Basic Math Textbook: the Napkin Project

Red flags of Math Academy:

- Centred around AI

- Seems geared around edutech (which is what I gather from the site)

Green flags for Napkin:

- Covers advanced undergraduate and graduate topics

- Encourages pencil & paper way of learning (took me way too long to learn this is the best appraoch)

+1 this is a great reference text

The preface has "I initially wrote this book with talented high-school students in mind, particularly those with math-olympiad type backgrounds."

Apparently the author tried to somewhat expand the audience from that, but to me it seems still mostly appropriate for smart high schoolers who have heard some pieces of lore from friends about these topics, but they can't put that puzzle in order in their minds yet.

It's most definitely not aimed at the average student. You need to be highly curious, motivated and find math fun already.

And I think that's a perfectly fine thing. It's great to have books for that kind of audience.

> that's way beyond what's called basic math instruction, e. g. in schools

I'm not saying you're wrong, I know for a fact that you aren't: unfortunately most high-school students fall extremely short of that bar, but it's not necessarily that way. Many teenagers can and do develop that kind of mathematical maturity.

In this context "basic" means "it doesn't require knowledge in the field", and by and large this book can indeed be followed with no other requirement than the mathematical maturity it talks about. Many classic books self-describe in similar way.

That's common with mathematics books. Weil's Basic Number Theory is enough to give the unsuspecting quite the fright, despite the name

It would make more sense to include the term "higher math" (from the author's own description) in the page title, like "Basic Higher Math Textbook" or "Introductory Higher Math Textbook".

Higher mathematics isn't necessarily very strictly defined anyway, but I guess most people who've heard the term would apply it to branches of math that are developed using formal definitions and at least moderately rigorous proofs, and that usually aim at a level of generality beyond their originally motivating examples.

I think it's not just some kind of humblebrag. I know this trope that college students feel like it says it's intro but it's hard so it's not an intro. But you only think this when you don't know the topic well. The "thing itself" is in the journals, at the conferences, and in the professional work of researchers, and (if applicable) the real-world applications of the content in various contexts. Any normal-sized book can really only be an introduction to all that for most topics taught in undergrad or master's level.

This is such a common misunderstanding it's worth explaining.

If you get a book in stem called "an introduction to x" it isn't claiming to be short or simple at all. What "introduction" means is that it is intended for a first course in that topic (ie it does not have prerequisites within that topic).

So if I get "an introduction to mechanics" by Kleppner and Kolenkow[1] for example (to pick one off my bookshelf), it is a challenging first course in classical mechanics but it doesn't require you to know any mechanics before reading it.

[1] This is a really good book in my opinion btw.

"undergrad math" might be a better phrase to use; "basic" and "advanced" mean very different things to people with different backgrounds

You need to be careful about this ... I believe that in France (for example) zero is regarded as both positive and negative. So in France:

Non-negative integers: 1, 2, 3, 4, 5, ...

Positive integers: 0, 1, 2, 3, 4, 5, ...

Similarly, for some countries "Whole Numbers" is equivalent to all the integers, while in other countries it's the set { 0, 1, 2, 3, 4, ... } while in still other countries it's { 1, 2, 3, 4, ... }

There is no approach that uses "natural language" and is universal, and being aware of this is both frustrating and useful. Whether it is important is up to the individual.

By convenience I mean "convenient from a technical perspective", and yes, you often need 0 in there.

Your other argument doesn't make much sense. I learnt both in school and at university ℕ, ℕ₀, and ℤ as THE symbols for the natural numbers, the natural numbers including 0, and the whole numbers.

Fuck convenience. ℕ, ℕ₀, and ℤ it is :-) It is just so much prettier (ℤ⁺ is a really ugly symbol for such a nice set). It is actually also not inconvenient if you don't use static types.

So this is aimed at somebody who has mathematical maturity but prefers... less content and detail? The point is that you are losing something in a shortened presentation. You're not just losing "unnecessary exercises" as you put it.

Bingo. I’m guessing you went through the same song and dance?

You are very welcome. The other resource I found excellent was Khan Academy which is free (donation supported) and has videos and supporting resources covering a very wide range on mathematical topics. I would recommend the “precalculus” playlist which is just a general grab bag of topics covering algebra, basic vectors and matrices and a few other things.

Def true. I often mark up math papers and books with DIY-hyperlinks. It's very easy for me to skip over an important, foundational clause just because some term isn't immediately familiar, and if that happens frequently in some reading, then I'm mentally checking out.

For the Napkin book, if the underlying metadata were in the latex source, we could have PDF annotations in a sidebar, e.g., ("def: p.123, key application: p.234, ..."), as well as live tooltips for a modern web experience. That would be totally wonderful for this text and its audience.