This post will likely make all of my math friends cringe. I asked this question early on in my grad school career, and was told that progress in mathematics comes about through proof, and that anything else is not really mathematical reasoning. Anyone who knows me professionally knows that I’m not as concerned with proofs as with concepts and evidence. It’s always preferable to be able to prove things, but most of the problems I think about are simply too complicated to fit that type of analysis.
So my question is one that I think a lot of applied mathematicians ask. If the model I’m using is too complicated to subject to rigorous theorem-proof type analysis, isn’t it possible to ask mathematical questions and make progress on them? Even though I’m working with meteorologists, and using meteorological models, the questions I’m asking are definitely math questions. For example, how do uncertainties in the initial and boundary conditions carry through to the model’s predictions? In the case of my model, it’s just not really tractable to try to answer this question rigorously, but I’m also not keen on just trying to think about it in the context of a single meteorological event. From my point of view, that’s definitely a mathematical question, not a physics question, because it’s a question about the model, rather than the processes. Naturally, there is a big impact of the answer on the amount of faith we put in any physical predictions, but fundamentally it’s still a math question.
These are really the sorts of questions that I envision myself working on throughout my career. What can we say about how much to trust existing model frameworks, and how can we improve them to make them more trustworthy, in a systematic, objective way? Put a different way, how much information about “reality” is contained in a particular model prediction/projection? Also, how does the information in each component of the modeling system combine together to give that final information tally? Do errors in each component cancel out to yield a better answer than they should (which is often the case with model tuning), or are they layering and interacting in a way that is complementary and positive?
Since the answers to these questions will never be of the theorem-proof variety for the class of models/problems I’m working on, the question I began with should be asked “How can mathematics treat empirically based truths (“accepted truths”)?” I think it would be really exciting to put together a framework onto which we can build a comprehensive theory, so that people can “do empiricism right” from a mathematical point of view, and don’t come to the wrong conclusion based on bad reasoning.
Sean M. R. Crowell 