On Models: On Mathematical Models (2010-2019)

"In mathematical models, a bifurcation occurs when a small change made to a parameter value of a system causes a sudden qualitative or topological change in its behavior." (Dmitriy Laschov & Michael Margaliot, "Mathematical Modeling of the λ Switch: A Fuzzy Logic Approach", 2010)

"System dynamics is an approach to understanding the behaviour of over time. It deals with internal feedback loops and time delays that affect the behaviour of the entire system. It also helps the decision maker untangle the complexity of the connections between various policy variables by providing a new language and set of tools to describe. Then it does this by modeling the cause and effect relationships among these variables." (Raed M Al-Qirem & Saad G Yaseen, "Modelling a Small Firm in Jordan Using System Dynamics", 2010)

"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)

"A catastrophe is a universal unfolding of a singular function germ. The singular function germs are called organization centers of the catastrophes. [...] Catastrophe theory is concerned with the mathematical modeling of sudden changes - so called 'catastrophes' - in the behavior of natural systems, which can appear as a consequence of continuous changes of the system parameters. While in common speech the word catastrophe has a negative connotation, in mathematics it is neutral." (Werner Sanns, "Catastrophe Theory" [Mathematics of Complexity and Dynamical Systems, 2012])

"An important aspect of the global theory of dynamical systems is the stability of the orbit structure as a whole. The motivation for the corresponding theory comes from applied mathematics. Mathematical models always contain simplifying assumptions. Dominant features are modeled; supposed small disturbing forces are ignored. Thus, it is natural to ask if the qualitative structure of the set of solutions - the phase portrait - of a model would remain the same if small perturbations were included in the model. The corresponding mathematical theory is called structural stability." (Carmen Chicone, "Stability Theory of Ordinary Differential Equations" [Mathematics of Complexity and Dynamical Systems, 2012])

"Descriptive statistics are built on the assumption that we can use a single value to characterize a single property for a single universe. […] Probability theory is focused on what happens to samples drawn from a known universe. If the data happen to come from different sources, then there are multiple universes with different probability models. If you cannot answer the homogeneity question, then you will not know if you have one probability model or many. [...] Statistical inference assumes that you have a sample that is known to have come from one universe." (Donald J Wheeler, "Myths About Data Analysis", International Lean & Six Sigma Conference, 2012)

"Models do not and need not match reality in all of its aspects and details to be adequate. A mathematical model is usually developed for a specific class of target systems, and its validity is determined relative to its intended applications. A model is considered valid within its intended domain of applicability provided that its predictions in that domain fall within an acceptable range of error, specified prior to the model’s development or identification." (Zoltan Domotor, "Mathematical Models in Philosophy of Science" [Mathematics of Complexity and Dynamical Systems, 2012])

"Simplified description of a real world system in mathematical terms, e. g., by means of differential equations or other suitable mathematical structures." (Benedetto Piccoli, Andrea Tosin, "Vehicular Traffic: A Review of Continuum Mathematical Models" [Mathematics of Complexity and Dynamical Systems, 2012])

"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])

"The four questions of data analysis are the questions of description, probability, inference, and homogeneity. [...] Descriptive statistics are built on the assumption that we can use a single value to characterize a single property for a single universe. […] Probability theory is focused on what happens to samples drawn from a known universe. If the data happen to come from different sources, then there are multiple universes with different probability models.  [...] Statistical inference assumes that you have a sample that is known to have come from one universe." (Donald J Wheeler," Myths About Data Analysis", International Lean & Six Sigma Conference, 2012)

"The standard view among most theoretical physicists, engineers and economists is that mathematical models are syntactic (linguistic) items, identified with particular systems of equations or relational statements. From this perspective, the process of solving a designated system of (algebraic, difference, differential, stochastic, etc.) equations of the target system, and interpreting the particular solutions directly in the context of predictions and explanations are primary, while the mathematical structures of associated state and orbit spaces, and quantity algebras – although conceptually important, are secondary." (Zoltan Domotor, "Mathematical Models in Philosophy of Science" [Mathematics of Complexity and Dynamical Systems, 2012])

