I went from only having done high school math 10 years ago to completing an MS in math and statistics at my local state university while working in an unrelated field. I would recommend NOT starting with calculus if you haven’t done it, instead, just learn how to do proofs - I used Chartrand “Mathematical proofs” - You don’t need to know any math beyond algebra in order to do that most of this book. If you need to revise or learn Algebra, then I would do Stroud “engineering math” first which is designed for self-learners with lots of solutions and feedback.
At some point, it would be good to get a a copy of Lyx and start to learn to write math in LaTeX - Then you can get feedback on your proofs online at math.stackexchange.com if you don’t know any math people locally.
Feel free to get in touch with me if you want to discuss further, happy to help!
I read a Murakami novel in high school, 1Q84. The protagonist is a math teacher who talked about math in a way that I had never seen before. I'd been told I was "good at math" beforehand(for whatever that means, I'm not a fields medalist or anything), but for ~6 months after reading that book, I was _really good_. Like, suddenly I did not have to do any homework in my sr. year calculus class. I loved sitting in class and watching my teacher work through problems, and it seemingly imprinted directly into my brain, because while doing no homework I could still ace the exams while writing with a pen (no erasing and re-do'ing with a pencil). All because of the way this fictional teacher from 1Q84 talked about math.
Has anyone else had an experience like that? (With math or other things?)
I loved last year being able to take university courses online. I knocked out analysis, topology, and quantum mechanics as a non matriculated student. I'd had those books for years but never could get through them alone. (The main thing being, you really don't have anything to gague whether you know it well enough or not).
I really wish there was more opportunity for that. I'd love to take a few more classes, mostly in pure math, but there's simply nothing on offer for remote study past the 200ish level. (There are some remote masters programs in applied math, but nothing for pure).
I don't think I'd enjoy doing a PhD full-time. One or two classes per semester while working seems just about right. But the closest university is an hour away, so in-person isn't a realistic option.
If doing math is essential to conceptual understanding and application, could the interface of math and physics be made more human-centered? For instance, the shift from Roman numerals to Arabic numerals made doing math easier. Based on your experience, might it be possible to increase accessibility by revising some of the arcane conventions of math and physics?
See Brett Victor’s 2011 proposal: http://worrydream.com/KillMath/
I've been on a Math journey since I retired a couple of years ago and I agree with all the books mentioned that I know and look forward to picking up some of the one I do not know. I agree baby Rudin is essential, but I find it tough going.
Some books I liked for self study because they have answers:
Introduction to Analysis, Mattock.
Elementary Differential Geometry, Pressley.
There is also recently Needham's Visual Differential Geometry and Forms, which is great.
Edit: I should also mention Topology without Tears (free, online, very good) https://www.topologywithouttears.net/
This felt like it was written by a physicist or engineer.
Too much emphasis on differential equations and not enough on things like topology, functional analysis and/or non-introductory parts of algebra like say representation theory.
Overall a pretty decent list, although I would suggest considering some tweaks.
For real analysis it recommends as essential Abbott's "Understanding Analysis" and Rudin's "Principles of Mathematical Analysis". If you "haven't gotten your fill of real analysis" from those it recommends Spivak's "Calculus".
I'd consider promoting Spivak to essential, but using it for calculus rather than real analysis, replacing their recommendation of Stewart's "Calculus: Early Transcendentals".
By doing calculus with a more rigorous, proof-oriented introductory calculus book like Spivak, there is a good chance you won't need a separate introduction to proofs book so can drop the recommended Vellemen's "How to Prove It: A Structured Approach".
Modern calculus (analysis) was invented because people shot themselves in the foot working with topology and wondering exactly what is a "curve" ? I am a big fan of this approach to learning mathematics, just forge ahead and when (if) things fall apart then go back and fix up the foundations. To this end I recommend a couple of books. "The Knot Book" by Adams is a very interesting exploration in topology (without requiring all the years of study at university before you are allowed to learn exactly what a topology is). And in another direction, group theory was invented because the study of symmetry gets very tricky! But if you want to dive in anyway then have a look at Conway's "The symmetries of things". It is a lot of fun. Most modern group theory (or algebra) books don't actually have any pictures of symmetric things, just endless formulas and lemmas. If you want to be a pro, then you gotta learn that stuff, but there's definitely pathways into higher mathematics that don't require you to learn that.
I am bit surprised there is nothing about graph theory in there. Also nothing about combinatorics or knot theory to mention two other subjects. If you want to make people dive into mathematics, it might be a good idea to show a broad range of subjects instead of focusing on the traditional subjects.
