Judging by the URL, this book was used for CMU's 15-151 / 21-128, which is a first-semester course for CS and math undergrads. Nowadays, the course uses [0].

If we're dropping links, here's the one [0] from my undergrad discrete math course that seems to have a lot of overlap with the topics in the CMU book. Our professor called it "notes" but it's more like a treatise...

That book was used in my "intro to higher math" class my freshman year. A very humbling experience, seeing that cover again gave me a bit of a knot in my stomach.

Every chapter of the OP book has an introduction which has the following subchapters: Objectives, Segue from previous chapter, Motivation, Goals and Warnings for the Reader. Which I really appreciate and I think its a great way to teach stuff, especially mathematics, since all those chapters contextualize the coming topic in various ways. (E.g. I imagine ‘Motivation’ does this with respect to the history and origins of the idea and ‘Segue from previous chapters’ contextualizes it with reference to the stuff reader has learnt so far. I didn’t read anything yet but it looks promising

As a CS major who went to CMU years ago, my favorite book from all of my professors was for my 15-213 class called Computer Systems: A Programmers Perspective

I remember at the time the book was in loose leaf paper so it warms my heart to see the book has a 3rd edition. It was used as a core part of teaching assembly, memory representations, and getting students ready for the operating systems class. When I help people learn to program, it's the only book I think is a must have:

We used this for my Computer Architecture class in college and it's one of the few textbooks that I wish I had kept around. I had a used copy so not selling/returning it to the bookstore really wouldn't have been too impactful at the time. Now that I'm wanting to get back into low-level programming, I miss it even more.

Is there a book or resource you'd recommend for operating systems? I recently finished CS:APP and really enjoyed it. Though I've been programming for ~7 years and felt that made it easier to digest.

Does anyone know of an entry level book that could take someone through say, high school math to college alegbra / calculus?

This is my singular biggest hurdle in going back to school to finish my degree and I'd love to fill the gaps I have around mathematics so I can not only finish my degree; I'd also like to participate in some more advanced computer science that rely heavily on underlying computation.

At Math Academy (https://mathacademy.com), we created a series of courses, Mathematical Foundations I, II, & III, that will take a student from basic arithmetic through calculus and prepare them for university-level courses like Linear Algebra, Multivariable Calculus, Probability & Statistics, etc. You can jump in at any with an adaptive diagnostic that will custom fit the course to you based on your individual strengths and weaknesses.

It's not free, but our adaptive, AI-driven algorithms makes it the most efficient way to learn math that you're going to find. We've had numerous students master 3-5 years of math in a single year.

We're still in beta and haven't done a proper Show HN yet, but we're getting there!

I'm the founder, so I'd be happy to answer any questions.

600 USD a year is definitely worth it to learn any highly technical topic: mathmematics, physics, chemistry, CS, engineering, etc.

BUT... I'm highly skeptical of any online math course that claims many students have mastered 3-5 years of math in a year. How many hours of study in what subjects? How was mastery measured... did they take the grad school math GRE and ace it? Mastery takes continued practice... I'm highly s

Most online math courses I've looked into [for my friends, my kids, etc.] are "paper thin" and contain less than 25% of the topical matter, descriptive detail, and depth of a good book on the subject... and I'm actually being generous.

I hope your courses are going at least as deep, or offer the capability to, as good books on the various topics. For instance, if linear algebra does not go as deep as Strang + VMLS[0]... folks should just get those two books (VMLS is free), plus watch some youtube, like 3blue1brown.

Hi, I'm Alex, the curriculum director at Math Academy.

I can completely understand the skepticism and agree that many online courses are paper thin. That's where we're different.

For example, our BC Calculus course comprises 302 topics, each containing 3-4 knowledge points, so ~1060 knowledge points in total. Students must master each knowledge point to move on to the next. Our spaced repetition algorithms ensure that students are repeatedly tested on the material (we have quizzes every 150 XP or so). If they fail a question on a quiz or topic review, the system requires that they retake the failed topic. Students _cannot_ complete a course without mastering the entire thing.

Each knowledge point is connected to key prerequisites in the same course and lower courses. If a student stumbles on a particular knowledge point, our system can determine the most likely point of confusion and refer them to the associated key prerequisite topic (which they must pass to continue making progress).

We also have a couple of dozen multistep questions, similar to those you'd find on the BC exam (although the BC exam has about 4-5 parts per question, ours have about 9-10).

Regarding results, we had an 11-year-old sit the BC exam recently, and it looks like they will get a 5, the top mark. (For those that are unaware, students usually sit the BC Calc exam at the end of high school in the US, so 18). I admit that's an extreme case, but it's not isolated. I could reel off many success stories of students achieving real results on real tests after self-studying using our curriculum. We also have an associated school district program in Pasadena, California, where dozens of 8th-graders have achieved 4s and 5s in the BC exam, mostly learning using our system.

In terms of the required effort - provided you have no issues with the necessary prerequisite knowledge, you can get through our entire BC Calculus course by committing 40-50 minutes per day, five days per week, for around 5-6 months. Of course, if there are gaps in the prerequisite knowledge, then it'd take a little longer - but thankfully, our algorithms can detect missing knowledge and fill the gaps. That’s one of the advantages of having an intelligent, interconnected system comprising over 3000 topics!

As for our higher-level courses - some of these are still in development. However, our linear algebra course is comparable to several high-quality books on the subject (I like Lay, Anthony & Harvey, and Axler, though we use others). It currently has 176 topics, but many foundations are laid out in our Integrated Math III / Precalculus courses (vectors, matrices, basic determinants, inverse matrices, linear transformations in the plane), so the real number is around 200.

(click on the "content" tab to get a complete list of topics).

