5 ms·

i always found it funny how much time was spent drilling techniques for different integral types or doing transforms for derivatives, yet the most important ide

by a-dub 5y ago

i always found it funny how much time was spent drilling techniques for different integral types or doing transforms for derivatives, yet the most important idea: the continuous nature of the reals and why this is the rug that pulls it all together, and the delta-epsilon definition of the limit, only saw about a grand total of 3 minutes of hand waving and an optional problem on one homework. ug calculus was about having algebraic/trig manipulations memorized along with a table of transforms and a handful of tricks; where the actually beautiful ideas that if we use an infinitely "elastic" representation of numbers, we can solve hard approximation problems both correctly and easily- get totally glossed over. physics has the same problem. basic physics without calculus is just a bunch of rote memorization. the idea that such a thing is taught and that is somehow "easier" is nuts. they should be taught together, as they were developed, as many of the expressions given to undergrads in physics are simply definitions of integrals and derivatives applied a few times. no student in calculus should ever be wondering what the constant is for in a computed integral and no student in physics should wonder where the constants come from in equations of motion. (there should be no equations of motion, just definitions in integral/derivative form and definitions of integrals and derivatives)

concreteblock 5y ago
As someone who has taught intro calculus a few times, one reason for the emphasis on computational techniques is simply that 90% of the students in such a class do not care/are not capable of grasping the epsilon-delta definition of the limit. Solution: spend more time on epsilon-delta so that students have time to wrap their minds around the idea. But I think the engineering departments would complain that the students who we send on to them cannot do basic computations. Also students would complain that we spend too much time on theory and not enough on application. There are probably other reasons that someone more experienced would know about.
- javajosh 5y ago
  I tutored calc too, but I had an unanswered complaint about the epsilon/delta definition of a limit - I found it to be circular. There was no impact on the computations required by the class, but the definition seemed rather circular, as it presupposes the reality of the infinity implicit in induction. Over time I recognized that my intuition was rebelling against unphysical math concepts. I was a physics major and so I was content to hand-wave my own concerns away as a merely philosophical problem, since you could always pick a sufficiently large number of induction iterations to satisfy any practical need. But yeah, it left a bad taste in my mouth that no-one seemed to care about this issue. (Oddly the use of induction for similar things, like Cantor's diagonal argument, didn't bother me at all. It was specifically epsilon delta that did, and still does, bother me.)
  - rsj_hn 5y ago
    The epsilon delta definition of limit is not circular and there is no infinity anywhere in it. Are you referring to the limit as something goes to +infinity on the real line? That's just bookeeping shorthand for some quantifiers over bounded numbers. I'm genuinely curious as to what you were getting hung up on, because odds are you learned the wrong definitions or somehow changed the definition to something that is incorrect, and then discovered the error later on. It's a fairly straightforward definition if you can do the proper bookkeeping over the quantifiers, and is designed to not have anything to do with infinities. But many people struggle with quantifiers.
    - a-dub 5y ago
      this is the point. they do not stress in beginning calculus that continuity is actually a profound property of a mathematical construction (the reals) that we use for modeling physical phenomena. it's not the delta-epsilon definition that is slightly weird, it's what it means to be truly continuous that is. (uncountable infinity, or as i like to call it, endless zooming everywhere) once that is clear, the delta-epsilon definition becomes obvious and clear.
      - rsj_hn 5y ago
        Continuity is just a formal property. I'm not sure if it is "profound", that's more of an emotional response, which is very personal. Mathematicians use words like "deep" to describe properties that are useful in many branches of math. Is continuity deep? I think what's deep is the notion of compactness and the essence of compactness is open covers and finite subcovers, so the deep idea really is about epsilons and deltas and not handwaving about "what it really means" to be continuous. For ruminations on what things "truly mean", you are not going to find that in a math book. Perhaps a philosophy book would better scratch that itch for you. Math is self-contained and provides its own meaning in terms of the relationships between formal properties and precise answers to precise questions. Some people truly love exploring the relationships between formal properties and discovering the answers to difficult questions that reveal new logical structures. If that is not interesting enough for you, perhaps you will get more satisfaction studying a different field.

by a-dub 5y ago

concreteblock 5y ago

As someone who has taught intro calculus a few times, one reason for the emphasis on computational techniques is simply that 90% of the students in such a class do not care/are not capable of grasping the epsilon-delta definition of the limit. Solution: spend more time on epsilon-delta so that students have time to wrap their minds around the idea. But I think the engineering departments would complain that the students who we send on to them cannot do basic computations. Also students would complain that we spend too much time on theory and not enough on application. There are probably other reasons that someone more experienced would know about.

javajosh 5y ago
I tutored calc too, but I had an unanswered complaint about the epsilon/delta definition of a limit - I found it to be circular. There was no impact on the computations required by the class, but the definition seemed rather circular, as it presupposes the reality of the infinity implicit in induction. Over time I recognized that my intuition was rebelling against unphysical math concepts. I was a physics major and so I was content to hand-wave my own concerns away as a merely philosophical problem, since you could always pick a sufficiently large number of induction iterations to satisfy any practical need. But yeah, it left a bad taste in my mouth that no-one seemed to care about this issue. (Oddly the use of induction for similar things, like Cantor's diagonal argument, didn't bother me at all. It was specifically epsilon delta that did, and still does, bother me.)
- rsj_hn 5y ago
  The epsilon delta definition of limit is not circular and there is no infinity anywhere in it. Are you referring to the limit as something goes to +infinity on the real line? That's just bookeeping shorthand for some quantifiers over bounded numbers. I'm genuinely curious as to what you were getting hung up on, because odds are you learned the wrong definitions or somehow changed the definition to something that is incorrect, and then discovered the error later on. It's a fairly straightforward definition if you can do the proper bookkeeping over the quantifiers, and is designed to not have anything to do with infinities. But many people struggle with quantifiers.
  - a-dub 5y ago
    this is the point. they do not stress in beginning calculus that continuity is actually a profound property of a mathematical construction (the reals) that we use for modeling physical phenomena. it's not the delta-epsilon definition that is slightly weird, it's what it means to be truly continuous that is. (uncountable infinity, or as i like to call it, endless zooming everywhere) once that is clear, the delta-epsilon definition becomes obvious and clear.
    - rsj_hn 5y ago
      Continuity is just a formal property. I'm not sure if it is "profound", that's more of an emotional response, which is very personal. Mathematicians use words like "deep" to describe properties that are useful in many branches of math. Is continuity deep? I think what's deep is the notion of compactness and the essence of compactness is open covers and finite subcovers, so the deep idea really is about epsilons and deltas and not handwaving about "what it really means" to be continuous. For ruminations on what things "truly mean", you are not going to find that in a math book. Perhaps a philosophy book would better scratch that itch for you. Math is self-contained and provides its own meaning in terms of the relationships between formal properties and precise answers to precise questions. Some people truly love exploring the relationships between formal properties and discovering the answers to difficult questions that reveal new logical structures. If that is not interesting enough for you, perhaps you will get more satisfaction studying a different field.