Bourbaki vs the Russian method as a lens on heuristic models

There are many approaches to teaching higher maths, but two popular ones, that are often held in contrast to each other, are the Bourbaki and Russian methods. The Bourbaki method is named after a fictional mathematician — a nom-de-plume used by a group of mostly French mathematicians in the middle of the 20th century — Nicholas Bourbaki, who is responsible for an extremely abstract and axiomatic treatment of much of modern mathematics in his encyclopedic work Éléments de mathématique. As a pedagogical method, it is very formalist and consists of building up clear and most general possible definitions for the student. Discussions of specific, concrete, and intuitive mathematical objects is avoided, or reserved for homework exercises, Instead, a focus on very general axioms that can apply to many specific structures of interest is favored.

The Russian method, in contrast, stresses specific examples and applications. The instructor gives specific, concrete, and intuitive mathematical objects and structures — say the integers — as a pedagogical examples of the abstract concept at hand — maybe rings, in this case. The student is given other specific instances of these general abstract objects as assignments — maybe some matrices, if we are looking at rings — and through exposure to many specific examples is expected to extract the formal axiomatic structure with which Bourbaki would have started. For the Russian, this overarching formalism becomes largely an afterthought; an exercise left to the reader.

As with many comparisons in education, neither method is strictly “better”. Nor should the names be taken as representative of the people that advocate for or are exposed to each method. For example, I am Russian but I feel like I learnt the majority of my maths following the Bourbaki method and was very satisfied with it. In fact, I am not sure where the ‘Russian’ in the name comes from, although I suspect it is due to V.I. Arnol’d‘s — a famous Russian mathematician from the second half of the 20th century — polemical attack on Bourbaki. Although I do not endorse Arnol’d attack, I do share his fondness for Poincaré and importance of intuition in mathematics. As you can guess from the title, in this article I will be stressing the Russian method as important to the philosophy of science and metamodeling.

I won’t be talking about science education, but about science itself. As I’ve stressed before, I think it a fool’s errand to provide a definition or categorization of the scientific method; it is particularly self-defeating here. But for the following, I will take the perspective that the scientific community, especially the theoretical branches that I work in, is engaged in the act of educating itself about the structure of reality. Reading a paper is like a lesson, I get to learn from what others have discovered. Doing research is like a worksheet: I try my hand at some concrete problems and learn something. Writing a paper is formalizing what I learned into a lesson for others. And, of course, as we try to teach, we end up learning more, so the act of writing often transforms what we learned in our ‘worksheet’.

Physics is often help up as an ideal, a field that other disciplines like biology should emulate. It seems that lots of people think that the way physics approaches science is the right way to do science — the only right way. These people are often physics undergrads, and past-Artem was among them. But present-Artem can’t agree. Physics provides a good way to do science, but not necessarily one that other fields should imitate or aspire to. The physics approach is like the Bourbaki scientific method: one among several options.

In the Bourbaki scientific method we start with a single (or a pair, if we want to keep GR and quantum separate) elegant and incredibly general theory. We proceed to understand broad general features of this theory — like conservation laws or the principle of least action. We then work within this theory as a general framework from which to generate specific models. We have reasons to believe that the specific models we generate from theory will have a good chance of matching up to reality, and so we proceed to go over the worksheets of testing them against reality. If the match fails then we first question our model: what did we leave out of the equation? what approximations did we make? And try with a variant model.

If the failure continues then we search for some fundamental general feature of our overarching theory that is being violated. Once found, we look for other failures of that feature. Once convinced that this feature does not reflect reality, we modify the framework. Usually by adding something to it.

These last steps seldom happen in modern physics, and that is why we have so much confidence in its foundations.

In biology, however, we don’t have a unified formal theory. We do have important guiding principles, like Dobzhansky’s “nothing in biology makes sense except in the light of evolution”. But such principles are not enough to generate reliable models.

Instead, we focus on building heuristic models.

As such, each model is its own adventures. We have some paradigmatic cases to which we turn for guidance and inspiration. But often models only have a formal resemblance to themselves. As researchers, we teach to and learn from each other concrete cases and not the general framework. In this way, it is like the Russian method.

The challenge here becomes not in implementing an overarching theory in the right model. Instead, the challenge is reading an overarching theory from a series of models. This means that there is less to be gained from figuring out what approximation a model makes, or what feature can be added to it. Instead, it becomes a question of identifying the essence of successful models. Of extracting that essence and formalizing it.

This is why steps 6 and 7 in the Noble Eightfold Path to Mathematical Biology are so important: analyse why model works and based on this create simplest, most general useful model. The hope is that by doing this, we can arrive at an essence which could form a unified framework for biology. And this reminds us, that steps 6 and 7 should not be restricted to our own models but to unifying whole families of vaguely related models.

In this way, we are doing the most important homework assignment of the Russian method: extracting the general structure from the particular structures we learnt in ‘class’.

I think that this embodies nicely with the naturalist’s urge to focus on the particular. But also allows for the mathematician’s urge for an elegant abstraction.

So although I might have been trained according to the Bourbaki method, I am very happy to be working in the Russian method.