Could one of our students ace the GRE? That's a great question. We still need content on several key areas required for the GRE (e.g., Abstract Algebra, Real Analysis, Complex Analysis, and Graph Theory). These courses are still in development - we already have a lot of this content behind the scenes. That said, I'm confident that our students have the necessary tools to succeed in the parts of the GRE we currently cover. We don't "teach to the test," not even with BC Calc, but equipping our students with the necessary knowledge and skills to go from 4th grade math right the way up to acing the GRE (just as we've done with BC Calc) is one of our medium to long-term goals.

Happy to answer any further questions about the curriculum you may have.

When I am ready to dive into this again, I will definitely look at this. I know I need concrete time dedicated to this sort of thing (repetition is the only way to master it really) but I'll circle back around to this soon!

Looks great and was ready to sign up but I surely wasn't expecting that price!
I am not saying it is not worth that, but as someone who has tried to start learning math on my own only to quit afterwards for whatever reason, it's a big risk to take.

How much would it be worth to you to learn 3-5 years of math in a single year without getting stuck? And I mean really learning it to the point where you're able to solve the more difficult problems and are not merely able to recognize some of the symbols and terminology and talk like you know it. If you're just kind of curious about some advanced math topics you see pop up on HN from time to time and aren't really willing to invest any real time, effort or money into learning the material, which is totally fine and is probably where most people reading this comment are, then sure, spending more than $40 on a book or watching some free online videos will seem expensive.

But the reality is that very few people will be able to learn a significant amount of math by simply working through some problems in a book. Eventually they'll get stuck or just run out of gas, and when I say eventually I mean probably in 2-3 weeks. But if you're that one student who successfully taught themselves multiple courses worth of mathematics on their own from a few books and outside of any educational institution, then hats off to you! You're like that guy who put on 30 pounds of muscle doing pushups and pull-ups at the local park. You know, ... that ONE guy. ;)

But if you want a sure fire way of mastering a large amount of mathematics as efficiently and painlessly as possible, then you want a system like Math Academy that will adapt to your individual learning curve and knowledge frontier and push you through the material using the most effective pedagogy available - careful scaffolding, active problem-based learning, spaced repetition, gamification, etc.

The bottom line is this. Our system is more effective than any course available and is much cheaper for what you get. In fact, we just had a group of students ages 11-13) start with basic pre-algebra in the fall of 2021 (as in Solve x - 4 = 10) and from what I've heard all did extremely well on the AP Calculus BC exam a couple weeks ago. That's like 6-7 academic years of math in 18 months and we're expecting mostly if not all of them to earn a 5 (the top score).

But take my word it. Try it out for yourself. You automatically get a full refund if you cancel in the first 30 days, so there's no risk. And we're always available to answer your questions and support your progress.

I’ve been a paying customer since October last year. I discovered it after someone recommended it in a hackernews comment.

I’m guessing you’re mentally comparing this to all the possible books you could buy instead for that price. But how many of those books would you actually read, let alone finish?
A better comparison is, having an MIT educated math tutor on call for $50 a month.

I have a bachelors in physics but it still feels great to learn new things that my education skipped. For example, we skipped singular value decomposition at my university in the interest of time. Mathacademy says, screw it, we’re teaching everything!

Also as someone with a physics degree, it's difficult for me to think of taking courses beyond sophomore year that didn't involve SVD to some extent or were using proximal solution strategies (solid but not crazy tough public state school, late aughts). It's not something skipped for time, it's a basic tool used in multiple branches of physics/math. I'll need to look further to validate some of the content/capabilities but as with most things, buyer beware.

What can I say. It simply wasn’t taught at our university. Instead the advanced linear algebra course focused more on abstract function spaces to prepare us for quantum mechanics. This was before the machine learning revolution.

Math Academy does not charge your card for the first 30 days. If you find it's not a good fit for then you can cancel within this period and you won't be charged. 30 days hopefully gives you enough time to determine whether it's a good fit or not.

My colleague informs me that, contrary to my previous message, you get charged immediately, but you get an automatic refund if you cancel within 30 days.

Geez, I'm trying to figure out how to describe in a short paragraph or two what it would take a book to explain. Here's my best shot.

We've created an extensive knowledge graph representing all of mathematics (3,000 topics and counting) from 4th Grade Math up through our university-level material, and our algorithms traverse the graph to identify the optimal learning tasks to assign to the the student at any point based on their performance on previously completed learning tasks: diagnostics, lessons, reviews, quizzes, etc.

There are actually multiple graphs, including one that defines the direct prerequisite relationships between topics as well as one that describes encompassing relationships (e.g. the topic on Solving Two-step Linear Equations fully encompasses the topic on Solving One-step Linear Equations Using Multiplication), but there are other graphs as well.

In addition, the algorithms have to deal with spaced repetition, which is vastly more complicated to sort out within the context of a hierarchical knowledge structure with both full and partial encompassings. (Without encompassing relationships, the backlog of reviews would quickly slow progress to a crawl).

We actually have some deep-dive writeup in the works that attempt to explain how all of this works at a level that will be accessible to most people, but it's more than I can describe here, unfortunately.

We should have those courses ready within the next year. Multivariable Calculus should be available in another few weeks, then Probability & Statistics at the end of July, then Methods of Proof, followed by Discrete Math, and Abstract Algebra later this fall. But courses in Number Theory, Graph Theory, Combinatorics, Real Analysis, etc. are all planned.

I think you might like my book "No Bullshit Guide to Math & Physics"[1,2,3], which contains a condensed review of high school math (a.k.a. algebra and precalculus), then explains PHYS and CALC topics in an integrated manner.

It's not one book, but for everything before calculus it would be difficult to beat the books in Israel Gelfand's High School Mathematics Correspondence Curriculum [0]. These are designed for self study and give a fresh perspective on topics they cover.