"Mathematical modeling is a mixed blessing for economics. Mathematical modeling provides real advantages in terms of precision of thought, in seeing how assumptions are linked to conclusions, in generating and communicating insights, in generalizing propositions, and in exporting knowledge from one context to another. In my opinion, these advantages are monumental, far outweighing the costs. But the costs are not zero. Mathematical modeling limits what can be tackled and what is considered legitimate inquiry. You may decide, with experience, that the sorts of models in this book do not help you understand the economic phenomena that you want to understand." (David M Kreps, "Microeconomic Foundations I: Choice and Competitive Markets", 2013) 

"Mathematical symmetry is an idealized model. However, slightly imperfect symmetry requires explanation; it’s not enough just to say ‘it’s asymmetric’." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"To put it simply, we communicate when we display a convincing pattern, and we discover when we observe deviations from our expectations. These may be explicit in terms of a mathematical model or implicit in terms of a conceptual model. How a reader interprets a graphic will depend on their expectations. If they have a lot of background knowledge, they will view the graphic differently than if they rely only on the graphic and its surrounding text." (Andrew Gelman & Antony Unwin, "Infovis and Statistical Graphics: Different Goals, Different Looks", Journal of Computational and Graphical Statistics Vol. 22(1), 2013)

"When confronted with multiple models, I find it revealing to pose the resulting uncertainty as a two-stage lottery. For the purposes of my discussion, there is no reason to distinguish unknown models from unknown parameters of a given model. I will view each parameter configuration as a distinct model. Thus a model, inclusive of its parameter values, assigns probabilities to all events or outcomes within the model’s domain. The probabilities are often expressed by shocks with known distributions and outcomes are functions of these shocks. This assignment of probabilities is what I will call risk. By contrast there may be many such potential models. Consider a two-stage lottery where in stage one we select a model and in stage two we draw an outcome using the model probabilities. Call stage one model ambiguity and stage two risk that is internal to a model." (Lars P Hansen, "Uncertainty Outside and Inside Economic Models", [Nobel lecture] 2013)

“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)

"But we also have to know that every model has its limitations. The model of natural numbers and their sums is very successful to determine the number of objects in the union of two different groups of well-distinguished objects. But as a mathematical model, the arithmetic of numbers is not generally true but only validated and confirmed for certain well-controlled situations. […] If a model makes valid predictions in many concrete cases, if it already has been applied and tested successfully in many situations, we have some right to trust in that model. By now, we believe in the model 'natural numbers and their arithmetic' and in its predictions without having to check it every time. We do not expect that the result might be wrong; hence the verification step is not needed any longer for validating the model. If the model had a flaw, it would have been eliminated already in the past." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 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)

"[…] the usefulness of mathematics is by no means limited to finite objects or to those that can be represented with a computer. Mathematical concepts depending on the idea of infinity, like real numbers and differential calculus, are useful models for certain aspects of physical reality." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

"There are several reasons why reaction-diffusion systems have been a popular choice among mathematical modelers of spatio-temporal phenomena. First, their clear separation between non-spatial and spatial dynamics makes the modeling and simulation tasks really easy. Second, limiting the spatial movement to only diffusion makes it quite straightforward to expand any existing non-spatial dynamical models into spatially distributed ones. Third, the particular structure of reaction-diffusion equations provides aneasy shortcut in the stability analysis (to be discussed in the next chapter). And finally, despite the simplicity of their mathematical form, reaction-diffusion systems can show strikingly rich, complex spatio-temporal dynamics. Because of these properties, reaction-diffusion systems have been used extensively for modeling self-organization of spatial patterns." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 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)

"Eventually, mechanical models failed too. They were duly abandoned, and replaced by much more abstract mathematical models. Compared to their predecessors, mathematical models are Spartan affairs. They consist of equations and formulas without the texture, the color, the visual detail - without the rich appeal - of their mechanical relatives. […] But what a mathematical model lacks in charm, it more than makes up for in generality and predictive power." (Hans C von Baeyer, "QBism: The future of quantum physics", 2016)

"The goal of physics is to explain the workings of the nonliving world. At first, philosophers described the properties of real objects: the wandering of planets across the night sky, the formation of ice, or the sound of a lyre. When attention turned to things that couldn’t be seen or measured so easily, physicists invented mechanical models to take the place of real things." (Hans C von Baeyer, "QBism: The future of quantum physics", 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)

