Approximation Methods in the Sciences

Approximation methods, such as perturbative expansions, Hartree-Fock methods, and numerical integration techniques, play a central role in the practice of many areas of mathematized science. Yet, where philosophers of science had much to say about theories and models, approximation methods have received comparatively little attention. In many contexts, approximation methods essentially contribute to the representational and predictive capacities of theories and models, and thus need to be integrated into a complete account of these functions. At the same time, approximation methods are often detachable from particular models, being transferred between different areas of applied mathematics, raising new questions in the debate about analogies in science. Furthermore, focusing on approximation methods can provide a new way of framing historical and interpretive questions about particular theories and models. In the case of quantum field theory, for instance, the dominance of weak coupling perturbative expansions has had a major impact on the evolution and interpretation of the theory, with central notions like that of intermediate virtual particles, arguably being tied to the perturbative approximation scheme. This topical collection brings together new historical and philosophical perspectives on the role played by approximation methods in the sciences.

Questions:

How should the notions of approximation and approximation method be defined, and how do they relate to neighbouring notions such as theory, model and idealization?

How has the use of particular approximations influenced the development of particular areas of applied mathematics? For instance, how has the dominance of perturbative expansions impacted on the historical development and physical interpretation of quantum field theory?

How does the role of approximation methods change in different historical periods and in different areas of mathematized science?

How should the transfer of approximation methods between different models and scientific fields be understood, and what factors condition these transfer events?

How do computer assisted techniques like numerical integration fit into the literature on simulation, and how do they relate to more traditional mathematical approximation methods?