5 ms·

Knowing and using a thing and proving it are (usually) different skills.

• ska 5y ago
  True; 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 ago
    The 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 ago
      Perhaps I was unclear, but we seem to be agreeing.