Perturbation theory starts with simplified problems that we understand, and then gradually adds corrections to bring the calculated solution into better alignment with the features of the system under study. It’s a “guess, check, fix” approach that copes with complex computation; calculating the moon’s orbit is one example, as it is affected by competing gravitational pulls from both the sun and the earth.

Fillion and his collaborators’ approach emphasizes that this form of modelling is best explained by ‘backward error analysis’. This is where, for example, calculating square roots without exact computation takes a ‘best guess’ approach, with successive tiny adjustments made until the desired result is obtained.

Backward error analysis is then seen as more than just a method of numerical analysis for studying approximation methods used to validate algorithms during in silico modelling, but a general scientific methodology to discuss error and uncertainty.

Currently, scientific method  relies mostly on two approaches to justify claims and inferences: logic, which asks whether conclusions follow from supportive claims as part of an argument, and probability/statistics, which assesses the likelihood and confidence levels of various hypotheses. Fillion suggests that the concepts of perturbation theory constitute an additional approach. Using perturbation theory as a third pillar could give researchers and philosophers of science better insight into the ‘why’ and ‘how’ of results they obtain.

“People think that science is about exact results and certainty,” Fillion explains. “But it isn’t. There is always error and uncertainty built in since there is no perfectly isolated environment to subject to experiment and to represent with models. Instead, science’s success comes from our learning to manage these errors and uncertainty. This research project will explain how science actually works in comparison to how we think or fantasize that it works.”

Grant Details: Nicolas Fillion  – principal investigator, with Robert Corless and Chris Smeenk (University of Western Ontario) as collaborating investigators

Keywords: philosophy of science and technology | epistemology | mathematics

Please see below for Courses taught by Nic Fillion this summer