"With each theory or model, our concepts of reality and of the fundamental constituents of the universe have changed." (Stephen Hawking & Leonard Mlodinow, "The Grand Design", 2010)
"A theory is a set of deductively closed propositions that explain and predict empirical phenomena, and a model is a theory that is idealized." (Jay Odenbaugh, "True Lies: Realism, Robustness, and Models", Philosophy of Science, Vol. 78, No. 5, 2011)
"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."
"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)
"Equations have hidden powers. They reveal the innermost secrets of nature. […] The power of equations lies in the philosophically difficult correspondence between mathematics, a collective creation of human minds, and an external physical reality. Equations model deep patterns in the outside world. By learning to value equations, and to read the stories they tell, we can uncover vital features of the world around us." (Ian Stewart, "In Pursuit of the Unknown", 2012)
"Complexity has the propensity to overload systems, making the relevance of a particular piece of information not statistically significant. And when an array of mind-numbing factors is added into the equation, theory and models rarely conform to reality." (Lawrence K Samuels, "Defense of Chaos: The Chaology of Politics, Economics and Human Action", 2013)
“Models do not only describe reality, they are also instruments for exploring reality. They are not only involved in the integration of known data, but also in the discovery of new data.” (Andreas Bartels, “The Standard Model of Cosmology as a Tool for Interpretation and Discovery”, 2013)
"One good experiment is worth a thousand models […]; but one good model can make a thousand experiments unnecessary." (David Lloyd & Evgenii I Volkov, "The Ultradian Clock: Timekeeping for Intracelular Dynamics" [in "Complexity, Chaos, and Biological Evolution", Ed. by Erik Mosekilde & Lis Mosekilde, 2013)
"Science does not live with facts alone. In addition to facts, it needs models. Scientific models fulfill two main functions with respect to empirical facts." (Andreas Bartels [in "Models, Simulations, and the Reduction of Complexity", Ed. by Ulrich Gähde et al, 2013)
"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)
"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)
"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)
"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)
"Equations have hidden powers. They reveal the innermost secrets of nature. […] The power of equations lies in the philosophically difficult correspondence between mathematics, a collective creation of human minds, and an external physical reality. Equations model deep patterns in the outside world. By learning to value equations, and to read the stories they tell, we can uncover vital features of the world around us." (Ian Stewart, "In Pursuit of the Unknown", 2012)
"Complexity has the propensity to overload systems, making the relevance of a particular piece of information not statistically significant. And when an array of mind-numbing factors is added into the equation, theory and models rarely conform to reality." (Lawrence K Samuels, "Defense of Chaos: The Chaology of Politics, Economics and Human Action", 2013)
“Models do not only describe reality, they are also instruments for exploring reality. They are not only involved in the integration of known data, but also in the discovery of new data.” (Andreas Bartels, “The Standard Model of Cosmology as a Tool for Interpretation and Discovery”, 2013)
"One good experiment is worth a thousand models […]; but one good model can make a thousand experiments unnecessary." (David Lloyd & Evgenii I Volkov, "The Ultradian Clock: Timekeeping for Intracelular Dynamics" [in "Complexity, Chaos, and Biological Evolution", Ed. by Erik Mosekilde & Lis Mosekilde, 2013)
"Science does not live with facts alone. In addition to facts, it needs models. Scientific models fulfill two main functions with respect to empirical facts." (Andreas Bartels [in "Models, Simulations, and the Reduction of Complexity", Ed. by Ulrich Gähde et al, 2013)
"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)
"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)
"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)
"Things which ought to be expected can seem quite extraordinary if you’ve got the wrong model." (David Hand, "Significance", 2014)
"A model is a simplified representation of a system. It can be conceptual, verbal, diagrammatic, physical, or formal (mathematical)." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)
"Design is the process of taking something that appears in the mind’s eye, modeling it in one or more of a number of ways, predicting how that thing will behave if it is made, and then making it, sometimes modifying the design as we make it. Design is what engineering is about. Furthermore, modeling is how engineering design is done. This includes mental models, mathematical models, computer models, plans and drawings, written language, and (sometimes) physical models." (William M Bulleit, "The Engineering Way of Thinking: The Idea", Structure [magazine], 2015)
"A mathematical model is never a completely accurate representation of a physical situation - it is an idealization. A good model simplifies reality enough to permit mathematical calculations but is accurate enough to provide valuable conclusions. It is important to realize the limitations of the model. In the end, Mother Nature has the final say." (James Stewart, "Calculus: Early Transcedentals" 8th Ed., 2016)
"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)
