# Models in Applied Math and Beyond

### A range of meanings for the word "model"

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When scientists talk about developing models as part of their research, they can refer to quite different things.  Even within industrial and applied mathematics, modeling can have a variety of meanings. To avoid confusion and clearly communicate the relevance of mathematical modeling, it's important to understand what are those different meanings.

To me, a mathematical model is a simplified description of a phenomenon that allows us to gain quantitative insight. In the best cases the model prunes out extraneous detail, leaving only the key factors that illuminate some previously puzzling behavior.

Perhaps the best example comes from Isaac Newton's masterpiece, the law of universal gravitation. Kepler had earlier proposed three laws of planetary motion motivated by data alone: e.g., he had no a-priori reason to expect orbits to be ellipses (his first law), but observations from Tycho Brahe led him to propose the shape. Newton took an idea that was debatably "in the air" at the time---the possibility of inverse square gravitational attraction---and worked carefully through the implications to show that Kepler's "curve fit" laws were actually inevitable consequences of this. (Newton also showed that much other astronomical data could all be understood via the new model.) The simplicity of the model and the depth of insight that came with it caused a revolution in our understanding of the universe.

Mathematical models can lead to new insights in many fields, not just physical systems. Alan Turing's bifurcation model for morphogenesis [1] helps us understand how body plans can be encoded in chemical interactions. Watts and Strogatz's "small world" network model [2] explains why information can travel fast on social networks even while they might remain clustered and cliquish. John Nash's "game theory" models for human economic behavior (e.g., [3]) show why and how cooperation could have evolved.

Not every model leads to revolutionary insight, but the aim of mathematical modeling (in my humble opinion) should be the same as Newton's goal: to elucidate a natural (or other) phenomenon by connecting it to ideas that are fundamental, understandable, and quantitatively testable.

This opinion puts me at odds with some others who describe their work as modeling of natural phenomena. For example: are machine learning methods that generate "nonparametric" models directly from data doing something similar? Research into machine learning is unquestionably valuable, but it does not appear to be modeling in the same sense of the word.

There are many other cases where "modeling" can take on differ meanings:

• Many biological, medical, and social scientific models propose fundamental and understandable ideas for phenomena in question, but lack the ability to make quantitative predictions that can be tested with data.These qualitative models may contain the seeds of a quantitative mathematical model.
• In some econometric, statistical, and health modeling, researchers aim to discover previously unknown correlations between variables in data. This can be extremely significant, but can still lack the connection to a fundamental quantitative mechanism (the insight) that I consider key to mathematical modeling.
• A distinction should be made between mathematical and computational modeling, which is important in a wide range of fields and goes hand in hand with mathematical modeling. Numerical experiments consisting of high accuracy simulations can offer tests of simplified intuitive models, and can provide useful predictions for engineering applications, but complicated models that throw in "everything and the kitchen sink" can also obscure key ideas rather than deliver insight.
• In medicine, a "model" may refer to a living creature used to test treatments or conduct experiments (e.g., mouse model of pancreatic cancer).
• The most popular sense of the word "model" is (I surmise from internet search) the fashion sense, with which there is not a lot of overlap or danger of confusion. Still, I can see at least two points of intersection: (1) Danica McKellar, the actress who has also worked on mathematical models for magnetic phenomena [4]; and (2) the study of draping [5], which can help us understand the folding of dresses that models like Danica wear on the red carpet.

---Danny Abrams

[1] Turing, A. M. (1952). The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of London B: Biological Sciences, 237(641), 37-72.
[2] Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of 'small-world' networks. Nature, 393(6684), 440-442.
[3] Nash, J. (1951). Non-cooperative games. Annals of mathematics, 286-295.
[4] Chayes, L., McKellar, D., & Winn, B. (1998). Percolation and Gibbs states multiplicity for ferromagnetic Ashkin-Teller models on. Journal of Physics A: Mathematical and General, 31(45), 9055.
[5] Cerda, E., Mahadevan, L., & Pasini, J. M. (2004). The elements of draping. Proceedings of the National Academy of Sciences, 101(7), 1806-1810.

Photo caption: Mathematical model? Actress and mathematical modeler Danica McKellar at the 2015 American Music Awards on November 22, 2015.   Dress shows folds of a characteristic wavelength that can be understood via mathematical modeling.   Image obtained from https://www.flickr.com/photos/disneyabc/23225679855 and used under Creative Commons license.

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