How to advise students who want to do a "Bourbaki"-style study?

Question

There are some good students who understand a lot and are very critical. Such students tend to think that they will only understand abstract algebra if they have followed a course about logic; or they think they can only understand an analysis course if they have attended a course about abstract topology, etc. - They tend to be only satisfied if they follow the structures of the books of Bourbaki where nothing is used unless if was proven before, a very strict deductive way of doing mathematics.

This is a good way of thinking and it is good to not believe something unless one has a proof for it. However, at most universities, the curriculum is not build in such a way. Mostly, you learn about the topology of $\mathbb{R}$ before attending a course about abstract topology where this is the main example everyone can deal with.

What is the advice I can give to students with that attitude?

Answer 1

I've run into a few students like this. I usually try to convey a few messages.

• It is great that you are so interested in foundations and there is absolutely a place in math for people with this perspective. Followed by a recommendation of books suited to their interests: At a variety of levels, I might include Spivak's Calculus, an intro set theory book, various Bourbaki volumes or the Stacks Project.

• Most mathematicians work at a higher level, and even mathematicians who work in foundations need to be aware of the work above them. You should spend some of your time reading work which takes a broader, less thorough view.

• Mathematical writing contains statements of theorems. You should be able to trust the theorems. It is perfectly all right to see a theorem in a reputable book and not read the proof, in order to spend time learning about how that result is used.

Answer 2

I would discuss with these students the rather healthy point of view of Terry Tao on the subject.

Summary

One can roughly divide mathematical education into three stages: The "pre-rigorous" stage, in which mathematics is taught in an informal, intuitive manner, based on examples, fuzzy notions, and hand-waving.

The "rigorous" stage, in which one is now taught that in order to do maths "properly", one needs to work and think in a much more precise and formal manner.

The "post-rigorous" stage, in which one has grown comfortable with all the rigorous foundations of one's chosen field, and is now ready to revisit and refine one's pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory.

All too often, one ends up discarding one's initial intuition and is only able to process mathematics at a formal level, thus getting stalled at the second stage of one's mathematical education.

The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition.

Answer 3

I was a bit like that in my first three years if study, then I read the first volume of Schwartz' Analysis, where he introduces the ZFC axioms. This is the point when I understood that one simply cannot make sense out of set theoretical axioms before having manipulated higher-level mathematics (here "higher-level" is to be understood in the sense of higher-level programming). The analogy with programming is quite good I think: who would first learn machine language, before learning higher-level languages?

Advanced mathematics consist in both building more complicated structure, and digging the foundations of what has already been built. Students in mathematics should realize that what they feel is well founded in their background knowledge in fact rests on a lot of maths they did not yet learned. Even the construction of the natural numbers is not as trivial as they might think, and in fact they have been used to reason from (high-level) axioms they have accepted. Accepting a Theorem because in some given course its proof is not central as its usage is not that different.

Also, the way one founds mathematics is not that important to higher-level maths. When we do mathematics, we all think in a strongly typed way while set theory is almost not typed at all (with the exception of classes vs sets). As a side note, I am very intrigued by the homotopy type theory as a strongly typed mathematical foundation, but have not have had enough time to dive in it.

As another side note, in the old debate as to which integration theory use in the first undergraduate years, my favorite answer is none at all: state the axioms of an integration theory we need for the course, and wait until Lebesgue integral is introduced. This is a healthy way to remind that we do rely on unproved thing almost all the time; better be warned about it.

At this point, I did not really answered the question. Why not let them read Bourbaki? Either they will end up hating it pretty fast and accept more easily a few result and hand-waving, or they will love it and will learn a lot from it.