Excellent recommendation; they are very good books to start with. Concepts are clearly explained and I wish every mathematics textbook was structured like this. Some people are biased against these books because they're Soviet, but I find that attitude parochial. If we're judging textbooks on their merits alone, these will get you to Calculus.

Check out Kahn Academy, they have a gamified course to guide you through the equivalent of a high school and early college math curriculum. AFAIK it's free?

Khan Academy is great. When I was taking AP calculus in highschool I failed to complete any homework and barely passed the class. But when it came time for the AP test I binged Khan Academy videos for 3 days beforehand and ended up getting a 5 (the max score). Great resource and even bingeable

Specifically these three courses [0][1][2] will take you from basic algebra to precalc. They're very thorough and I've found them extremely useful in upgrading my high school level math skills. I have heard that their calculus courses aren't sufficient though, and that it should be learned from somewhere else.

The Math 1-3 courses intersperse all of those courses and provide a more streamlined path. I would personally recommend using those 3 courses to learn up to pre-calc.

The aops books [0] will take you from prealgebra all the way to calculus and discrete math and will give you a foundation strong enough to enter any undergraduate math program in the world.

And the books all have complete solutions manuals available so you can get immediate feedback.

You could use programs like Anki to schedule your review of defintions you've understood and problems you've solved to supercharge your learning as well.

Very fast calculus: Consider a standard car with a speedometer (reports how fast are going) and an odometer (reports how far have gone).

Easily enough we can take the speedometer readings, say, 1 time each second, and calculate a good approximation to the odometer readings. That is a 1 second approximation to the calculus operation of integration.

Similarly we can take the odometer readings, say, 1 time each second, and calculate a good approximation to the speedometer readings. That is a 1 second approximation to the calculus operation of differentiation.

If we use smaller time intervals than just 1 second, then we will usually get a more accurate approximation. It is a theorem that, under mild assumptions, as we let the lengths of the time intervals shrink toward 0, the results of the operations will reach limits and quit changing.

Those limiting values are the actual definitions of differentiation and integration.

No big surprise, under mild assumptions, if we start with the odometer readings, differentiate to get the speedometer readings, and integrate to get back the odometer readings, then we really will get back the odometer readings. That is the fundamental theorem of calculus.

Some common mild assumptions are basically that the speedometer readings change only continuously (no jumps) over time and we are working over only finitely long time intervals.

Newton's second law of motion

force = mass x acceleration

essentially guarantees the continuity of the speedometer readings and, thus, justifies the integration back to the odometer readings.

Of course, calculus and Newton's second law of motion are close cousins in both theory and applications -- no big surprise since Newton essentially created both (might mention Leibniz and some others).

Can quickly show that if we integrate time t, we get (1/2)t^2. So if we differentiate (1/2)t^2 we will get back t.

A calculus course will show how to differentiate and integrate a wide variety of mathematical expressions, polynomials, sines and cosines, products, quotients, composite expressions, etc.. E.g., differentiate sine(t) and get cosine(t). Differentiate cosine(t) and get -sine(t). Can also find many cases of arc lengths, areas, volumes.

Suppose we are starting a business. At time t, let the revenue be y(t). Suppose we have argued that as we reach all our target customers, our monthly revenue will be b. Suppose we argue that due to word of mouth advertising the rate of growth is proportional to both the number of happy customers talking and the number of target customers not yet customers listening. Denote the rate of growth of y(t), that is the derivative, by y'(t). Then for some constant of proportionality we should have

y'(t) = k y(t) ( b - y(t) )

Of course we know current revenue, say, at time t = 0, that is, y(0).

Then by the first weeks of calculus, can show that, with TeX syntax,

y(t) = { y(0) b e^{bkt} \over y(0) \big
( e^{bkt} - 1 \big ) + b }

More generally

y'(t) = k y(t) ( b - y(t) )

is an example of an initial value problem
of a first order, linear, ordinary differential
equation and an introduction to a course in ordinary differential equations.

Calculus has wide applications to physical science, engineering, economics, finance,
spread of diseases, etc.

