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Comment by ajross | Hacker News Reader

You never got to the parts of the section about integration that are wrong.

Broadly: your criticism is a mathematician's nitpicking. I don't know that that really has much value to someone learning "Calculus" in a practical sense about understanding and working with functions. And frankly... I mean, the fundamental theorem of calculus is a historically important result and a relatively profound truth, but it's not that big a deal. If you really want to play the "you really need to know this important truth more deeply" game with amateurs, start with the incompleteness theorem or something.

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I think I did, but I'll be more precise. The first problem is that the FTC is reduced to a tautology. But the more serious problem is that the definite integral is _defined_ via the antiderivative. This is wrong, as the existence of integrable functions without antiderivative shows.

I'm all for simplifying material, and I'm not saying this needs to be a mathematically rigorous treatise, but it should get the basic maths right, because there is already too much confusion out there.

FWIW, I think the FTC is deeply important and used all the time, but the first part of it - the one about writing antiderivatives as integral - could probably be omitted for non-mathematicians.

It would probably enough to establish: - Here's what differentiation is. Here's some rules. - The inverse of that is antidifferentiation. You compute it by going in reverse, but note that this is harder than differentiation (which is mechanical) - Then we have the integral via Riemann sums. Maybe you can show some simple example. - Now comes the deep result: the integral can be expressed via the antiderivative, if it exists.

You don't have to go into full mathematical detail, but this presentation wouldn't be getting it wrong anymore.

So... if it were made clear that the discussion was specific to continuous functions, your criticism would disappear? Is that really worth the level of your dismissal?

I mean, that's how it's actually taught to high school students, after all. Complexities like discontinuities are absolutely not part of the initial curriculum.

Alternatively, if you think a simple introduction to calculus requires a discussion of discontinuous functions and the fundamental theorem of calculus before it describes how to compute a definite integral, I retreat to my earlier point: that's ridiculous nitpicking.

Basically: you want a textbook. They make those. This isn't that, it's an intuitive guide for people who found the treatments in textbooks opaque. Those people exist, they want to learn this stuff too, and they aren't well served by people like you just yelling at them to read the textbooks they already have.

I don't feel you've actually read and understood my criticism.

The fact that the theory falls apart for discontinuous functions is just a visible symptom of the deeper problem that the author confuses a definition with a theorem. I don't think this is just theoretically wrong, it also makes no pedagogical sense. To say "we define the integral as being the difference of the antiderivative at the two endpoints, but we can also define it as the Riemann sum" doesn't tell the student anything except "mathematics is magic, don't even bother to understand it". Whereas to be genuine and say "we can prove that the integral can be expressed through the antiderivative (though we choose not to do this here)" doesn't treat the reader like an idiot.

I don't think it's impossible to provide an intuitive and non-proof-based account without sacrificing getting things right conceptually. In fact, I provided a template for doing so above.

> To say "we define the integral as being the difference of the antiderivative at the two endpoints, but we can also define it as the Riemann sum" is [like, really bad!]

How about if there was a footnote on that point explaining that these two definitions are actually equivalent (for the continuous functions we're talking about) and that the proof is really interesting and can be found in your textbook? Would that meet your requirements?

Again, this is just nitpicking. In fact those two definitions are equivalent, as you keep pointing out. You don't have to prove everything in an introductory treatment, and in fact even high school textbooks on this subject don't even try.

The article purports to teach calculus, so it should get the basics right. The Fundamental Theorem of Calculus is called fundamental for a reason: it's the most important concept in the entire subject. It's not just a technicality. The article defines things in a non-standard way so as to reduce FTC to a tautology, and nowhere does it correctly explain the connection between differentiation and integration (which can -- and should -- be understood visually and intuitively).

In short, I'm not convinced that the author knows what he's talking about.

Yeah well - next time I read a proof by a mathematician that skips several steps that to mathematicians are unimportant or "obvious" yet fundamentally blocked my understanding of the proof overall - I'll think on this reaction and smile...

Or maybe even just write some mathematical overviews myself that leave out stuff that drives mathematicians insane with frustration...

It would be justice...

Would you write code with syntax errors and blame the compiler for not understand how smart you are, or would you just learn the language and write correct code? Your frustration shows lack of understanding and appreciation for mathematics, when in reality maths and computer science are two sides of the same coin. If you can be good at one, you can be good at the other :)

> Would you write code with syntax errors and blame the compiler for not understand how smart you are

No, but you might tolerate example code with mild syntax errors (like using a "..." to skip unrelated material, or skipping import declarations and boilerplate to better clarify the point being made). You don't demand that every example in a programming tutorial be a fully buildable piece of software, why are you demanding that a simpified calculus tutorial exhibit full rigor?

> why are you demanding that a simpified calculus tutorial exhibit full rigor?

Aren't they just suggesting a marginal amount of rigor to communicate that the "definition" is in fact based on justifications which render it non-arbitrary?

I sympathise with your point of view and I agree that sometimes mathematicians are not keeping their audience in mind enough. You don't always need completely rigorous proofs for non-mathematicians.

But I think that it's a false dichotomy to only consider "extremely rigorous" or "simplified to the point of being wrong" as the only teaching philosophies. There is a middle ground where you can be somewhat hand-wavy and inexact without getting the conceptual broad strokes wrong.

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