My research focuses on reliable mathematical methods for the gaining of knowledge and insight into the behaviour of natural phenomena. At one level this involves work on generating such mathematical methods in the areas of numerical and symbolic scientific computing. At another level this involves studying the processes of development of and application of mathematical methods in science to better understand both the nature and meaning of our scientific knowledge as well as its limitations.
Below are descriptions of the two main approaches I take in my research in philosophy and my current project in applied math.
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feasible scientific epistemology
An epistemology of science (view of truth, scientific knowledge and their limitations) based upon the examination of feasible, i.e., actually possible, inferential and computational processes. I argue in my philosophy dissertation that such an epistemology must be developed through the scientific study of knowledge and methods in science. Doing so will allow us to construct reliable, valid models of reasoning and methodology in contemporary scientific practice that give us better understanding of actual content of theories and models in science and insight into the actual limitations of their applicability.
rational scientific discovery
An analysis of the process of theory discovery in science that comprehends its rationality in terms of processes of conversion of empirical data (or their symbolic surrogates) into new symbolic frameworks through processes of symbolic or structural variation. I approach this problem through historical study of the development of scientific theories and by proposing a model of how theory discovery processes in science work that explains the rationality of the methods scientists actually used to discover new theories. My work in this area has resulted in two papers that take small steps toward building a fuller picture of theory discovery in science.
continuity and stability in symbolic and numeric computation
A study of scientific computation from the point of view of the omnipresence of error in mathematical models that are used to describe real-world phenomena. Reliable computational methods must produce stable results in the presence of physical, modeling and computational error. The properties computational methods must have to be reliable in this way are different in each of the cases of numerical and symbolic computation. In the case of numerics, the relevant goodness test for an algorithm is given in terms of its numerical stability. The corresponding condition in symbolics is continuity, so that the symbolic result of a computation varies continuously with a varied or perturbed input. My current research in this general area is concerned with the development of methods of symbolic integration that return a continuous integral of a function whenever the integral is continuous. In particular, I aim to ensure continuous expressions of integrals of continuous functions.