)

"We may be thinking out a chain of reasoning in abstract Geometry, but if we draw a figure, as we usually must do in order to fix our ideas and prevent our attention from wandering owing to the difficulty of keeping a long chain of syllogisms in our minds, it is excusable if we are apt to forget that we are not in reality reasoning about the objects in the figure, but about objects which ore their idealizations, and of which the objects in the figure are only an imperfect representation. Even if we only visualize, we see the images of more or less gross physical objects, in which various qualities irrelevant for our specific purpose are not entirely absent, and which are at best only approximate images of those objects about which we are reasoning." (Ernest W Hobson, "Squaring the Circle", 1913)

"Science is reduction. Mathematics is its ideal, its form par excellence, for it is in mathematics that assimilation, identification, is most perfectly realized. The universe, scientifically explained, would be a certain formula, one and eternal, regarded as the equivalent of the entire diversity and movement of things." (Émile Boutroux, "Natural law in Science and Philosophy", 1914)

"It is difficult, however, to learn all these things from situations such as occur in everyday life. What we need is a series of abstract and quite impersonal situations to argue about in which one side is surely right and the other surely wrong. The best source of such situations for our purposes is geometry. Consequently we shall study geometric situations in order to get practice in straight thinking and logical argument, and in order to see how it is possible to arrange all the ideas associated with a given subject in a coherent, logical system that is free from contradictions. That is, we shall regard the proof of each proposition of geometry as an example of correct method in argumentation, and shall come to regard geometry as our ideal of an abstract logical system. Later, when we have acquired some skill in abstract reasoning, we shall try to see how much of this skill we can apply to problems from real life." (George D Birkhoff & Ralph Beately, "Basic Geometry", 1940)

"There is a logic of language and a logic of mathematics. The former is supple and lifelike, it follows our experience. The latter is abstract and rigid, more ideal. The latter is perfectly necessary, perfectly reliable: the former is only sometimes reliable and hardly ever systematic. But the logic of mathematics achieves necessity at the expense of living truth, it is less real than the other, although more certain. It achieves certainty by a flight from the concrete into abstraction." (Thomas Merton, "The Secular Journal of Thomas Merton", 1959)

"[…] mathematics is not portraying laws inherent in the design of the universe but is merely providing man-made schemes or models which we can use to deduce conclusions about our world only to the extent that the model is a good idealization." (Morris Kline, "Mathematics for the Nonmathematician", 1967)

"This distinction between regular and catastrophic points is obviously somewhat arbitrary because it depends on the fineness of the observation used. One might object, not without reason, that each point is catastrophic to sufficiently sensitive observational techniques. This is why the distinction is an idealization, to be made precise by a mathematical model, and to this end we summarize some ideas of qualitative dynamics." (René F Thom, "Structural Stability and Morphogenesis", 1972)

"For the mathematician, the physical way of thinking is merely the starting point in a process of abstraction or idealization. Instead of being a dot on a piece of paper or a particle of dust suspended in space, a point becomes, in the mathematician's ideal way of thinking, a set of numbers or coordinates. In applied mathematics we must go much further with this process because the physical problems under consideration are more complex. We first view a phenomenon in the physical way, of course, but we must then go through a process of idealization to arrive at a more abstract representation of the phenomenon which will be amenable to mathematical analysis." (Peter Lancaster, "Mathematics: Models of the Real World", 1976)

"In the limit of idealization, all of mathematics can be regarded as the collection of grammatically correct potential texts in a formal language." (Yuri I Manin, "Mathematics and Physics", 1981)

"Since geometry is the mathematical idealization of space, a natural way to organize its study is by dimension. First we have points, objects of dimension O. Then come lines and curves, which are one-dimensional objects, followed by two-dimensional surfaces, and so on. A collection of such objects from a given dimension forms what mathematicians call a 'space'. And if there is some notion enabling us to say when two objects are 'nearby' in such a space, then it's called a topological space." (John L Casti, "Five Golden Rules", 1995)

"The real numbers are one of the most audacious idealizations made by the human mind, but they were used happily for centuries before anybody worried about the logic behind them. Paradoxically, people worried a great deal about the next enlargement of the number system, even though it was entirely harmless. That was the introduction of square roots for negative numbers, and it led to the 'imaginary' and 'complex' numbers. A professional mathematican should never leave home without them […]" (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"Free will is an idealization of human beings that makes the ethics game playable. Euclidean geometry requires idealizations like infinite straight lines and perfect circles, and its deductions are sound and useful even though the world does not really have infinite straight lines or perfect circles. The world is close enough to the idealization that the theorems can usefully be applied. Similarly, ethical theory requires idealizations like free, sentient, rational, equivalent agents whose behavior is uncaused, and its conclusions can be sound and useful even though the world, as seen by science, does not really have uncaused events. As long as there is no outright coercion or gross malfunction of reasoning, the world is close enough to the idealization of free will that moral theory can meaningfully be applied to it." (Steven Pinker, "How the Mind Works", 1997)

"Models can be: formulations, abstractions, replicas, idealizations, metaphors - and combinations of these. [...] Some mathematical models have been blindly used - their presuppositions as little understood as any legal fine print one ‘agrees to’ but never reads - with faith in their trustworthiness. The very arcane nature of some of the formulations of these models might have contributed to their being given so much credence. If so, we mathematicians have an important mission to perform: to help people who wish to think through the fundamental assumptions underlying models that are couched in mathematical language, making these models intelligible, rather than (merely) formidable Delphic oracles." (Barry Mazur, "The Authority of the Incomprehensible" , 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)

