On Models: On Mathematical Models (1990-1999)

"Scaling invariance results from the fact that homogeneous power laws lack natural scales; they do not harbor a characteristic unit (such as a unit length, a unit time, or a unit mass). Such laws are therefore also said to be scale-free or, somewhat paradoxically, 'true on all scales'. Of course, this is strictly true only for our mathematical models. A real spring will not expand linearly on all scales; it will eventually break, at some characteristic dilation length. And even Newton's law of gravitation, once properly quantized, will no doubt sprout a characteristic length." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"There are at least three (overlapping) ways that mathematics may contribute to science. The first, and perhaps the most important, is this: Since the mathematical universe of the mathematician is much larger than that of the physicist, mathematicians are able to go beyond existing frameworks and see geometrical or analytic structures unavailable to tie physicist. Instead of using the particular equations used previously to describe reality, a mathematician has at his disposal an unused world of differential equations, to be studied with no a priori constraints. New scientific phenomena, new discoveries, may thus generated. Understanding of the present knowledge may be deepened via the corresponding deductions. [...] The second way [...] has to do with the consolidation of new physical ideas. One may express this as the proof of consistency of physical theories. [...] mathematical foundations of quantum mechanics with Hilbert space, its operator theory, and corresponding differential equations. [...] The third way [...] is by describing reality in mathematical terms, or by simply constructing a mathematical model." (Steven Smale, "What is chaos?", 1990)

"Game theory can be defined as the study of mathematical models of conflict and cooperation between intelligent rational decision-makers." (Roger B Myerson, "Game Theory: Analysis of Conflict", 1991)

"Statistical models for data are never true. The question whether a model is true is irrelevant. A more appropriate question is whether we obtain the correct scientific conclusion if we pretend that the process under study behaves according to a particular statistical model." (Scott Zeger, "Statistical reasoning in epidemiology", American Journal of Epidemiology, 1991)

"The rise of statistical reasoning was a key step in the birth of many empirical sciences, especially epidemiology. The ability to focus on the aggregate behavior amidst apparently chaotic variation across autonomous individuals has dramatically increased our understanding of disease processes that affect the health of the public. Simple statistical models based upon the laws of probability provide the language for this population perspective." (Scott Zeger, "Statistical reasoning in epidemiology", American Journal of Epidemiology, 1991)

"[…] a model is a mathematical representation of the modeler's reality, a way of capturing some aspects of a particular reality within the framework of a mathematical apparatus that provides us with a means for exploring the properties of the reality mirrored in the model." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

“Mathematical modeling is about rules - the rules of reality. What distinguishes a mathematical model from, say, a poem, a song, a portrait or any other kind of ‘model’, is that the mathematical model is an image or picture of reality painted with logical symbols instead of with words, sounds or watercolors.” (John L Casti, "Reality Rules, The Fundamentals", 1992)

“Nature behaves in ways that look mathematical, but nature is not the same as mathematics. Every mathematical model makes simplifying assumptions; its conclusions are only as valid as those assumptions. The assumption of perfect symmetry is excellent as a technique for deducing the conditions under which symmetry-breaking is going to occur, the general form of the result, and the range of possible behaviour. To deduce exactly which effect is selected from this range in a practical situation, we have to know which imperfections are present” (Ian Stewart & Martin Golubitsky, “Fearful Symmetry”, 1992)

"Pedantry and sectarianism aside, the aim of theoretical physics is to construct mathematical models such as to enable us, from the use of knowledge gathered in a few observations, to predict by logical processes the outcomes in many other circumstances. Any logically sound theory satisfying this condition is a good theory, whether or not it be derived from ‘ultimate’ or ‘fundamental’ truth." (Clifford Truesdell & Walter Noll, "The Non-Linear Field Theories of Mechanics" 2nd Ed., 1992)

"Just as few concrete physical systems are strictly deterministic in their behavior, so very few are strictly linear. The great importance of linearity lies in a combination of two circumstances. First, many tangible phenomena behave approximately linearly over restricted periods of time or restricted ranges of the variables, so that useful linear mathematical models can simulate their behavior. A pendulum swinging through a small angle is a nearly linear system. Second, linear equations can be handled by a wide variety of techniques that do not work with nonlinear equations." (Edward N Lorenz, "The Essence of Chaos", 1993)