Curious if I could find another interesting way to learn math for someone who hasn't gone to college, I asked chatGPT to cluster 20 animal emojis by taxonomy and use them to explain Affinity Propagation like I'm 5 year old. (I'm an idiot.) More or less, the lion emoji wants to be friends with the tiger emoji more than the dog emoji with some explanation of the math and math symbols in between.

```
For example, when Scar wanted to be king, he sent a "responsibility" message to the other big cats, trying to convince them that he should be the leader. However, the "availability" message he received back was weak because most animals didn't trust him.

Meanwhile, Simba sent out a strong "responsibility" message showing he could be a good leader, and in return, he got a strong "availability" message back with many animals showing support. That's why Simba was a better leader for the Pride Lands, according to Affinity Propagation!

Same here! May I ask which field of programming you went into? And how you feel about being a programmer?

I often wonder how it might have been if I just had stayed with math. Especially, after years of doing regular programming for the Web, recently I had to develop a computational geometry library. It had been years since I graduated but really made me feel nostalgic. I think I lost something when I left the field, but then again life is a lot easier now...

At some point I was interested in learning to read and write proofs.

I did the "Introduction to Mathematical Thinking" MOOC from Keith Devlin. The curriculum is available as a book as well.

The class is basically how to write and read proofs for non-math majors. It starts pretty slow, but gets harder at some point. The number theory proofs were fun.

You 'got to' grade others proofs online, and they graded yours which was an interesting way to get familiar with reading and writing proofs.

I recommend it because instead of an area of math it focuses on what it means to prove something. And the teacher is pretty entertaining.

That book has a completely different focus... breadth instead of understanding argumentation.

I agree that the Programmer's Introduction to Mathematics is more likely to contain useful content (instead of being about how to develop the ability to reason carefully). It also has a LOT more breadth than the OP.

The free part of Chapter 3 "On Pace and Patience" is a key very important attitude toward learning from this book (especially on your own). If you are thinking about studying from this book, make sure you philosophically agree.

I tried to copy and paste a paragraph here, but it looks like it has been ROT-13 encoded in the PDF (or something)!

> [...] mathematical culture requires being comfortable being almost continuously in a state of little to no understanding It's a humble life [...]

With that title and reception I can imagine people bookmarking this for „later“ and feeling good about it. But who reads that stuff really?

To each their own, but 700+ pages for material that is done in my experience in the first 2-3 weeks of undergraduate math is more disheartening than empowering for a student, in my opinion.

If you can open a math book anywhere in the last 20% of pages and just start reading, you are looking at pop science and not lecture notes.

FWIW, Barr and Wells in a well respected reference on category theory
[1] use a different definition of a function than the one in this
book. Essentially they avoid insisting on the concrete model of a
function as a set of ordered pairs (which they call its graph), and
would regard two functions having identical graphs as being
nevertheless distinct if they have different co-domains (section
1.2). IMO their definition is better thought out and ultimately less
troublesome when it comes to reasoning about functions. The definition
of a function is always the first thing I check in any book about math
for a rough assessment of whether reading the rest of it might make me
dumber or smarter.

Indeed. The question "given a function as a set of ordered pairs, write code to decide if it is injective" has an answer, but replace that with "surjective" and suddenly you need to drag the codomain along as an extra parameter or the question makes no sense. This asymmetry bothers me.

EDIT: to be clear, "code" only works if your set is finite, but the principle of "does this question even have an answer" remains "yes" for injective and "no" for surjective without the codomain being named explicitly.

What I got was one of the best bound paperback books I've ever held.

<rant>
I wish all publishers would just sell eBooks (and publish settings like "optimal paper size") so that I could print my own copy using a service like lulu. One of my biggest gripes with hardcopy books nowadays is that the paper stock they use is so thin that the ink on the other side of the page shows through. Lulu let's you choose the paper stock. I really can't say enough about how happy I was with the quality of the printed book lulu sent me.
</rant>

In another recent thread, somebody mentioned using some Amazon feature for doing this. I might be wrong, but I think you sign up as an author, upload the PDF like it's your book, then buy one copy (or request a proof) for yourself, and then delete it. Maybe somebody who has used this facility can chime in with more (or better) details...

I print them myself and take them to FedEx to do the binding. Letter size/A4, though, result in big books and a lot of paper. So I wrote a script that produces a new PDF and reorders the pages. In the new PDF, you have two pages per side of paper (4 pages per sheet). The reordering is done so that FedEx can cut the sheet right in the middle, and put the left half on top of the right half to get the usual ordering. Then they just bind it.

Not counting the cost of my paper, they charge about $11 (probably can get it a bit cheaper but I add some extras in the service when I bind them).

There might be a print shop in your area which offers similar services. Or services which explicitly offer to print thesis, but they typically require a minimum number of prints.

Note that these are 700 pages which could take multiple years (if you're doing it in your spare time) to go through. I'm not sure who would have the motivation and discipline for this.

Most of the books are meant to be consumed in parts. This book explicitly mentions "Segue from previous chapter" for most chapters indicating what key concepts are continued in exploring/building new ones. Also, I believe any reader on HN can make sense quickly of the basic definitions needed to understand a concept in isolation.

However, mathematics is especially known to require continuous dedication for years to attain any sort of mastery.

It depends on how rigorous/detailed you want to be. This is at most two semesters worth of material, and has a very shallow learning curve (hence the large PDF). If you've been exposed to these ideas, you could cover it in a semester. I'd say it's quite doable in a year in your spare time.

If you drop the combinatorics chapter, it is definitely doable in one semester.

For example, Chapter 7 is 100 pages. In Tao's analysis book, he covers the same material in 42 pages.

