"The model [of reality] takes on a life of its own, in which its future is under perpetual construction through the micro interactions of the diverse entities comprising it. The "final" form toward which it moves is not given in the model itself, nor is it being chosen from outside the model. The forms continually emerge in an unpredictable way as the system moves into the unknown. However, there is nothing mysterious or esoteric about this. What emerges does so because of the transformative cause of the process of the micro interactions, the fluctuations themselves." (Ralph D Stacey et al, "Complexity and Management: Fad or Radical Challenge to Systems Thinking?", 2000)
"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)
"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 isolates one or a few causal connections, mechanisms, or processes, to the exclusion of other contributing or interfering factors - while in the actual world, those other factors make their effects felt in what actually happens. Models may seem true in the abstract, and are false in the concrete. The key issue is about whether there is a bridge between the two, the abstract and the concrete, such that a simple model can be relied on as a source of relevantly truthful information about the complex reality." (Uskali Mäki, "Fact and Fiction in Economics: Models, Realism and Social Construction", 2002)
"Modeling involves a style of scientific thinking in which the argument is structured by the model, but in which the application is achieved via a narrative prompted by an external fact, an imagined event or question to be answered." (Uskali Mäki, "Fact and Fiction in Economics: Models, Realism and Social Construction", 2002)
"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)
"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)
"Modeling, in a general sense, refers to the establishment of a description of a system (a plant, a process, etc.) in mathematical terms, which characterizes the input-output behavior of the underlying system. To describe a physical system […] we have to use a mathematical formula or equation that can represent the system both qualitatively and quantitatively. Such a formulation is a mathematical representation, called a mathematical model, of the physical system." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)
"[…] interval mathematics and fuzzy logic together can provide a promising alternative to mathematical modeling for many physical systems that are too vague or too complicated to be described by simple and crisp mathematical formulas or equations. When interval mathematics and fuzzy logic are employed, the interval of confidence and the fuzzy membership functions are used as approximation measures, leading to the so-called fuzzy systems modeling." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)
"A model isolates one or a few causal connections, mechanisms, or processes, to the exclusion of other contributing or interfering factors - while in the actual world, those other factors make their effects felt in what actually happens. Models may seem true in the abstract, and are false in the concrete. The key issue is about whether there is a bridge between the two, the abstract and the concrete, such that a simple model can be relied on as a source of relevantly truthful information about the complex reality." (Uskali Mäki, "Fact and Fiction in Economics: Models, Realism and Social Construction", 2002)
"Modeling involves a style of scientific thinking in which the argument is structured by the model, but in which the application is achieved via a narrative prompted by an external fact, an imagined event or question to be answered." (Uskali Mäki, "Fact and Fiction in Economics: Models, Realism and Social Construction", 2002)
"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)
"Always remember that the model is not the diagram. The diagram’s purpose is to help communicate and explain the model. The code can serve as a repository of the details of the design." (Eric Evans, "Domain-Driven Design: Tackling complexity in the heart of software", 2003)
"Knowledge is encoded in models. Models are synthetic sets of rules, and pictures, and algorithms providing us with useful representations of the world of our perceptions and of their patterns." (Didier Sornette, "Why Stock Markets Crash: Critical Events in Complex Systems", 2003)
"Knowledge is encoded in models. Models are synthetic sets of rules, and pictures, and algorithms providing us with useful representations of the world of our perceptions and of their patterns." (Didier Sornette, "Why Stock Markets Crash: Critical Events in Complex Systems", 2003)
"Maps are models, and every model represents some aspect of reality or an idea that is of interest. A model is a simplification. It is an interpretation of reality that abstracts the aspects relevant to solving the problem at hand and ignores extraneous detail." (Eric Evans, "Domain-Driven Design: Tackling complexity in the heart of software", 2003)
"Useful models seldom lie on the surface. As we come to understand the domain and the needs of the application, we usually discard superficial model elements that seemed important in the beginning, or we shift their perspective. Subtle abstractions emerge that would not have occurred to us at the outset but that pierce to the heart of the matter." (Eric Evans, "Domain-Driven Design: Tackling complexity in the heart of software", 2003)
"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)
"All models are mental projections of our understanding of processes and feedbacks of systems in the real world. The general approach is that models are as good as the system upon which they are based. Models should be designed to answer specific questions and only incorporate the necessary details that are required to provide an answer." (Hördur V Haraldsson & Harald U Sverdrup, "Finding Simplicity in Complexity in Biogeochemical Modelling" [in "Environmental Modelling: Finding Simplicity in Complexity", Ed. by John Wainwright and Mark Mulligan], 2004)
"Alternative models are neither right nor wrong, just more or less useful in allowing us to operate in the world and discover more and better options for solving problems." (Andrew Weil," The Natural Mind: A Revolutionary Approach to the Drug Problem", 2004)
"The definition of a ‘good model’ is when everything inside it is visible, inspectable and testable. It can be communicated effortlessly to others. A ‘bad model’ is a model that does not meet these standards, where parts are hidden, undefined or concealed and it cannot be inspected or tested; these are often labelled black box models." (Hördur V Haraldsson & Harald U Sverdrup, "Finding Simplicity in Complexity in Biogeochemical Modelling" [in "Environmental Modelling: Finding Simplicity in Complexity", Ed. by John Wainwright and Mark Mulligan, 2004])