"Systems that vary deterministically as time progresses, such as mathematical models of the swinging pendulum, the rolling rock, and the breaking wave, and also systems that vary with an inconsequential amount of randomness - possibly a real pendulum, rock, or wave - are technically known as dynamical systems." (Edward N Lorenz, "The Essence of Chaos", 1993)

"When the pinball game is treated as a flow instead of a mapping, and a simple enough system of differential equations is used as a model, it may be possible to solve the equations. A complete solution will contain expressions that give the values of the variables at any given time in terms of the values at any previous time. When the times are those of consecutive strikes on a pin, the expressions will amount to nothing more than a system of difference equations, which in this case will have been derived by solving the differential equations. Thus a mapping will have been derived from a flow." (Edward N Lorenz, "The Essence of Chaos", 1993)

"An artificial neural network is an information-processing system that has certain performance characteristics in common with biological neural networks. Artificial neural networks have been developed as generalizations of mathematical models of human cognition or neural biology, based on the assumptions that: 1. Information processing occurs at many simple elements called neurons. 2. Signals are passed between neurons over connection links. 3. Each connection link has an associated weight, which, in a typical neural net, multiplies the signal transmitted. 4. Each neuron applies an activation function" (usually nonlinear) to its net input" (sum of weighted input signals) to determine its output signal." (Laurene Fausett, "Fundamentals of Neural Networks", 1994)

"All the [mathematical] models that we have considered […] have been rough and ready. They have all been obviously crude approximations, and no one supposes that they are anything more. This does not mean that they are useless - far from it - but it does mean that the answers they give to practical questions are also approximations. There is a pragmatic payoff here between the use of simple models which give good-enough answers which are good value for money, and the use of much more sophisticated models which are more powerful, but also more complex to use, perhaps requiring more advanced mathematics and the use of computers." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Chaos and catastrophe theories are among the most interesting recent developments in nonlinear modeling, and both have captured the interests of scientists in many disciplines. It is only natural that social scientists should be concerned with these theories. Linear statistical models have proven very useful in a great deal of social scientific empirical analyses, as is evidenced by how widely these models have been used for a number of decades. However, there is no apparent reason, intuitive or otherwise, as to why human behavior should be more linear than the behavior of other things, living and nonliving. Thus an intellectual movement toward nonlinear models is an appropriate evolutionary movement in social scientific thinking, if for no other reason than to expand our paradigmatic boundaries by encouraging greater flexibility in our algebraic specifications of all aspects of human life." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)

"[…] 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." (David Cox, "Comment on ‘Model uncertainty, data mining and statistical inference’", Journal of the Royal Statistical Society, Series A 158, 1995)

"Most statistical models assume error free measurement, at least of independent (predictor) variables. However, as we all know, measurements are seldom if ever perfect. Particularly when dealing with noisy data such as questionnaire responses or processes which are difficult to measure precisely, we need to pay close attention to the effects of measurement errors. Two characteristics of measurement which are particularly important in psychological measurement are reliability and validity." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995)

"Probably the most important reason that catastrophe theory received as much popular press as it did in the mid-1970s is not because of its unchallenged mathematical elegance, but because it appears to offer a coherent mathematical framework within which to talk about how discontinuous behaviors - stock market booms and busts or cellular differentiation, for instance - might emerge as the result of smooth changes in the inputs to a system, things like interest rates in a speculative market or the diffusion rate of chemicals in a developing embryo. These kinds of changes are often termed bifurcations, and playa central role in applied mathematical modeling. Catastrophe theory enables us to understand more clearly how - and why - they occur." (John L Casti, "Five Golden Rules", 1995)

"System dynamics models are not derived statistically from time-series data. Instead, they are statements about system structure and the policies that guide decisions. Models contain the assumptions being made about a system. A model is only as good as the expertise which lies behind its formulation. A good computer model is distinguished from a poor one by the degree to which it captures the essence of a system that it represents. Many other kinds of mathematical models are limited because they will not accept the multiple-feedback-loop and nonlinear nature of real systems." (Jay W Forrester, "Counterintuitive Behavior of Social Systems", 1995)

"The science of statistics may be described as exploring, analyzing and summarizing data; designing or choosing appropriate ways of collecting data and extracting information from them; and communicating that information. Statistics also involves constructing and testing models for describing chance phenomena. These models can be used as a basis for making inferences and drawing conclusions and, finally, perhaps for making decisions." (Fergus Daly et al, "Elements of Statistics", 1995)

