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Comment by jszymborski | Hacker News Reader

I've discovered Calculus Made Easy recently and it's wonderful. The edition edited by Martin Gardner is particularly good, with amazing preliminary chapters [0].

I've spent many years "pretending to understand" calculus, but things I remember gnawing at me, like limits & infinitesmals, are accompanied with context and history such that you can finally put yourself into the conversation and understand that my confusion is simply due to only getting a fraction of the story.

You can read the full text for free here [1]

[0] https://openlibrary.org/books/OL351037M/Calculus_made_easy

[1] http://calculusmadeeasy.org/

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[a critique, not a request for help] I found it relied too much on faith: you can ignore this small quantity, but not this one. What's the threshold? Why?

Fractional and negative powers are assumed to work as a generalization of positive integer powers, without proof.

But, TBF, all maths education requires a lot of faith. e.g. the unique prime factorization theorem is assumed in high school, not proven.

If I recall correctly, the author explicitly states in the introduction that mathematicians will hate the book precisely because it skips over proofs and takes a pragmatic approach. If you're the kind of person who wants to understand the proofs before using them, you will need to supplement this book with other material.

He spends much time on "minute" quantities at the start, but the explanation doesn't really make sense. To me, it's a mental model I can't trust, like rickety stairs.

The book is explicitly calculus-as-a-bag-of-tricks; monkey-see, monkey-do. As he says:

  What one fool can do, another can.
  (Ancient Simian Proverb.)

Fair enough on proofs. BTW in the free gutenburg edition (maybe MG differs): "prologue" doesn't mention proofs, but that textbook writers make it difficult (no "introduction" - also no "preface", except to the 2nd ed):

> The fools who write the textbooks of advanced mathematics ... seem to desire to impress you with their tremendous cleverness by going about it in the most difficult way.

Yeah, this sounds right. I think MG contextualizes it a little bit more. I actually like the book for the exercises and MG's footnotes especially, but as I worked through it I had to consult other resources to make sure I wasn't missing anything. The Khan Academy Youtube channel was very helpful for this.

Yes, but if you think faith alone is not enough you can always drop back to the sacred texts of the proofs.

But it is weird not being shown the proofs for things like generalization for other situations. I studied Calculus while in engineering at college and we always got the proofs for the tricks we were using.

Why is it weird? What did having the proofs change?

If you just want to use a tool at a basic level, you do not need to be explained how it works. If you want to master a tool and be able to use it in novel situations you need to understand how the tool was created. Same with math proofs - without knowing why something is true and how it was discovered, how can you utilize its full power?

Personally, working through the proof until I have a flash of insight is how I develop intuition that allows me to know when and where the "trick" should be applied, which is often more broad than the narrow context in which it is presented through the materials. You may even develop some new tricks of your own.

You want to prove it to yourself at least once I think. Even if you only need the result, it’s nice to have at least once in your life a complete understanding of what you’re using.

I don’t remember many proofs that I did, but I’m happy to know the maths that I still remember really works.

While not a math wiz myself— I understand why you would want proofs, but that can easily be taken to the extreme. Should you be measuring the gravitational constant every time you need to use it as well?

Seems like a good path to learn high level abstractions. As you progress in understanding you dig deeper.

Maybe I'm even still too much a sprite—but when I first learned anything about computers the first program I wrote was in a high level, simpler language. I wasn't moving bits around with explicit knowledge of where they were going.

Then again, maybe it's not a fair comparison on my part?

To some extent you're right, the benefit of a higher-level abstraction is using it without knowing the details. For standard usage, on the "happy path", this is fine. But if you need to modify techniques, or debug them, it's kind of impossibly frustrating without actually knowing what you're doing!

BTW computers have much cleaner abstractions than mathematics. e.g. the JLS defines Java independently of hardware; IEEE 754 is similar for fp arithmetic. There are specifications all over the place.

But my experience with mathematics is completely different - you have to understand the lower level to understand the next level.

In my personal journey, I started off with your perspective of just learning the higher levels that I directly needed. It was very difficult, but after heroic efforts, I made breakthroughs! After a while, I noticed these "breakthroughs" were mostly entirely to do with material from lower levels... So I went back to them. This happened again and again, going lower and lower. Now I'm basically re-doing high school maths.

Heheh. That’s not too far off from me. Though my progress has been delayed I was starting to circle back through many basics even taking high school courses to prep for a return to university. Hasn’t panned out so far but I’d still like to do it.

Your point about specifications makes sense. I think that was a point that made some aspects of maths harder for me— contextual differences in notation all over the place. Thanks for the input.

I can relate to that, because it's something that's bothered me about most texts. The preliminary chapter in the Gardner ed. addresses this to an extent, I believe.

On the topic of infinitesmals, Gardner addresses their existence/utility as historically controversial. By providing both sides of the argument I was more able to understand how both parties were correct, and it has given me better insight as to why and when small numbers are ghosts and when they corporealise.

Check out Bartlett’s papers and textbook if you want something based on differentials,

https://arxiv.org/abs/1811.03459

https://arxiv.org/abs/1801.09553

https://amzn.com/1944918027

Unable to look inside to see the textbook content, I went to the Bartlett Publishing website where it’s listed along with alternative textbooks debunking evolutionary theory and fossil forensics. Sorry, I’ll pass.

You can read the two arxiv papers if you like. The book doesn’t really have any technical content that isn’t in those; just more elaboration and exercises pitched at students coming to the subject for the first time.

I certainly wouldn’t recommend this as a sole introductory textbook; it’s obviously produced on a shoestring budget (a LaTeX file with the default template sent over to a print shop) and is somewhat limited in many aspects compared to established calculus textbooks. But the pedagogical idea of focusing on differentials seems sound. I haven’t ever taught an introductory calculus course, but I think it seems entirely plausible that this approach would save some confusion for many students (this is something which could be tested empirically, if any math-ed researcher has the time and budget for a study).

It’s not clear why the guy’s ideas about evolutionary theory (which I know nothing about) have anything to do with his ideas about derivatives. From what I understand he’s a computer programmer and math teacher without extensive training in biology; I wouldn’t expect him to have any insight into evolutionary theory.

I'm reading that now! It was suggested on another HN thread a few months ago. It's really given me a new appreciation of calculus. I run a data science department but there are parts of calculus that never really made sense before.

Thank you for the links. I always did well with math classes in school, but my retention was always so so after passing the final. Will give this a try for a more lasting understanding of the subject!

Calculus Made Easy is the best!

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