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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