"A mathematical model is never a completely accurate representation of a physical situation - it is an idealization. A good model simplifies reality enough to permit mathematical calculations but is accurate enough to provide valuable conclusions. It is important to realize the limitations of the model. In the end, Mother Nature has the final say." (James Stewart, "Calculus: Early Transcedentals" 8th Ed., 2016)
"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)
"A model may be defined as a substitute of any object or system. […] A mental image used in thinking is a model, and it is not the real system. A written description of a system is a model that presents one aspect of reality. The simulation model is logically complete and describes the dynamic behaviour of the system. Models can be broadly classified as (a) physical models and (b) abstract models [..] Mental models and mathematical models are examples of abstract models." (Bilash K Bala et al, "System Dynamics: Modelling and Simulation", 2017)
"An all-inclusive model would be like the map in the famous story by Borges - perfect and inclusive because it was identical to the territory it was mapping." (Reuben Hersh,” Mathematics as an Empirical Phenomenon, Subject to Modeling”, 2017)
"Different models are both competitive and complementary. Their standing will depend on their benefits in practice. If philosophy of mathematics were seen as modeling rather than as taking positions, it might consider paying attention to mathematics research and mathematics teaching as testing grounds for its models." (Reuben Hersh, ”Mathematics as an Empirical Phenomenon, Subject to Modeling", 2017)
"An all-inclusive model would be like the map in the famous story by Borges - perfect and inclusive because it was identical to the territory it was mapping." (Reuben Hersh,” Mathematics as an Empirical Phenomenon, Subject to Modeling”, 2017)
"Different models are both competitive and complementary. Their standing will depend on their benefits in practice. If philosophy of mathematics were seen as modeling rather than as taking positions, it might consider paying attention to mathematics research and mathematics teaching as testing grounds for its models." (Reuben Hersh, ”Mathematics as an Empirical Phenomenon, Subject to Modeling", 2017)
"For a scientist, a model is useful if it generates insight into the structure of [the] real system, makes correct prediction and stimulates meaningful questions for future research. For the public and political leaders, a model is useful if it explains the causes of important problems and provides a basis for designing policy to improve the behaviour of the system. Validity meaning confidence in a model’s usefulness is inherently relative concepts. One must choose between competing models." (Bilash K Bala et al, "System Dynamics: Modelling and Simulation", 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)
"The crucial concept that brings all of this together is one that is perhaps as rich and suggestive as that of a paradigm: the concept of a model. Some models are concrete, others are abstract. Certain models are fairly rigid; others are left somewhat unspecified. Some models are fully integrated into larger theories; others, or so the story goes, have a life of their own. Models of experiment, models of data, models in simulations, archeological modeling, diagrammatic reasoning, abductive inferences; it is difficult to imagine an area of scientific investigation, or established strategies of research, in which models are not present in some form or another. However, models are ultimately understood, there is no doubt that they play key roles in multiple areas of the sciences, engineering, and mathematics, just as models are central to our understanding of the practices of these fields, their history and the plethora of philosophical, conceptual, logical, and cognitive issues they raise. "(Otávio Bueno, [in" Springer Handbook of Model-Based Science", Ed. by Lorenzo Magnani & Tommaso Bertolotti, 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)
"The crucial concept that brings all of this together is one that is perhaps as rich and suggestive as that of a paradigm: the concept of a model. Some models are concrete, others are abstract. Certain models are fairly rigid; others are left somewhat unspecified. Some models are fully integrated into larger theories; others, or so the story goes, have a life of their own. Models of experiment, models of data, models in simulations, archeological modeling, diagrammatic reasoning, abductive inferences; it is difficult to imagine an area of scientific investigation, or established strategies of research, in which models are not present in some form or another. However, models are ultimately understood, there is no doubt that they play key roles in multiple areas of the sciences, engineering, and mathematics, just as models are central to our understanding of the practices of these fields, their history and the plethora of philosophical, conceptual, logical, and cognitive issues they raise. "(Otávio Bueno, [in" Springer Handbook of Model-Based Science", Ed. by Lorenzo Magnani & Tommaso Bertolotti, 2017])
"There is nothing in either physical or social science about which we have perfect knowledge and information. We can never say that a model is a perfect representation of the reality. On the other hand, we can say that there is nothing of which we know absolutely nothing. So, models should not be judged on an absolute scale but on relative scale if the models clarify our knowledge and provide insights into systems." (Bilash K Bala et al, "System Dynamics: Modelling and Simulation", 2017)
"The different classes of models have a lot to learn from each other, but the goal of full integration has proven counterproductive. No model can be all things to all people." (Olivier Blanchard, "On the future of macroeconomic models", Oxford Review of Economic Policy Vol. 34 (1–2), 2018)
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