4 ms·

Just curious, what exactly do you consider the necessary rigor in this case?

RiderOfGiraffes 17y ago
Well, there's nothing in this article that really requires rigor, because there aren't really any problems. It's talking about the concepts of limiting processes and infintesimals, but not how to use them. In the one example given, sin(x)/x, the difficult bit is glossed over when it talks about sin(x) being more-or-less equal to x when x is "close to zero." The rigor there is missing entirely, because that's not the point.
- kalid 17y ago
  Thanks for the comment. I did contemplate mentioning the definition of sin(x) = x - x^3/3! + ... but thought it would be too leading & tangential -- I wanted to show visually that most curves can in fact be modeled by a straight line over a small distance, without the sledgehammer of showing sine's expansion.
  - RiderOfGiraffes 17y ago
    Better would simply be to say that sin is opposite over hypotenuse, and when the angle is really, really small, the opposite is effectively the same as the arc length. Since the acr length is the angle in degrees, that means that sin(x) ~ x when x is very small. Then you can be more rigorous with the epsilon/delta, and say "you pick how close, and I'll get it closer by making x small enough." Similar arguments can show, for example, that acceleration in a circle is v^2/R. "Calculus" was, or course, done for decades, if not centuries, before "calculus proper" was (somewhat) formalised by Leibniz and Newton. This whole discussion should probably be taken to email if it's to be taken further. The question is, how can we have this sort of discussion, and produce something worth publishing back onto HN?
    - kalid 17y ago
      Yep, definitely -- actually I have an article on radians which does just that (comparing height on the circle, sin, to distance traveled around, x). Always happy to discuss on email (kalid.azad@gmail.com) -- I'm not sure if there's something to produce out of it necessarily, but I enjoy the discussion / investigation of personal philosophies & teaching approaches.

Well, there's nothing in this article that really requires rigor, because there aren't really any problems. It's talking about the concepts of limiting processes and infintesimals, but not how to use them. In the one example given, sin(x)/x, the difficult bit is glossed over when it talks about sin(x) being more-or-less equal to x when x is "close to zero." The rigor there is missing entirely, because that's not the point.

kalid 17y ago
Thanks for the comment. I did contemplate mentioning the definition of sin(x) = x - x^3/3! + ... but thought it would be too leading & tangential -- I wanted to show visually that most curves can in fact be modeled by a straight line over a small distance, without the sledgehammer of showing sine's expansion.
- RiderOfGiraffes 17y ago
  Better would simply be to say that sin is opposite over hypotenuse, and when the angle is really, really small, the opposite is effectively the same as the arc length. Since the acr length is the angle in degrees, that means that sin(x) ~ x when x is very small. Then you can be more rigorous with the epsilon/delta, and say "you pick how close, and I'll get it closer by making x small enough." Similar arguments can show, for example, that acceleration in a circle is v^2/R. "Calculus" was, or course, done for decades, if not centuries, before "calculus proper" was (somewhat) formalised by Leibniz and Newton. This whole discussion should probably be taken to email if it's to be taken further. The question is, how can we have this sort of discussion, and produce something worth publishing back onto HN?
  - kalid 17y ago
    Yep, definitely -- actually I have an article on radians which does just that (comparing height on the circle, sin, to distance traveled around, x). Always happy to discuss on email (kalid.azad@gmail.com) -- I'm not sure if there's something to produce out of it necessarily, but I enjoy the discussion / investigation of personal philosophies & teaching approaches.