It might, but it shouldn’t. There are 8 chapters and most of the content in the first 7 shows up in any run of the mill “Introduction to Higher Mathematics” or whatever that university decides to call their first actual math class, although occasionally combinatorics will replace number theory content.

Far from being "everything" anyone would want or need, it's rather "some (fundamental) things" you must know. A couple of pages a day, on average, would get you there within one year.

Still, I'm in doubt that a significant proportion of us would have the discipline for this. One reason why some things are pretty much only learned in university is that they provide the necessary motivation.

I've been very interested in this method for a while. How do you find the math that would solve them? Is it well-understood, or is it easy to figure out in practice? Would love to hear more about how this looks for you practically.

If I remember correctly, Brendan Sullivan had a reputation as a TA for Concepts of Mathematics at CMU as "Math Jesus", not sure if that was a testament to his pedagogical skills or just due to the long hair and beard...

> The Doctor of Arts degree shares all requirements and standards with the Ph.D., except with regard to the thesis. The D.A. thesis is not expected to display the sort of original research required for a Ph.D. thesis, but rather to demonstrate an ability to organize, understand, and present mathematical ideas in a scholarly way, usually with sufficient innovation and worth to produce a publishable work. Whenever practical, the department provides D.A. candidates with the opportunity to use materials developed to teach a course. While a typical Ph.D. recipient will seek a position that has a substantial research component, the D.A. recipient will usually seek a position where research is not central.

Well, there's too much of (modern) mathematics to touch everything - even everything important... One of the most important notions is that of compact sets, and it's not even mentioned. Neither are groups (well, they are mentioned, but not much else), etc.

From a quick skim of the ToC this looks very well thought-out. Looks like an excellent book!

If you want to go deeply into formalized mathematics, take a look at the Metamath Proof Explorer <https://us.metamath.org/mpeuni/mmset.html>. It defines a set of axioms, and formally verifies every proof showing every step (hiding nothing).

How do people here like to read pdfs? It doesn't work on my ebook reader and I never actually read anything long-form on a computer screen, even when I think I will. Someone below mentioned using a service to print and bind into a book. Anyone have a better workflow for this kind of thing?

I bought an iPad specifically for reading PDFs. It does get used for other things too, but the primary driver was PDFs. Most tablets seem to be optimised for horizontal video content, which makes then too narrow for A4 PDF, whereas the 4:3 of the iPad is perfect.

One thing I've always wanted is a comprehensive guide to mathematical notation, which tells you what symbols mean or at least what field of mathematics they come from.

I frequently come across all sorts of weird mathematical symbols in papers, and of course these symbols are virtually never explained, so I have no idea what they mean.

Even better would be if there was some way an LLM could read through a paper itself and then explain the equations.

One problem (feature?) of math notation is that paper authors don't follow a consistent convention for symbols, so what you're asking might not even be possible... It doesn't help that different math domains might use different symbols for the same concept! That being said, there is the ISO 80000-2 standard that defines recommendations for many of the math symbols, with mentions of other variations, see https://web.archive.org/web/20210705180417/https://people.en...
You might want to read through that as a starting point.

Unfortunately, just knowing the notation (being able to read the symbols) is not usually enough. Understanding each symbol/concept usually requires knowing the math context and other related definitions of the domain where the notation is used. In other words, knowing the notation is not a shortcut for learning math... But still, I hear you about the need to select some symbols then right-click and choose "read this to me" or "explain this to me."

One thing I love about programmers is that they rarely use complex symbols. They are the complete opposite to mathematicians in this regard. Basically all programming languages consist just of ASCII keywords. So when in doubt about some keyword, you can just use Google and type it in. Together with the name of the language.

I guess symbols look fancier and are more concise.

Also, mathematicians do love their PDFs. No doubt because it supports their beloved symbols so well. Of course PDFs are terrible for the Web and screens in general, but only a programmer would care about that.

GPT-4 can probably explain LaTeX formulas pretty well, but when they are compiled PDFs, the encoded PDF plain text is often garbage, so GPT-4 probably couldn't understand much.

many books will have a page in the preface that lays out the notation, even for very common objects. it's hard to make a cheat sheet for something like this because many symbols are used in different fields, and even if different subfields of the same field.

If I could read, and understand, a textbook like this in a day, I would be so happy. Alas, I must only look with longing at the links provided by HN, thinking wistfully 'someday...'

Fair enough -- to appreciate it really, one needs a motivated undergrad. But, it's similar in pedagogical style. It's not run of the mill concepts, but rather like a conversation that builds up with strong hints to form intuition.

Susskind's series does a pretty decent job walking the reader through topics in (theoretical) physics. At a somewhat more advanced level, you can try Schwichtenberg's No-Nonsense books. I also enjoyed Stevens' The Six Core Theories of Modern Physics.

Do you mean basic physics like mechanics (PHYS 101) ? If this is what you're interested, check out my MATH & PHYS book. Posted link to it in another comment this thread.

[0] https://infinitedescent.xyz/

[0] https://www.goodreads.com/book/show/445059.Mathematical_Thin...

[0] https://www.cs.yale.edu/homes/aspnes/classes/202/notes.pdf

That said, OP's book looks more conversational in tone, which I personally have a slight preference for.

I remember at the time the book was in loose leaf paper so it warms my heart to see the book has a 3rd edition. It was used as a core part of teaching assembly, memory representations, and getting students ready for the operating systems class. When I help people learn to program, it's the only book I think is a must have:

https://www.amazon.com/Computer-Systems-Programmers-Perspect...

What are the other books that keep around or wish to keep around?

This is my singular biggest hurdle in going back to school to finish my degree and I'd love to fill the gaps I have around mathematics so I can not only finish my degree; I'd also like to participate in some more advanced computer science that rely heavily on underlying computation.

