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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?

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I'm not satisfied with the title I gave the question. If someone has a better idea, don't hesitate to change. –  Markus Klein Mar 31 at 11:38
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Maybe change it to "who want to study mathematics linearly"? –  Zhou Fang Mar 31 at 19:48

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up vote 12 down vote accepted

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.

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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.

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Nice answer! I felt similar and then attended a course about mathematical logic, but felt very lost in the middle of the course since all the examples from abstract algebra were unknown to me. To clarify: I did not want to say that we should/or shold not let them read Bourbaki, but that was the best buzz word coming to my mind summarizing what I meant by the question (and you did answer it very good and on-topic :)) –  Markus Klein Mar 31 at 21:15
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A trivial issue: is it "Schwarz", or "Schwartz"? I hesitated to edit, not knowing exactly what you meant. –  paul garrett Mar 31 at 22:39
    
@paulgarrett Honestly, I can never remember which of "them" has that extra "t"... Do you have a trick to remember this? –  Zhou Fang Apr 1 at 2:25
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@ZhouFang: Laurent Schwartz has a 't' in both his first name and last name, while Schwarz Lemma has no 't' in either of them. –  Benoît Kloeckner Apr 1 at 9:19
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@ZhouFang My previous comment was written when I did not have enough coffee, and I just realized how silly it was. Let me try again: There's a 't' in Distribution, but no 't' in Cauchy. –  Roland Apr 1 at 13:49

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

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All excellent answers, but this is (or at least points at) the best answer by far. –  vonbrand Apr 11 at 2:33

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