Everything you always wanted to know about mathematics (2013) [pdf]
math.cmu.edu
math.cmu.edu
That said, OP's book looks more conversational in tone, which I personally have a slight preference for.
[0] https://www.goodreads.com/book/show/445059.Mathematical_Thin...
[0] https://www.cs.yale.edu/homes/aspnes/classes/202/notes.pdf
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.
[0]https://www.khanacademy.org/math/math1 [1]https://www.khanacademy.org/math/math2 [2]https://www.khanacademy.org/math/math3
[0] https://www.goodreads.com/series/318605-gelfand-corresponden...
``` 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!
```
The explanations are great and the examples and excises are such that you can just do them in your head.
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.
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.
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!
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.
Can 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.
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.
It assumes you have some algebra, but does not require college algebra.
[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/
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.
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 really takes you from the ground up all the way to advanced subjects. He published multiple books on various levels of mathematics.
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.
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...
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...
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>
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).
However, mathematics is especially known to require continuous dedication for years to attain any sort of mastery.
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.
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.
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.
From a quick skim of the ToC this looks very well thought-out. Looks like an excellent book!
It is intended to develop that skill, not introduce you to a breadth of topics.
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.
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.
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.
Programming languages have their own symbols, and unfortunately search engines are petty awful at searching for them.
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.
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 properly appreciate the material, you need training in Physics. Otherwise you will be getting much less.