https://mathacademy.com/courses/mathematical-foundations-i https://mathacademy.com/courses/mathematical-foundations-ii https://mathacademy.com/courses/mathematical-foundations-iii

We also have courses on Linear Algebra and Mathematics for Machine Learning:

https://mathacademy.com/courses/linear-algebra https://mathacademy.com/courses/mathematics-for-machine-lear...

It's not free, but our adaptive, AI-driven algorithms makes it the most efficient way to learn math that you're going to find. We've had numerous students master 3-5 years of math in a single year.

We're still in beta and haven't done a proper Show HN yet, but we're getting there!

I'm the founder, so I'd be happy to answer any questions.

BUT... I'm highly skeptical of any online math course that claims many students have mastered 3-5 years of math in a year. How many hours of study in what subjects? How was mastery measured... did they take the grad school math GRE and ace it? Mastery takes continued practice... I'm highly s

Most online math courses I've looked into [for my friends, my kids, etc.] are "paper thin" and contain less than 25% of the topical matter, descriptive detail, and depth of a good book on the subject... and I'm actually being generous.

I hope your courses are going at least as deep, or offer the capability to, as good books on the various topics. For instance, if linear algebra does not go as deep as Strang + VMLS[0]... folks should just get those two books (VMLS is free), plus watch some youtube, like 3blue1brown.

[0] https://web.stanford.edu/~boyd/vmls/

Edit: btw... not trying to be overly harsh, just skeptical. If your courses end up being half as good as advertised I'll 100% sign up at some point.

I can completely understand the skepticism and agree that many online courses are paper thin. That's where we're different.

For example, our BC Calculus course comprises 302 topics, each containing 3-4 knowledge points, so ~1060 knowledge points in total. Students must master each knowledge point to move on to the next. Our spaced repetition algorithms ensure that students are repeatedly tested on the material (we have quizzes every 150 XP or so). If they fail a question on a quiz or topic review, the system requires that they retake the failed topic. Students _cannot_ complete a course without mastering the entire thing.

Each knowledge point is connected to key prerequisites in the same course and lower courses. If a student stumbles on a particular knowledge point, our system can determine the most likely point of confusion and refer them to the associated key prerequisite topic (which they must pass to continue making progress).

We also have a couple of dozen multistep questions, similar to those you'd find on the BC exam (although the BC exam has about 4-5 parts per question, ours have about 9-10).

Regarding results, we had an 11-year-old sit the BC exam recently, and it looks like they will get a 5, the top mark. (For those that are unaware, students usually sit the BC Calc exam at the end of high school in the US, so 18). I admit that's an extreme case, but it's not isolated. I could reel off many success stories of students achieving real results on real tests after self-studying using our curriculum. We also have an associated school district program in Pasadena, California, where dozens of 8th-graders have achieved 4s and 5s in the BC exam, mostly learning using our system.

In terms of the required effort - provided you have no issues with the necessary prerequisite knowledge, you can get through our entire BC Calculus course by committing 40-50 minutes per day, five days per week, for around 5-6 months. Of course, if there are gaps in the prerequisite knowledge, then it'd take a little longer - but thankfully, our algorithms can detect missing knowledge and fill the gaps. That’s one of the advantages of having an intelligent, interconnected system comprising over 3000 topics!

As for our higher-level courses - some of these are still in development. However, our linear algebra course is comparable to several high-quality books on the subject (I like Lay, Anthony & Harvey, and Axler, though we use others). It currently has 176 topics, but many foundations are laid out in our Integrated Math III / Precalculus courses (vectors, matrices, basic determinants, inverse matrices, linear transformations in the plane), so the real number is around 200.

https://mathacademy.com/courses/linear-algebra

(click on the "content" tab to get a complete list of topics).

Could one of our students ace the GRE? That's a great question. We still need content on several key areas required for the GRE (e.g., Abstract Algebra, Real Analysis, Complex Analysis, and Graph Theory). These courses are still in development - we already have a lot of this content behind the scenes. That said, I'm confident that our students have the necessary tools to succeed in the parts of the GRE we currently cover. We don't "teach to the test," not even with BC Calc, but equipping our students with the necessary knowledge and skills to go from 4th grade math right the way up to acing the GRE (just as we've done with BC Calc) is one of our medium to long-term goals.

Happy to answer any further questions about the curriculum you may have.

You would probably get more traction if you offered a free month up front because so many platforms before you have failed to deliver on the hype.

How much would it be worth to you to learn 3-5 years of math in a single year without getting stuck? And I mean really learning it to the point where you're able to solve the more difficult problems and are not merely able to recognize some of the symbols and terminology and talk like you know it. If you're just kind of curious about some advanced math topics you see pop up on HN from time to time and aren't really willing to invest any real time, effort or money into learning the material, which is totally fine and is probably where most people reading this comment are, then sure, spending more than $40 on a book or watching some free online videos will seem expensive.

But the reality is that very few people will be able to learn a significant amount of math by simply working through some problems in a book. Eventually they'll get stuck or just run out of gas, and when I say eventually I mean probably in 2-3 weeks. But if you're that one student who successfully taught themselves multiple courses worth of mathematics on their own from a few books and outside of any educational institution, then hats off to you! You're like that guy who put on 30 pounds of muscle doing pushups and pull-ups at the local park. You know, ... that ONE guy. ;)

But if you want a sure fire way of mastering a large amount of mathematics as efficiently and painlessly as possible, then you want a system like Math Academy that will adapt to your individual learning curve and knowledge frontier and push you through the material using the most effective pedagogy available - careful scaffolding, active problem-based learning, spaced repetition, gamification, etc.