"Set theory has a dual role in mathematics. In pure mathematics, it is the place where questions about infinity are studied. Although this is a fascinating study of permanent interest, it does not account for the importance of set theory in applied areas. There the importance stems from the fact that set theory provides an incredibly versatile toolbox for building mathematical models of various phenomena." (Jon Barwise & Lawrence Moss, "Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena", 1996)

"From a physical point of view, time is very different from space - and this difference is often built into mathematical systems that model the physical world. Sometimes, however, we can exploit the very general nature of mathematics, and from this point of view a spatial distance and an interval of time are simply the values of certain numerical quantities, or variables. The difference between space and time then becomes a matter of interpretation - the same underlying mathematics can have several different meanings." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"Mathematical models are continually invoking ideas of infinitely smooth surfaces, weightless strings, weightless beams, perfectly spherical balls, projectiles flying through airless space, gases which are perfectly compressible and liquids which are perfectly incompressible, and so on. The purpose of such simplifications is, in theory, to understand the world better despite the oversimplification, which you hope either will not matter or will be corrected when you construct a second (better) model." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Science focuses on the study of the natural world. It seeks to describe what exists. Focusing on problem finding, it studies and describes problems in its various domains. The humanities focus on understanding and discussing the human experience. In design, we focus on finding solutions and creating things and systems of value that do not yet exist.   The methods of science include controlled experiments, classification, pattern recognition, analysis, and deduction. In the humanities we apply analogy, metaphor, criticism, and (e)valuation. In design we devise alternatives, form patterns, synthesize, use conjecture, and model solutions." (Béla H Bánáthy, "Designing Social Systems in a Changing World", 1996)

"The methods of science include controlled experiments, classification, pattern recognition, analysis, and deduction. In the humanities we apply analogy, metaphor, criticism, and (e)valuation. In design we devise alternatives, form patterns, synthesize, use conjecture, and model solutions." (Béla H Bánáthy, "Designing Social Systems in a Changing World", 1996)

"Why don't the chemicals take up the fully symmetric uniform state? Because it is unstable. Any tiny lack of uniformity grows, and destroys the uniform pattern. And in the real world there are always tiny lacks of uniformity - dust motes, bubbles, even just a few molecules vibrating because of heat. (All molecules vibrate because of heat - or more accurately 'heat' is what you get when molecules vibrate - but it only takes a few of them to trigger instability.) The instability is not intuitively obvious, but it's what happens both in the real world and in mathematical models, and here we can take it as given." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"In mathematical models, usually the qualitative effects are at least partially understood. Quantitative results are often unknown. When quantitative results are known (perhaps due to precise experiments), then mathematical models are desirable in order to discover which mechanisms best account for the known data, i.e., which quantities are important and which can be ignored. In complex problems sometimes two or more effects interact. Although each by itself is qualitatively and quantitatively understood, their interaction may need mathematical analysis in order to be understood even qualitatively." (Richard Haberman, "Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow", 1998)

"One cannot underestimate the importance of good experiments in developing mathematical models. However, mathematical models are important in their own right, aside from an attempt to mimic nature. This occurs because the real world consists of many interacting processes. It may be impossible in an experiment to entirely eliminate certain undesirable effects. Furthermore one is never sure which effects may be negligible in nature. A mathematical model has an advantage in that we are able to consider only certain effects, the object being to see which effects account for given observations and which effects are immaterial." (Richard Haberman, "Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow", 1998)

"The main problem that hampered the development of neural network models was the absence of a mathematical model for adjusting the weights of neurons situated somewhere in the middle of the network, and not directly connected to the input or output." (Paul Cilliers, "Complexity and Postmodernism: Understanding Complex Systems", 1998)

"Use of the term 'model' makes it easier to keep in mind this distinction between theory and reality. By its very nature a model cannot include all the details of the reality it seeks to represent, for then it would be just as hard to comprehend and describe as the reality we want to model. At best, our model should give a reasonable picture of some small part of reality. It has to be a simple (even crude) description; and we must always be ready to scrap or improve a model if it fails in this task of accurate depiction. That having been said, old models are often still useful. The theory of relativity supersedes the Newtonian model, but all engineers use Newtonian mechanics when building bridges or motor cars, or probing the solar system." (David Stirzaker, "Probability and Random Variables: A Beginner’s Guide", 1999)