Posted by: Jeremy Fox | May 10, 2012

On the use and care of mathematical models

A wonderful passage from Simon Levin (1975):

Most models which find their way into the pages of journals such as this one are not meant as literal descriptions of particular situations, and thus generate predictions which are not to be tested against precise population patterns. On the other hand, such models, if handled skillfully, can lead to robust qualitative predictions which do not depend critically on para- meter values and can stimulate new directions of investigation. Such is often the role of theory throughout the basic sciences.

Further, that which mathematical models give up in reality is offset by the tightness of logic and the precision characteristic of mathematics. The author of mathematical papers thus has the same responsibility for the accurate presentation of theorems and reasoning as does the experimentalist in presenting data. Those who observe mathematicians at work often wonder at their fascination with precise statements and endless refinements of theorems and results already approximately known. The experimentalist can, however, easily appreciate this pride of craft, which has a direct analogue in the care one lavishes on the presentation of data.

As one who indulges in mathematical models and analysis, I am thus especially concerned by the tendency to state and use mathematics imprecisely. Not only do folk theorems become more and more garbled as they are handed down, but also the results which such theorems engender are suspect.

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Responses

  1. I believe I’ve slowly come to the conclusion that the main source of error and confusion in scientific analysis in general comes from people not really understanding what they are doing when they engage in quantitative analyses of various types.

    I will also add that the statisticians and mathematicians haven’t helped anything by making their subjects so opaque and not at well framed with reference to the types of questionss that scientists typically are trying to answer.

  2. [...] stability concepts by Jeremy Fox. A compendium of ways to describe the properties of fixed points. On the use and care of mathematical models by Simon Levin (1975) as quoted by Jeremy [...]


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