Feasible Computation in Symbolic and Numeric Integration

Two central concerns in scientific computing are the reliability and efficiency of algorithms. We introduce the term feasible computation to describe algorithms that are reliable and efficient given the contextual constraints imposed in practice. The main focus of this dissertation then, is to bring greater clarity to the forms of error introduced in computation and modeling, and in the limited context of symbolic and numeric integration, to contribute to integration algorithms that better account for error while providing results efficiently. Chapter 2 considers the problem of spurious discontinuities in the symbolic integration problem, proposing a new method to restore continuity based on a pair of unwinding numbers. Computable conditions for the unwinding numbers are specified, allowing the computation of a variety of continuous integrals. Chapter 3 introduces two structure-preserving algorithms for the symbolic-numeric integration of rational functions on exact input. A structured backward and forward error analysis for the algorithms shows that they are a posteriori backward and forward stable, with both forms of error exhibiting tolerance proportionality. Chapter 4 identifies the basic logical structure of feasible inference by presenting a logical model of stable approximate inference, illustrated by examples of modeling and numerical integration. In terms of this model it is seen that a necessary condition for the feasibility of methods of abstraction in modeling and complexity reduction in computational mathematics is the preservation of inferential structure, in a sense that is made precise. Chapter 5 identifies a robust pattern in mathematical sciences of transforming problems to make solutions feasible. It is showed that computational complexity reduction methods in computational science involve chains of such transformations. It is argued that the structured and approximate nature of such strategies indicates the need for a higher-order model of computation and a new definition of computational complexity

Structures in Real Theory Application: A Study in Feasible Epistemology

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

Idealization in Scientific Explanation

Many phenomena pose interesting “fundamental” questions for both physics and philosophy of science. Understanding and explanation often seem to require non-Galilean, essential idealizations. But idealizations are false. This fact suggests that we need to give up on the view that truth is a necessary condition for explanation