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There is a fundamental tension here because a) doing this stuff properly takes a long time and b) lots of undergraduate math teaching is "service" teaching, mea
by ska 5y ago
There is a fundamental tension here because a) doing this stuff properly takes a long time and b) lots of undergraduate math teaching is "service" teaching, meant to get students up to a level where they can do the manipulations needed in other courses.
This is why there is typically separate Calculus and Real Analysis streams in undergraduate - the former is needed for all the hard sciences and engineering, the latter for math students and fellow travelers who really want to understand what is going on and be able to prove things properly.
Typically the analysis stream will start with a pretty in-depth look at what real numbers actually are (spoiler: weirder than you thought) and the implications of that. You get a much better understanding this way but you don't cover ground nearly as quickly.
- smallnamespace 5y agoKnowing and using a thing and proving it are (usually) different skills.
- ska 5y agoTrue; but so are understanding a thing well and being able to use it. I wasn't really talking about proof-based vs. not though - it's true that an analysis course will be more proof based (than Calculus) because that is also a skill you are expected to be developing. However most calculus graduates, even the ones at the top of their class, still have at best a somewhat superficial understanding of the set of real numbers and functions on it, regardless of whether or not they can regurgitate an epsilon-delta proof. This isn't a knock on calculus courses, there is an opportunity cost to the time. By the time a continuous math student hits measure theory and understands why they need a(nother) different definition of an integral, most calculus students have been happily chugging along calculating things they need, blissfully ignorant of the "problems" with the integrals they barely remember being defined.
- sidr 5y agoThe thing is, you really don't neeed to consider non-Riemann integrable functions if you're actually trying to integrate or measure things. At some point, trying to find non-Borel measurable, but Lebesgue measurable sets serves no practical purpose other than being written down in an analysis textbook. More power to pure mathematicians and their students who enjoy playing these games, but the more applied folks can get around those "problems" by just making some niceness assumptions about your model (like f is Riemann integrable!) and moving on to the more pressing issues in their fields.
- ska 5y agoPerhaps I was unclear, but we seem to be agreeing.