The bottom line is this. Our system is more effective than any course available and is much cheaper for what you get. In fact, we just had a group of students ages 11-13) start with basic pre-algebra in the fall of 2021 (as in Solve x - 4 = 10) and from what I've heard all did extremely well on the AP Calculus BC exam a couple weeks ago. That's like 6-7 academic years of math in 18 months and we're expecting mostly if not all of them to earn a 5 (the top score).

But take my word it. Try it out for yourself. You automatically get a full refund if you cancel in the first 30 days, so there's no risk. And we're always available to answer your questions and support your progress.

I’m guessing you’re mentally comparing this to all the possible books you could buy instead for that price. But how many of those books would you actually read, let alone finish? A better comparison is, having an MIT educated math tutor on call for $50 a month.

I have a bachelors in physics but it still feels great to learn new things that my education skipped. For example, we skipped singular value decomposition at my university in the interest of time. Mathacademy says, screw it, we’re teaching everything!

Math Academy does not charge your card for the first 30 days. If you find it's not a good fit for then you can cancel within this period and you won't be charged. 30 days hopefully gives you enough time to determine whether it's a good fit or not.

Apologies for any confusion.

> our adaptive, AI-driven algorithms makes it the most efficient way to learn math that you're going to findCan you elaborate on this? What do these algorithms do?

We've created an extensive knowledge graph representing all of mathematics (3,000 topics and counting) from 4th Grade Math up through our university-level material, and our algorithms traverse the graph to identify the optimal learning tasks to assign to the the student at any point based on their performance on previously completed learning tasks: diagnostics, lessons, reviews, quizzes, etc.

There are actually multiple graphs, including one that defines the direct prerequisite relationships between topics as well as one that describes encompassing relationships (e.g. the topic on Solving Two-step Linear Equations fully encompasses the topic on Solving One-step Linear Equations Using Multiplication), but there are other graphs as well.

In addition, the algorithms have to deal with spaced repetition, which is vastly more complicated to sort out within the context of a hierarchical knowledge structure with both full and partial encompassings. (Without encompassing relationships, the backlog of reviews would quickly slow progress to a crawl).

We actually have some deep-dive writeup in the works that attempt to explain how all of this works at a level that will be accessible to most people, but it's more than I can describe here, unfortunately.

Anyway, I hope this helps a little.

[1] website https://minireference.com/ [2] PDF preview and sample chapter = https://minireference.com/static/excerpts/noBSmathphys_v5_pr... [3] concept map = https://minireference.com/static/conceptmaps/math_and_physic...

If you prefer something focussed on a review of high school math topic, then you might prefer the "green book" instead, see https://nobsmath.com/

[0] https://www.goodreads.com/series/318605-gelfand-corresponden...

[0]https://www.khanacademy.org/math/math1 [1]https://www.khanacademy.org/math/math2 [2]https://www.khanacademy.org/math/math3

And the books all have complete solutions manuals available so you can get immediate feedback.

You could use programs like Anki to schedule your review of defintions you've understood and problems you've solved to supercharge your learning as well.

[0] https://artofproblemsolving.com/store

I recommended a lot of people these courses and myself went over a few videos to revise Trigonometry.

I can vouch for the quality.

It assumes you have some algebra, but does not require college algebra.

it really takes you from the ground up all the way to advanced subjects. He published multiple books on various levels of mathematics.

The explanations are great and the examples and excises are such that you can just do them in your head.

Easily enough we can take the speedometer readings, say, 1 time each second, and calculate a good approximation to the odometer readings. That is a 1 second approximation to the calculus operation of

integration.Similarly we can take the odometer readings, say, 1 time each second, and calculate a good approximation to the speedometer readings. That is a 1 second approximation to the calculus operation of

differentiation.If we use smaller time intervals than just 1 second, then we will usually get a more accurate approximation. It is a theorem that, under mild assumptions, as we let the lengths of the time intervals shrink toward 0, the results of the operations will reach limits and quit changing.

Those limiting values are the actual definitions of differentiation and integration.

No big surprise, under mild assumptions, if we start with the odometer readings, differentiate to get the speedometer readings, and integrate to get back the odometer readings, then we really will get back the odometer readings. That is the fundamental theorem of calculus.

Some common mild assumptions are basically that the speedometer readings change only continuously (no jumps) over time and we are working over only finitely long time intervals.

Newton's second law of motion

force = mass x acceleration

essentially guarantees the continuity of the speedometer readings and, thus, justifies the integration back to the odometer readings.

Of course, calculus and Newton's second law of motion are close cousins in both theory and applications -- no big surprise since Newton essentially created both (might mention Leibniz and some others).

Can quickly show that if we integrate time t, we get (1/2)t^2. So if we differentiate (1/2)t^2 we will get back t.

A calculus course will show how to differentiate and integrate a wide variety of mathematical expressions, polynomials, sines and cosines, products, quotients,

compositeexpressions, etc.. E.g., differentiate sine(t) and get cosine(t). Differentiate cosine(t) and get -sine(t). Can also find many cases of arc lengths, areas, volumes.Suppose we are starting a business. At time t, let the revenue be y(t). Suppose we have argued that as we reach all our target customers, our monthly revenue will be b. Suppose we argue that due to word of mouth advertising the rate of growth is proportional to both the number of happy customers talking and the number of target customers not yet customers listening. Denote the rate of growth of y(t), that is the derivative, by y'(t). Then for some constant of proportionality we should have

y'(t) = k y(t) ( b - y(t) )

Of course we know current revenue, say, at time t = 0, that is, y(0).

Then by the first weeks of calculus, can show that, with TeX syntax,

y(t) = { y(0) b e^{bkt} \over y(0) \big ( e^{bkt} - 1 \big ) + b }

More generally

y'(t) = k y(t) ( b - y(t) )

is an example of an

initial value problem of a first order, linear, ordinary differential equationand an introduction to a course in ordinary differential equations.Calculus has wide applications to physical science, engineering, economics, finance, spread of diseases, etc.

