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Ecologists attempt to understand the diversity of life with mathematical models. Often, mathematical models contain simplifying idealizations designed to cope with the blooming, buzzing confusion of the natural world. This strategy frequently issues in models whose predictions are inaccurate. Critics of theoretical ecology argue that only predictively accurate models are successful and contribute to the applied work of conservation biologists. Hence, they think that much of the mathematical work of ecologists is poor science. Against this view, I argue that model building is successful even when models are predictively inaccurate for at least three reasons: models allow scientists to explore the possible behaviors of ecological systems; models give scientists simplified means by which they can investigate more complex systems by determining how the more complex system deviates from the simpler model; and models give scientists conceptual frameworks through which they can conduct experiments and fieldwork. Critics often mistake the purposes of model building, and once we recognize this, we can see their complaints are unjustified. Even though models in ecology are not always accurate in their assumptions and predictions, they still contribute to successful science.

Theoretical biology and economics are remarkably similar in their reliance on mathematical models, which attempt to represent real world systems using many idealized assumptions. They are also similar in placing a great emphasis on derivational robustness of modeling results. Recently philosophers of biology and economics have argued that robustness analysis can be a method for confirmation of claims about causal mechanisms, despite the significant reliance of these models on patently false assumptions. We argue that the power of robustness analysis has been greatly exaggerated. It is best regarded as a method of discovery rather than confirmation.

Ecologist Richard Levins argues population biologists must trade-off the generality, realism, and precision of their models since biological systems are complex and our limitations are severe. Steven Orzack and Elliott Sober argue that there are cases where these model properties cannot be varied independently of one another. If this is correct, then Levins's thesis that there is a necessary trade-off between generality, precision, and realism in mathematical models in biology is false. I argue that Orzack and Sober's arguments fail since Levins's thesis concerns the pragmatic features of model building not just the formal properties of models.

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