Robert H. C. Moir

The paper discusses to what extent the conceptual issues involved in solving the simple harmonic oscillator model fit Wigner’s famous point that the applicability of mathematics borders on the miraculous. We argue that although there is ultimately nothing mysterious here, as is to be expected, a careful demonstration that this is so involves unexpected difficulties. Consequently, through the lens of this simple case we derive some insight into what is responsible for the appearance of mystery in more sophisticated examples of the Wigner problem.

The traditional approach in philosophy of using logic to reconstruct scientific theories and methods operates by presenting or representing a scientific theory or method in a specialized formal language. The logic of such languages is deductive, which makes this approach effective for those aspects of science that use deductive methods or for which deductive inference provides a good idealization. Many theories and methods in science, however, use non-deductive forms of approximation. Approximate inferences, which produce approximately correct conclusions and do so only under restricted conditions before becoming unreliable, behave in a fundamentally different way. In the interest of developing accurate models of the structure of inference methods in scientific practice, the focus of this paper, we need conceptual tools that can faithfully represent the structure and behaviour of inference in scientific practice. To this end I propose a generalization of the traditional notion of logical validity, called effective validity, that captures the form of approximate inferences typically used in applied mathematics and computational science. I provide simple examples of approximate inference in mathematical modeling to show how a logic based on effectively valid inference can directly, faithfully represent a wide variety of the forms of inference used in scientific practice. I conclude by discussing how such a generalized logic of scientific inference can provide a richer understanding of problem-solving and mathematical modeling processes.

We argue that two powerful error-theoretic concepts provide a general framework that satisfactorily accounts for key aspects of the explanation of physical patterns. This method gives an objective criterion to determine which mathematical models in a class of neighboring models are just as good as the exact one. The method also emphasizes that abstraction is essential for explanation and provides a precise conceptual framework that determines whether a given abstraction is explanatorily relevant and justified. Hence, it increases our epistemological understanding of how one should go about reconstructing scientific practices by making clear that, at a fundamental level, a key aspect of mathematical modeling consists in exactly solving nearby problems.

This thesis considers the following problem: What methods should the epistemology of science use to gain insight into the structure and behaviour of scientific knowledge and method in actual scientific practice? After arguing that the elucidation of epistemological and methodological phenomena in science requires a method that is rooted in formal methods, I consider two alternative methods for epistemology of science. One approach is the classical approaches of the syntactic and semantic views of theories. I show that typical approaches of this sort are inadequate and inaccurate in their representation of scientific knowledge by showing how they fail to account for and misrepresent important epistemological structure and behaviour in science. The other method for epistemology of science I consider is modeled on the methods used to construct valid models of natural phenomena in applied mathematics. This new epistemological method is itself a modeling method that is developed through the detailed consideration of two main examples of theory application in science: double pendulum systems and the modeling of near-Earth objects to compute probability of future Earth impact. I show that not only does this new method accurately represent actual methods used to apply theories in applied mathematics, it also reveals interesting structural and behavioural patterns in the application process and gives insight into what underlies the stability of methods of application. I therefore conclude that for epistemology of science to develop fully as a scientific discipline it must use methods from applied mathematics, not only methods from pure mathematics and metamathematics as traditional formal epistemology of science has done.