"As explained in the preface, the main prerequisite is some amount of mathematical maturity. This means I expect the reader to know how to read and write a proof, follow logical arguments, and so on."
Yeah, that's way beyond what's called basic math instruction, e. g. in schools. A more specific, as in accurate, subtitle (or description) is in order.
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'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.
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.
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.
The author says that this is largely aimed at high school students who are doing self-study, which is a realistic audience but not a context where a lot of people would naturally apply the word "basic". But this material is basic for mathematicians, I guess (although even a lot of mathematicians may not have quite as broad a knowledge of mathematics as the author does!).
Hell yeah!
I've agonised over this quite a lot over the decades. Not including 0 is more intuitive, but including 0 is more convenient. Of course, both approaches are correct. My main reason for not including 0 is that I hate seeing sequences numbered starting with 0.
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.
That would cause all kinds of problems, so I'd be pretty surprised if it turned out to be true.
I note that this is the heading of the relevant wikipedia page:
> Un nombre négatif est un nombre réel qui est inférieur à zéro, comme −3 ou −π.
( https://fr.wikipedia.org/wiki/Nombre_n%C3%A9gatif )
It'd be hard to be more explicit that zéro is not a negative number.
> "Zéro est le seul nombre qui est à la fois réel, positif, négatif et imaginaire pur."
From: https://fr.wikipedia.org/wiki/Z%C3%A9ro#Propri.C3.A9t.C3.A9s...
It's hard to be more explicit that it is considered both.
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Added in edit
In speaking with a French colleague, he says that "inférieur" often means "less-than-or-equal-to" rather than "strictly-less-than", so the passage you quote would still imply that 0 is negative (and most likely also positive).
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Second edit:
> In France, "positive" means "supérieur à 0", and "supérieur à " means "greater than or equal to". Similarly, "négative" means "inférieur à 0", that is "less than or equal to 0".
> (We have the similar reaction towards the anglosaxon world and the introduction of nonnegative…)
-- https://mathstodon.xyz/@antoinechambertloir/1153275891164575...
From a pure convenience perspective, it doesn't make sense to assign ℕ to the positive integers when they're already called ℤ⁺. Now you have two convenient names for the smaller set and none for the larger set.
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.
And round we go.
https://news.ycombinator.com/item?id=20168936
Need to see how this looks on my Kindle Scribe --- I suspect that it will push me over to updating to the newly announced colour model when it becomes available.
Sorry to be negative Nancy over here, but you're going to need more than 54 pages to cover calculus. There is value in organizing the major theorems in the different disciplines. But, to be honest, this doesn't really serve the beginner.
1. I don't think it is at all intended to serve the beginner.
It's geared towards readers wait a reasonable amount of mathematical maturity already (it explicitly says that in the landing page).
2. Many, many of the pages of most introductory calculus textbooks are spent on exercises and on the specifics of computing integrals and derivatives of particular functions - none of this is necessary to understand the concepts themselves.
For example, Baby Rudin (the standard textbook for Analysis for math majors) covers Sequences, Series, Continuity, Differentiation, and the Riemann integral in less than 100 pages (including exercises).
> Philosophy behind the Napkin approach
> As far as I can tell, higher math for high-school students comes in two flavors:
> • Someone tells you about the hairy ball theorem in the form “you can’t comb the hair on a spherical cat” then doesn’t tell you anything about why it should be true, what it means to actually “comb the hair”, or any of the underlying theory, leaving you with just some vague notion in your head.
> • You take a class and prove every result in full detail, and at some point you stop caring about what the professor is saying.
> Presumably you already know how unsatisfying the first approach is. So the second approach seems to be the default, but I really think there should be some sort of middle ground here. Unlike university, it is not the purpose of this book to train you to solve exercises or write proofs, or prepare you for research in the field. Instead I just want to show you some interesting math. The things that are presented should be memorable and worth caring about. For that reason, proofs that would be included for completeness in any ordinary textbook are often omitted here, unless there is some idea in the proof which I think is worth seeing. In particular, I place a strong emphasis over explaining why a theorem should be true rather than writing down its proof.
And yes, with more mathematical maturity you definitely don't need as much detail. The proofs get terser as you're expected to be able to fill out the more straightforward details yourself.