As a mathematics major, I find it encouraging to see non-mathematicians sing the praises of the subject. So I applaud that. But I think the author is overstating/slightly wrong about a few things, perhaps because her exposure to mathematics has been through the lens of physics.
Maybe a more humble rewording of some of her statements e.g., "Anyone that follows and completes this curriculum will walk away with the knowledge equivalent to an undergraduate degree in mathematics." would be helpful.
Her suggested curriculum doesn't include anything from Number Theory, which is a foundational part of an advanced mathematics education. It is also one of, if not the most, beautiful topics one can study in mathematics.
I find it odd to call out "Introduction to Proofs" as a topic in and of itself. Proofs aren't really a topic in the way analysis or number theory is. At advanced levels, devising theorems and theirs proofs is what mathematics is.
I have a decades-old math degree and ended up working in tech as an engineer. Are there options, like a "Math Camp for the Middle-aged" where I could get a chance to re-learn everything I've forgotten?
Going from Strang to D&F seems like a steep jump. The former is an applied textbook for non-mathematicians and the latter is a proof-based text for advanced undergraduate / graduate-level math students.
I would suggest working through a proof-based linear algebra book in between to ease the transition. Axler's is a good one. Alternatives include Hoffman and Kunze and the more modern Friedberg, Insel, and Spence.
I find it hard to believe that the author started to appreciate physics by reading The Feynman Lectures on Physics before any exposure to physics or even algebra, and in less than three years went from barely knowing high school math to enjoying advanced mathematical physics and graduate-level quantum physics. It looks this is one-in-a-million level brilliance as learning the sheer amount of requirement knowledge in such a short time is amazingly challenging: analysis, functional analysis, complex analysis, linear algebra, abstract algebra, differential equations, mathematical statistics, and all the physics: mechanics, electromagnetism, thermodynamics, optics, statistical mechanics, relativity, and of course quantum physics, all in less than three years.
Kudos if the author is this talented.
Spivak’s Calculus reignited my interest and appreciation in math. Sad to discover the author passed away quite recently. The way of explaining principles and making you do the hard work via problems which I believe is a must with this book, is profoundly astonishing. There’s a lot of mathematical insight packed into those problems, it almost feels you can build up the entire high school and the early uni curriculum from the ground up, for instance there are a number of popular formulas you’d arrive at and derive accidentally while working on those problems. Furthermore it really works your brains by making sure you can reason within the established framework and exercise great doubt. I’m taking this book very slowly.
The website mentions some good courses, personally I love Richard Borcherds' YouTube channel[1] for both undergraduate and graduate courses. No frills, exceptionally clear, (mostly) bite-sized lectures that cover a good range of material (especially in geometry).
Something that might interest HN's demographic is Kevin Buzzard's Xena Project[2], centered around proof systems (in Lean). The natural numbers game [3] is particularly fun IMHO. I don't know if it counts as learning materials per se but it's certainly instructive.
[1] https://www.youtube.com/channel/UCIyDqfi_cbkp-RU20aBF-MQ/pla...
[2] https://xenaproject.wordpress.com/
[3] https://www.ma.imperial.ac.uk/~buzzard/xena/natural_number_g...
To the section "Popular Math Books" I would add almost anything by Julian Havil. John D. Cook referred to him as a writer of "serious recreational mathematics" [1]. I would probably put his "Nonplussed!" and "Impossible?" books in the "Level: Easy" group, with the others at least in "Level: Medium". "Gamma" is one of my favorite serious recreational math books.
If you like getting into the nitty-gritty of problem solving then check out the books of Paul Nahin. They vary between "Level: Medium" and "Level: Difficult", with many of them reveling in the solution of equations, and integrals in particular. Although he recognizes the need for proofs, he makes a point of avoiding them in his books.
[1] https://www.johndcook.com/blog/2019/09/29/a-sort-of-mathemat...
You don't need so many books. many of the old texts will cover many important college-level concepts in a single source.
https://www.gwern.net/docs/statistics/1957-feller-anintroduc...
this is linear algebra + combinatorics + probability + stats
If you understand the material in this one book it's reasonable to say that you are pretty good at math
As a math PhD I have to say the only way you're going to learn mathematics is if you actually have a pressing need to do so. i.e. You have a project at work that needs some math, you have a hobby that needs some math. In this case you just learn what you need. Just learning math for its own sake outside of a University STEM track is just too hard (I wouldn't be able to do it and I've tried).
