Interest in the computational aspects of modeling has been steadily growing in philosophy of science. This paper aims to advance the discussion by articulating the way in which modeling and computational errors are related and by explaining the significance of error management strategies for the rational reconstruction of scientific practice. To this end, we first characterize the role and nature of modeling error in relation to a recipe for model construction known as Euler’s recipe. We then describe a general model (...) that allows us to assess the quality of numerical solutions in terms of measures of computational errors that are completely interpretable in terms of modeling error. Finally, we emphasize that this type of error analysis involves forms of perturbation analysis that go beyond the basic model-theoretical and statistical/probabilistic tools typically used to characterize the scientific method; this demands that we revise and complement our reconstructive toolbox in a way that can affect our normative image of science. (shrink)

We demonstrate via several examples how the backward error viewpoint can be used in the analysis of solutions obtained by perturbation methods. We show that this viewpoint is quite general and offers several important advantages. Perhaps the most important is that backward error analysis can be used to demonstrate the validity of the solution, however obtained and by whichever method. This includes a nontrivial safeguard against slips, blunders, or bugs in the original computation. We also demonstrate its utility in deciding (...) when to truncate an asymptotic series, improving on the well-known rule of thumb indicating truncation just prior to the smallest term. We also give an example of elimination of spurious secular terms even when genuine secularity is present in the equation. We give short expositions of several well-known perturbation methods together with computer implementations. We also give a generic backward error based method that is equivalent to iteration for regular perturbation. (shrink)

Despite being one of the most dependable methods used by applied mathematicians and engineers in handling complex systems, the finite element method commits variational crimes. This paper contextualizes the concept of variational crime within a broader account of mathematical practice by explaining the tradeoff between complexity and accuracy involved in the construction of numerical methods. We articulate two standards of accuracy used to determine whether inexact solutions are good enough and show that, despite violating the justificatory principles of one, the (...) finite element method nevertheless succeeds in obtaining its legitimacy from the other. (shrink)

Speaking from our experience as department chairs in fields in which women are traditionally underrepresented, we offer reflections and advice on how one might move beyond the chilly climate and create a warmer environment for women students and faculty members.

ACMES is a multidisciplinary conference series that focuses on epistemological and mathematical issues relating to computation in modern science. This volume includes a selection of papers presented at the 2015 and 2016 conferences held at Western University that provide an interdisciplinary outlook on modern applied mathematics that draws from theory and practice, and situates it in proper context. These papers come from leading mathematicians, computational scientists, and philosophers of science, and cover a broad collection of mathematical and philosophical topics, including (...) numerical analysis and its underlying philosophy, computer algebra, reliability and uncertainty quantification, computation and complexity theory, combinatorics, error analysis, perturbation theory, experimental mathematics, scientific epistemology, and foundations of mathematics. By bringing together contributions from researchers who approach the mathematical sciences from different perspectives, the volume will further readers' understanding of the multifaceted role of mathematics in modern science, informed by the state of the art in mathematics, scientific computing, and current modeling techniques. (shrink)