I think rigor is imposing to people when you try and combine it with high-level theory. The concepts in that article are built on top of so much theory, that to explain it all would require the equivalent of a course on analysis.

But if you supply rigor in an appropriate dose, it can really be eye-opening. For me, calculus was a nightmare because none of the terms used were well defined. I was just presented with all this highly developed theory and had to take much of it on faith. And I couldn't enjoy nor care about something I felt was just forced memorization of computational techniques for solving contrived problems. But if they'd done it rigorously, the classes would have begun with questions like "what is distance?", then answering that question would lead into "what is a space?", etc. This is rigor and I feel like exploring these types of questions is a lot more intuitive than forcing kids to memorize how to compute the volume of some cone.

I think we're pretty much in agreement. I would rather, though, see a real example with proper rigor, but introduced and explained starting with intuition and then developed forwards into the detail.

The problem is, as I'm sure you're aware, just how many traps and pitfalls there are for the insufficiently paranoid. Working "intuitively" has a tendency not to instill that paranoia.

And, as you say, working just on the rigor tends to result in rote work with no understanding.

I think only time and practice is the real solution. The article, though, is good for someone who already has the "rote" understanding, and needs to "grok" the whole thing.

Similar criticisms can be aimed at my stuff: http://news.ycombinator.com/item?id=940169

Yep, I agree with your points about pure intuition leading you down blind alleys unless you have some structure in place.

For the examples, my approach is definitely tailored for people who need to grok the subject after getting a rote understanding. It's not so much to become an expert but to realize what insights would help on the path to becoming an expert.

I figure there are thousands of other sites for the formal definition and practice problems :). I'll check out the article now, looks interesting.

I studied calculus on my own in my twenties, and I think the textbooks' focus on rigor led me down a lot of blind alleys before I started trying to come to an intuitive, often graphical, understanding of the problem before attacking the rigorous view. (There is a good textbook from 1977 that was reprinted by Dover in 1998, Morris Kline's "Calculus: An Intuitive and Physical Approach", that I wish I had available when I was studying.)

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

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.

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.

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?

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 ([email protected]) -- 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.