"We feel we are discovering mathematics. The results are there, waiting for us. They have been inherent in the figures all along. We are not inventing them. […] we are discovering facts that already exist, that are inherent in the objects we study. Although we have creative freedom to invent the objects themselves - to create idealizations like perfect spheres and circles and cylinders - once we do, they take on lives of their own." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

On Idealization: Models

"A material model is the representation of a complex system by a system which is assumed simpler and which is also assumed to have some properties similar to those selected for study in the original complex system. A formal model is a symbolic assertion in logical terms of an idealised relatively simple situation sharing the structural properties of the original factual system." (Arturo Rosenblueth & Norbert Wiener, "The Role of Models in Science", Philosophy of Science Vol. 12 (4), 1945)

"This model will be a simplification and an idealization, and consequently a falsification. It is to be hoped that the features retained for discussion are those of greatest importance in the present state of knowledge." (Alain M Turing, "The Chemical Basis of Morphogenesis" , Philosophical Transactions of the Royal Society of London, Series B: Biological Sciences, Vol. 237 (641), 1952)

"The word model is used as a noun, adjective, and verb, and in each instance it has a slightly different connotation. As a noun 'model' is a representation in the sense in which an architect constructs a small-scale model of a building or a physicist a large-scale model of an atom. As an adjective 'model' implies a degree or perfection or idealization, as in reference to a model home, a model student, or a model husband. As a verb 'to model' means to demonstrate, to reveal, to show what a thing is like." (Russell L Ackoff, "Scientific method: optimizing applied research decisions", 1962)

"This other world is the so-called physical world image; it is merely an intellectual structure. To a certain extent it is arbitrary. It is a kind of model or idealization created in order to avoid the inaccuracy inherent in every measurement and to facilitate exact definition." (Max Planck, "The Philosophy of Physics", 1963)

"[…] mathematics is not portraying laws inherent in the design of the universe but is merely providing man-made schemes or models which we can use to deduce conclusions about our world only to the extent that the model is a good idealization." (Morris Kline, "Mathematics for the Nonmathematician", 1967)

"This distinction between regular and catastrophic points is obviously somewhat arbitrary because it depends on the fineness of the observation used. One might object, not without reason, that each point is catastrophic to sufficiently sensitive observational techniques. This is why the distinction is an idealization, to be made precise by a mathematical model, and to this end we summarize some ideas of qualitative dynamics." (René F Thom, "Structural Stability and Morphogenesis", 1972)

"To call a model an idealization is to suggest that the model is a simplification of what occurs in reality, usually a simplification which omits some relevant features, such as the extended mass of the planets or, in the example of the circuit model, the resistance in the bypass capacitor. Sometimes the omitted factors make only an insignificant contribution to the effect under study. But that does not seem to be essential to idealizations, especially to the idealizations that in the end are applied by engineers to study real things. In calling something an idealization it seems not so important that the contributions from omitted factors be small, but that they be ones for which we know how to correct. If the idealization is to be of use, when the time comes to apply it to a real system we had better know how to add back the contributions of the factors that have been left out. In that case the use of idealizations does not seem to counter realism: either the omitted factors do not matter much, or in principle we know how to treat them." (Nancy Cartwright, "How the Laws of Physics Lie", 1983)

"The assumption of rationality has a favored position in economics. It is accorded all the methodological privileges of a self-evident truth, a reasonable idealization, a tautology, and a null hypothesis. Each of these interpretations either puts the hypothesis of rational action beyond question or places the burden of proof squarely on any alternative analysis of belief and choice. The advantage of the rational model is compounded because no other theory of judgment and decision can ever match it in scope, power, and simplicity." (Amos Tversky & Daniel Kahneman, "Rational Choice and the Framing of Decisions", The Journal of Business Vol. 59 (4), 1986)

"[…] if a system is sufficiently complicated, the time it takes to return near a state already visited is huge (think of the hundred fleas on the checkerboard). Therefore if you look at the system for a moderate amount of time, eternal return is irrelevant, and you had better choose another idealization." (David Ruelle, "Chance and Chaos", 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)

"Through modeling, scientists manipulate symbols with meanings to represent an environment with structure. Such manipulations take place to fulfill a human need, solve a problem, or create a product. When constructing a model, one works in the cognitive space of ideas. Models are used to encapsulate, highlight, replicate or represent patterns of events and the structures of things. Of course, no model provides an exact duplication of the subject matter being modeled. Details are hidden, features are skewed, and certain properties are emphasized. Models are abstract and idealized. As an abstraction, a model omits some features of the subject matter, while retaining only significant properties. As an idealization, a model depicts a subject's properties in a more perfect form." (Daniel Rothbart [Ed.], "Modeling: Gateway to the Unknown", 2004)

"Abstract formulations of simply stated concrete ideas are often the result of efforts to create idealized models of complex systems. The models are 'idealized' in the sense that they retain only the most fundamental properties of the original systems. The vocabulary is chosen to be as inclusive as possible so that research into the model reveals facts about a wide variety of similar systems. Unfortunately, it is often the case that over time the connection between a model and the systems on which it was based is lost, and the interested reader is faced with something that looks as if it were created to be deliberately complicated - deliberately confusing - but the original intention was just the opposite. Often, the model was devised to be simpler and more transparent than any of the systems on which it was based." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"Models can be: formulations, abstractions, replicas, idealizations, metaphors - and combinations of these. [...] Some mathematical models have been blindly used - their presuppositions as little understood as any legal fine print one ‘agrees to’ but never reads - with faith in their trustworthiness. The very arcane nature of some of the formulations of these models might have contributed to their being given so much credence. If so, we mathematicians have an important mission to perform: to help people who wish to think through the fundamental assumptions underlying models that are couched in mathematical language, making these models intelligible, rather than (merely) formidable Delphic oracles." (Barry Mazur, "The Authority of the Incomprehensible" , 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)