Defending Chaos: An Examination and Defense of the Models Used in Chaos Theory

Dissertation, The Ohio State University (1997)

Jeffrey Koperski
Saginaw Valley State University
The indispensable role of models in science has long been recognized by philosophers. In contemporary dynamics, the models are often simply sets of equations. Bridging the gap between pure mathematics and real-world phenomenon is especially difficult when the model is chaotic. I address the charge that this bridge has not, in fact, been built and that chaos remains "just math." Although the problems discussed have become acute with the rise of modern chaos theory, their roots were recognized nearly a century ago by Pierre Duhem. The skeptical attacks are both foundational and epistemic. ;Targeting the foundations of chaos theory, the skeptic claims that the geometry instantiated by some chaotic models cannot be representative of real-world processes. Specifically, mathematical fractals are said to have more complexity than can be captured in a material world of discrete atoms. Furthermore, there is what physicists call "the problem of quantum chaos:" quantum mechanics appears to forbid full-blown chaotic evolutions. I argue that the counterfactual nature of chaotic models can be used to resolve these tensions. Moreover, I show that the skeptic cannot attack the foundations of chaos theory without calling into question unproblematic areas of mathematical science. ;The epistemological problems are driven by an essential property of chaotic dynamics, viz., sensitive dependence on initial conditions. This sensitivity confounds the confirmation schemes familiar to philosophers by precluding future predictions of the state of a system. The skeptic argues that without such predictions, the models cannot be tested. I show how researchers in chaos theory can overcome these skeptical objections via a recently developed method of model construction. This new method not only yields a decisive answer to the skeptic, but reveals a lacuna in the philosophical literature on models in science. I present a new taxonomy of models that incorporates these recent discoveries and conforms more closely to contemporary scientific practice than do previous classifications
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