``` For example, when Scar wanted to be king, he sent a "responsibility" message to the other big cats, trying to convince them that he should be the leader. However, the "availability" message he received back was weak because most animals didn't trust him.

Meanwhile, Simba sent out a strong "responsibility" message showing he could be a good leader, and in return, he got a strong "availability" message back with many animals showing support. That's why Simba was a better leader for the Pride Lands, according to Affinity Propagation!

```

I often wonder how it might have been if I just had stayed with math. Especially, after years of doing regular programming for the Web, recently I had to develop a computational geometry library. It had been years since I graduated but really made me feel nostalgic. I think I lost something when I left the field, but then again life is a lot easier now...

I did the "Introduction to Mathematical Thinking" MOOC from Keith Devlin. The curriculum is available as a book as well.

The class is basically how to write and read proofs for non-math majors. It starts pretty slow, but gets harder at some point. The number theory proofs were fun.

You 'got to' grade others proofs online, and they graded yours which was an interesting way to get familiar with reading and writing proofs.

I recommend it because instead of an area of math it focuses on what it means to prove something. And the teacher is pretty entertaining.

https://www.amazon.ca/Introduction-Mathematical-Thinking-Kei...

https://www.coursera.org/learn/mathematical-thinking

For geometry, get a book like Art of Problem Solving's Introduction to Geometry. That will cover many beautiful topics in a question and answer style.

[1] https://artofproblemsolving.com/store/book/intro-geometry

I agree that the Programmer's Introduction to Mathematics is more likely to contain useful content (instead of being about how to develop the ability to reason carefully). It also has a LOT more breadth than the OP.

The free part of Chapter 3 "On Pace and Patience" is a key very important attitude toward learning from this book (especially on your own). If you are thinking about studying from this book, make sure you philosophically agree.

I tried to copy and paste a paragraph here, but it looks like it has been ROT-13 encoded in the PDF (or something)!

> [...] mathematical culture requires being comfortable being almost continuously in a state of little to no understanding It's a humble life [...]

To each their own, but 700+ pages for material that is done in my experience in the first 2-3 weeks of undergraduate math is more disheartening than empowering for a student, in my opinion.

If you can open a math book anywhere in the last 20% of pages and just start reading, you are looking at pop science and not lecture notes.

I'm in this comment and I don't like it.

> If you can open a math book anywhere in the last 20% of pages and just start reading, you are looking at pop science and not lecture notes.

What do you mean?

https://www.lesswrong.com/posts/xg3hXCYQPJkwHyik2/the-best-t...

[1] https://abstractmath.org/CTCS/CTCS.pdf

EDIT: to be clear, "code" only works if your set is finite, but the principle of "does this question even have an answer" remains "yes" for injective and "no" for surjective without the codomain being named explicitly.

I used lulu.com to print a copy of OnLisp [1]

What I got was one of the best bound paperback books I've ever held.

<rant> I wish all publishers would just sell eBooks (and publish settings like "optimal paper size") so that I could print my own copy using a service like lulu. One of my biggest gripes with hardcopy books nowadays is that the paper stock they use is so thin that the ink on the other side of the page shows through. Lulu let's you choose the paper stock. I really can't say enough about how happy I was with the quality of the printed book lulu sent me. </rant>

[1] https://www.lurklurk.org/onlisp/onlisp.html

thinkyou sign up as an author, upload the PDF like it's your book, then buy one copy (or request a proof) for yourself, and then delete it. Maybe somebody who has used this facility can chime in with more (or better) details...I print them myself and take them to FedEx to do the binding. Letter size/A4, though, result in big books and a lot of paper. So I wrote a script that produces a new PDF and reorders the pages. In the new PDF, you have two pages per side of paper (4 pages per sheet). The reordering is done so that FedEx can cut the sheet right in the middle, and put the left half on top of the right half to get the usual ordering. Then they just bind it.

Not counting the cost of my paper, they charge about $11 (probably can get it a bit cheaper but I add some extras in the service when I bind them).

[1] https://www.printonweb.in

However, mathematics is especially known to require continuous dedication for years to attain any sort of mastery.

If you drop the combinatorics chapter, it is definitely doable in one semester.

For example, Chapter 7 is 100 pages. In Tao's analysis book, he covers the same material in 42 pages.

Far from being "everything" anyone would want or need, it's rather "some (fundamental) things" you

mustknow. A couple of pages a day, on average, would get you there within one year.So does that mean this is his PhD thesis? What's a Doctor of Arts?

> The Doctor of Arts degree shares all requirements and standards with the Ph.D., except with regard to the thesis. The D.A. thesis is not expected to display the sort of original research required for a Ph.D. thesis, but rather to demonstrate an ability to organize, understand, and present mathematical ideas in a scholarly way, usually with sufficient innovation and worth to produce a publishable work. Whenever practical, the department provides D.A. candidates with the opportunity to use materials developed to teach a course. While a typical Ph.D. recipient will seek a position that has a substantial research component, the D.A. recipient will usually seek a position where research is not central.

It is intended to develop that skill, not introduce you to a breadth of topics.

compact sets, and it's not even mentioned. Neither are groups (well, theyarementioned, but not much else), etc.From a quick skim of the ToC this looks very well thought-out. Looks like an excellent book!

I frequently come across all sorts of weird mathematical symbols in papers, and of course these symbols are virtually never explained, so I have no idea what they mean.

Even better would be if there was some way an LLM could read through a paper itself and then explain the equations.

Check out this excerpt of definitions of basic math notation I use in my books: https://minireference.com/static/excerpts/set_notation.pdf It has some examples of the "alien symbols" ∀ (for all), ∃ (there exists), etc.

One problem (feature?) of math notation is that paper authors don't follow a consistent convention for symbols, so what you're asking might not even be possible... It doesn't help that different math domains might use different symbols for the same concept! That being said, there is the ISO 80000-2 standard that defines recommendations for many of the math symbols, with mentions of other variations, see https://web.archive.org/web/20210705180417/https://people.en... You might want to read through that as a starting point.

Unfortunately, just knowing the notation (being able to read the symbols) is not usually enough. Understanding each symbol/concept usually requires knowing the math context and other related definitions of the domain where the notation is used. In other words, knowing the notation is not a shortcut for learning math... But still, I hear you about the need to select some symbols then right-click and choose "read this to me" or "explain this to me."

I guess symbols look fancier and are more concise.

Also, mathematicians do love their PDFs. No doubt because it supports their beloved symbols so well. Of course PDFs are terrible for the Web and screens in general, but only a programmer would care about

that."Basically all programming languages consist just of ASCII keywords. So when in doubt about some keyword, you can just use Google and type it in."Programming languages have their own symbols, and unfortunately search engines are petty awful at searching for them.

1. ∇ = nabla see https://mathworld.wolfram.com/Nabla.html

2. ...

3. ...

would improve your understanding. Notation encodes ideas, it is the understanding of the ideas that is tricky, not the encoding.

It would also help me figure out what to ask LLMs (or people) to explain to me.

I frequently look at PDFs online and am looking for a tool that reformats the PDF into a single column, so I can just scroll, absorbing the content.

It would be golden if you can create a study group that meets weekly. Just start a discord/Element room/Zulip/IRC chat and go from there.

I am learning Topology as an adult and learned Category Theory this way.

This truly works.

Setting aside time 4-5 days a week works. Make it a habit.

[1]: https://www.feynmanlectures.caltech.edu/

Feynman Lectures, to be properly understood requires one to be at least an advanced undergrad.

Most people just use it as home decor and many use it like a novel.

To

properlyappreciate the material, you need training in Physics. Otherwise you will be getting much less.