The style of this textbook does seem to primarily skip the "techniques for evaluating" stuff, on the basis that you just wanted to understand what each branch of mathematics is about and what kinds of theorems it has that might relate to the larger edifice of mathematics.
I highly appreciate this approach: "As i have ranted about before, linear algebra is done wrong by the extensive use of matrices to obscure the structure of a linear map. Similar problems occcur with multivariable calculus, so here I would like to set the record straight"
Math education and textbooks are doing an awesome job obscuring simple ideas by focusing on weird details and bad notation. Always good to see people trying to counter this :)
However, a small critique to the author: the audience of this book is not clear. It says “basic” math, but then in chapter 1, the group's explanation starts with this sentence: “The additive group of integers (Z,+) and the cyclic group Z/Zm.” Maybe it was a draft note. To be fair the paragraphs that follow attempt a more basic explanation of groups, but even my “Algebra I” book at the university was friendlier than that.
I recently tried to go for a math degree in my free time using my countries’ remote learning option, and even though the attempt didn’t last long because the format is hopelessly broken (Mediterranean bureaucracy), I’m still engaging in self learning through his books and they’re an absolute goldmine.
Most basic math books assume no knowledge of the subject but a familiarity with general math that is unreasonable - it’s like saying you don’t need to know what a deadlift is but you need a back that resists 200kg… It’s a borderline fictional audience in practice.
Cummings manages to understand the novice far, far better.
I did it to get my very rusty high-school maths back up to snuff before starting to self-study for a maths degree and it helped a lot. The problems are really excellent and since it's Serge Lang, he treats you like a mathematician right from the beginning even though he really is doing basic stuff.
Learning and internalizing higher math is largely about connecting lots of ideas, terms, definitions, named theorems, lemmas, etc. If the book were instead built for the modern web stack with heavy use of tooltips, it would be lots more engaging and fun, supporting a more active learning process.
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.
Qwen3 recommends
Blurb Lulu BookBaby Mixam
For a 1000 page book, it suggest pricing of ≈$120 for single copies, down to $15-25 for a run of 1000.
Maybe cheap child labour is called for…
tocs3•4mo ago
_hao•4mo ago
commandersaki•4mo ago
- 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)
ptsneves•4mo ago
Where do you see the centered around AI? I have used it a lot and have not touched a single subject around AI.
> - Seems geared around edutech (which is what I gather from the site)
What is edutech and why is it unsuitable?
Finally, have you _used_ MathAcademy at all?
commandersaki•4mo ago
From https://www.mathacademy.com/how-it-works:
> Math Academy is an AI-powered, fully-automated online math-learning platform. Math Academy meets each student where they are via an adaptive diagnostic assessment and introduces and reinforces concepts based on each student’s individual strengths and weaknesses.
What is edutech and why is it unsuitable?
I don't want a computer in the loop when I learn math, plain and simple. My preferred style of learning is instructor led with a mix of Socratic method and hand holding. But bar that, reading texts and using a pen and paper.
Finally, have you _used_ MathAcademy at all?
Nope, doesn't look like my cup of tea.
delichon•4mo ago
yorwba•4mo ago
See also How Math Academy Creates its Knowledge Graph https://www.justinmath.com/how-math-academy-creates-its-know... "We do it manually, by hand."
qwertytyyuu•4mo ago
barrenko•4mo ago
https://grizzle.robotics.umich.edu/education/rob201 - "ROB 201 Calculus for the Modern Engineer"
BeetleB•4mo ago
j2kun•4mo ago
eapriv•4mo ago
j2kun•4mo ago
rramadass•4mo ago
1) The Princeton Companion to Mathematics by Timothy Gowers et al. and The Princeton Companion to Applied Mathematics by Nicholas Higham et al. - The closest you have to a Modern Encyclopedia of Mathematics. You get unmatched breadth after which you can move on to dedicated books as needed. Well worth the money.
2) Mathematics: Its Content, Methods and Meaning by Aleksandrov, Kolmogorov et al. - Absolutely brilliant overview of Basic Mathematics. Published by Dover and hence very affordable.
3) Elements of Mathematics: From Euclid to Gödel by John Stillwell - Written as sort of an update to the great Felix Klein's Elementary Mathematics from an Advanced Standpoint books. The Topics are particularly well chosen given modern advances; they include Arithmetic, Computation, Algebra, Geometry, Calculus, Combinatorics, Probability, Logic.