My research interests in philosophy of science centre on questions concerning the relationships between our theories and models of natural phenomena, the natural world and the methods of computing we use to connect them. From one point of view, my principal current focus, I investigate these relationships through study of the methods used in contemporary physics and applied mathematics to generate valid descriptions, predictions and explanations of the behaviour of natural phenomena. From another, I investigate the same relationships through study of the methods used historically to develop scientific theories. The combination of these approaches helps to clarify the interplay between our experiences of natural phenomena and the theories and models we develop to describe them.
My research interests in applied mathematics centre on the development of methods that incorporate assurances of valid results into the design of the algorithm. In the context of symbolic computation, my current focus, the results of a computation by a computer algebra system (CAS) sometimes fail to be general without informing the user. An example of this is symbolic integrals of continuous functions that fail to be continuous. My interest in this area is the development of methods for which the symbolic result has the same generality as the analysis dictates or that inform the user of the limitations. In the context of numerical computation, the results of computations are typically either assumed to be valid or are validated using benchmark methods. An example of this is numerical solution of differential equations, which, for individual computations in practice, rarely have efficient a posteriori validity checks. My interest in this area is the development of methods that incorporate checks of validity into to the computational method.