"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)
"We tackle a multifaceted universe one face at a time, tailoring our models and equations to fit the facts at hand. Whatever mechanical conception proves appropriate, that is the one to use. Discovering worlds within worlds, a practical observer will deal with each realm on its own terms. It is the only sensible approach to take." (Michael Munowitz, "Knowing: The Nature of Physical Law", 2005)
"A model is called a model of a theory exactly if the theory is entirely true if considered with respect to this model alone. (Figuratively: the theory would be true if this model was the whole world.)" (Martin Thomson‐Jones, "Models and the Semantic View", 2006)
"Effective models require a real world that has enough structure so that some of the details can be ignored. This implies the existence of solid and stable building blocks that encapsulate key parts of the real system’s behavior. Such building blocks provide enough separation from details to allow modeling to proceed."(John H. Miller & Scott E. Page," Complex Adaptive Systems: An Introduction to Computational Models of Social Life", 2007)
"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)
"Models need to be judged by what they eliminate as much as by what they include - like stone carving, the art is in removing what you do not need." (John H. Miller & Scott E. Page, "Complex Adaptive Systems: An Introduction to Computational Models of Social Life", 2007)
"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)
"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)
"All models are mental projections of our understanding of processes and feedbacks of systems in the real world. The general approach is that models are as good as the system upon which they are based. Models should be designed to answer specific questions and only incorporate the necessary details that are required to provide an answer." (Hördur V Haraldsson & Harald U Sverdrup, "Finding Simplicity in Complexity in Biogeochemical Modelling" [in "Environmental Modelling: Finding Simplicity in Complexity", Ed. by John Wainwright and Mark Mulligan], 2004)
"Alternative models are neither right nor wrong, just more or less useful in allowing us to operate in the world and discover more and better options for solving problems." (Andrew Weil," The Natural Mind: A Revolutionary Approach to the Drug Problem", 2004)
"The definition of a ‘good model’ is when everything inside it is visible, inspectable and testable. It can be communicated effortlessly to others. A ‘bad model’ is a model that does not meet these standards, where parts are hidden, undefined or concealed and it cannot be inspected or tested; these are often labelled black box models." (Hördur V Haraldsson & Harald U Sverdrup, "Finding Simplicity in Complexity in Biogeochemical Modelling" [in "Environmental Modelling: Finding Simplicity in Complexity", Ed. by John Wainwright and Mark Mulligan, 2004])
"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)
"We tackle a multifaceted universe one face at a time, tailoring our models and equations to fit the facts at hand. Whatever mechanical conception proves appropriate, that is the one to use. Discovering worlds within worlds, a practical observer will deal with each realm on its own terms. It is the only sensible approach to take." (Michael Munowitz, "Knowing: The Nature of Physical Law", 2005)
"A model is called a model of a theory exactly if the theory is entirely true if considered with respect to this model alone. (Figuratively: the theory would be true if this model was the whole world.)" (Martin Thomson‐Jones, "Models and the Semantic View", 2006)
"Effective models require a real world that has enough structure so that some of the details can be ignored. This implies the existence of solid and stable building blocks that encapsulate key parts of the real system’s behavior. Such building blocks provide enough separation from details to allow modeling to proceed."(John H. Miller & Scott E. Page," Complex Adaptive Systems: An Introduction to Computational Models of Social Life", 2007)
"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)
"Models need to be judged by what they eliminate as much as by what they include - like stone carving, the art is in removing what you do not need." (John H. Miller & Scott E. Page, "Complex Adaptive Systems: An Introduction to Computational Models of Social Life", 2007)
"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)
"A model is a representation in that it (or its properties) is chosen to stand for some other entity (or its properties), known as the target system. A model is a tool in that it is used in the service of particular goals or purposes; typically these purposes involve answering some limited range of questions about the target system." (Wendy S Parker, "Confirmation and Adequacy-for-Purpose in Climate Modelling", Proceedings of the Aristotelian Society, Supplementary Volumes, Vol. 83, 2009)
"In order to deal with these phenomena, we abstract from details and attempt to concentrate on the larger picture - a particular set of features of the real world or the structure that underlies the processes that lead to the observed outcomes. Models are such abstractions of reality. Models force us to face the results of the structural and dynamic assumptions that we have made in our abstractions." (Bruce Hannon and Matthias Ruth, "Dynamic Modeling of Diseases and Pests", 2009)
"In order to deal with these phenomena, we abstract from details and attempt to concentrate on the larger picture - a particular set of features of the real world or the structure that underlies the processes that lead to the observed outcomes. Models are such abstractions of reality. Models force us to face the results of the structural and dynamic assumptions that we have made in our abstractions." (Bruce Hannon and Matthias Ruth, "Dynamic Modeling of Diseases and Pests", 2009)
"Much of the recorded knowledge of physics and engineering is written in the form of mathematical models. These mathematical models form the foundations of our understanding of the universe we live in. Furthermore, nearly all of the existing technology, in one way or another, rests on these models. To the extent that we are surrounded by evidence of the technology working and being reliable, human confidence in the validity of the underlying mathematical models is all but unshakable." (Jerzy A Filar, "Mathematical Models", 2009)
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