"Data almost always contain uncertainty. This uncertainty may arise from selection of the items to be measured, or it may arise from variability of the measurement process. Drawing general conclusions from data is the basis for increasing knowledge about the world, and is the basis for all rational scientific inquiry. Statistical inference gives us methods and tools for doing this despite the uncertainty in the data. The methods used for analysis depend on the way the data were gathered. It is vitally important that there is a probability model explaining how the uncertainty gets into the data." (William M Bolstad & James M Curran, "Introduction to Bayesian Statistics" 3rd Ed., 2017)

"Different models serve different purposes. Setting up a model involves focusing on features of the phenomenon that are compatible with the methodology being proposed, and neglecting features that are not compatible with it. A mathematical model in applied science explicitly refrains from attempting to be a complete picture of the phenomenon being modeled." (Reuben Hersh, ”Mathematics as an Empirical Phenomenon, Subject to Modeling”, 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)

"Model-building requires much more than just technical knowledge of statistical ideas. It also requires care and judgment, and cannot be reduced to a flowchart, a table of formulas, or a tidy set of numerical summaries that wring every last drop of truth from a data set. There is almost never a single 'right' statistical model for some problem. But there are definitely such things as good models and bad models, and learning to tell the difference is important. Just remember: calling a model good or bad requires knowing both the tool and the task." (James G Scott, "Statistical Modeling: A Gentle Introduction", 2017)

"The lack of direct control means the outside factors will be affecting the data. There is a danger that the wrong conclusions could be drawn from the experiment due to these uncontrolled outside factors. The important statistical idea of randomization has been developed to deal with this possibility. The unidentified outside factors can be 'averaged out' by randomly assigning each unit to either treatment or control group. This contributes variability to the data. Statistical conclusions always have some uncertainty or error due to variability in the data. We can develop a probability model of the data variability based on the randomization used. Randomization not only reduces this uncertainty due to outside factors, it also allows us to measure the amount of uncertainty that remains using the probability model. Randomization lets us control the outside factors statistically, by averaging out their effects." (William M Bolstad & James M Curran, "Introduction to Bayesian Statistics" 3rd Ed., 2017)

"The scientific method searches for cause-and-effect relationships between an experimental variable and an outcome variable. In other words, how changing the experimental variable results in a change to the outcome variable. Scientific modeling develops mathematical models of these relationships. Both of them need to isolate the experiment from outside factors that could affect the experimental results. All outside factors that can be identified as possibly affecting the results must be controlled." (William M Bolstad & James M Curran, "Introduction to Bayesian Statistics" 3rd Ed., 2017)

"When we use algebraic notation in statistical models, the problem becomes more complicated because we cannot 'observe' a probability and know its exact number. We can only estimate probabilities on the basis of observations." (David S Salsburg, "Errors, Blunders, and Lies: How to Tell the Difference", 2017)

"Some scientists (e.g., econometricians) like to work with mathematical equations; others (e.g., hard-core statisticians) prefer a list of assumptions that ostensibly summarizes the structure of the diagram. Regardless of language, the model should depict, however qualitatively, the process that generates the data - in other words, the cause-effect forces that operate in the environment and shape the data generated." (Judea Pearl & Dana Mackenzie, "The Book of Why: The new science of cause and effect", 2018)

"A neural-network algorithm is simply a statistical procedure for classifying inputs" (such as numbers, words, pixels, or sound waves) so that these data can mapped into outputs. The process of training a neural-network model is advertised as machine learning, suggesting that neural networks function like the human mind, but neural networks estimate coefficients like other data-mining algorithms, by finding the values for which the model’s predictions are closest to the observed values, with no consideration of what is being modeled or whether the coefficients are sensible." (Gary Smith & Jay Cordes,The 9 Pitfalls of Data Science", 2019)

"Mathematicians love math and many non-mathematicians are intimidated by math. This is a lethal combination that can lead to the creation of wildly unrealistic mathematical models. [...] A good mathematical model starts with plausible assumptions and then uses mathematics to derive the implications. A bad model focuses on the math and makes whatever assumptions are needed to facilitate the math." (Gary Smith & Jay Cordes, "The 9 Pitfalls of Data Science", 2019)