On Idealization: Models

"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)

"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)