I would personally recommend The Princeton Companion to Mathematics as an excellent introduction to mathematics from someone no pursuing a degree.
See https://press.princeton.edu/books/hardcover/9780691118802/th...
I am fairly confident that Susan Rigetti is a future president of the USA. In addition to becoming somewhat well-known as a household name early on in her adult life, she has achieved so many difficult and impressive things (publishing multiple books, studying physics and philosophy at graduate level, working for a top-tier tech company, taking down the CEO of a top-tier tech company and damaging the company's reputation, being asked to work for the New York Times, publishing curricula in graduate Physics, graduate Philosophy, and undergraduate mathematics). Furthermore, she seems to have a gift for or knack with the public eye.
I love that the author highlighted Prof. Robert Ghrist's great material. I took his calculus classes through Coursera maybe about 8-9 years ago. He just makes everything exciting and his visuals are just beautiful. For example, within the first few lectures, he made it feel like Taylor series was like the coolest thing ever. Highly recommend checking his lectures out. Check out his website: https://www2.math.upenn.edu/~ghrist/
The curriculum guides Susan Rigetti provides are an amazing resource for self-study. And the fact that she worked through all of this is truly inspiring.
Not to be greedy, but do any of you know of other thorough curriculum guides like this? I know about https://teachyourselfcs.com already -- another amazing guide. Are there others? I would love to find one for statistics especially, but really any subject would be interesting.
I'm still stuck at "wait, sets can contain other sets, and sets can contain themselves?" part of Russell's paradox, and I'm close to retirement!
I don't want to study math. I want to know enough of it to solve some well-understood problems I've wanted to solve for decades. Simply learning how to diagonalize a matrix (and how to use such a thing) meant more than understanding a bunch of complicated matrix theory.
I never got beyond algebra/geometry in High school. I think I had to take one 100 level math class in college, but it was basically a review of HS math. Oh, and I had to take a stats class for non-technical people in graduate school. That was my worst graduate class by far. But, I would like to learn some more math, like calculus. I’m hoping to get to it when I retire in a decade or so.
I question just how realistic is is to have "Proofs from the Book" at the start. While the art is wonderful, the background required to read it is not realistically at a lower level than the textbooks listed.
The discussion here has been much more interesting than the actual list, to me. If you wanted to master all of that material, I think a master's program is the way to go, not self-study.
I would be interested in hearing from people who _do_ successfully self-study. What makes it work?
The least interesting (from my point of view) is "already successful in a related field, applied my skills". That would include CS professionals studying maths, I think.
More interesting would be "unable to attend university because of X, did Y and really enjoyed it." Are you a person who completes MOOCs and get something out of them?
As a fellow penn alum, I can totally vouch for Ghrist's approach to calculus. Check out his youtube channel: https://www.youtube.com/c/ProfGhristMath
I studied theoretical math.
Step #0: make sure you know what math is
I can't stress it enough. I know this sounds funny, but when I went to study math a lot of people would drop out very quickly because they did not realise that what most people call math and what is taught at high school is completely different from actual theoretical math.
What most people think math is is a collection of formulas that you need to learn to "know math".
Math is actually a dynamic activity and is 100% about solving problems.
Just like programming is not about knowing programming languages. Programming is about solving problems (and knowing programming languages is necessary but not sufficient to be programming).
It's true that a standard undergraduate curriculum in mathematics will contain a lot of analysis (including calculus), a lot of ODEs/PDEs and some algebra. But if I wanted to get someone interested in math I would also point them towards some of the more fun (and less known stuff), such as combinatorics, probability, topology, differential geometry and number theory. Some of these are also much more applicable today than, say, complex analysis.
Not to be too negative but I wouldn't have super high hopes for this if it's anything like, “So you want to study philosophy”[1][2]
[1] website :- https://www.susanrigetti.com/philosophy
[2] discussion :- https://news.ycombinator.com/item?id=28367416
A shame I missed out on that discussion.
Calculus: I suggest just forget about "precalculus" and, instead, just get a good calculus book and dig in.
There are two main parts of calculus, and both can be well illustrated by driving a car. In the first part, we take the data on the odometer and from that construct the data on the speedometer. The speedometer values are called the (first) derivative of the odometer values. In the second part we take the speedometer values and construct the odometer values. The odometer values are the integral of the speedometer values. In notation, let t denote time measured in, say, seconds, and d(t) the distance, odometer value, at time t. Let s(t) be the speed at time t. Then in calculus
s(t) = d'(t) = d/dt d(t)
And d(t) is the integral of speed s(t) from time t = 0 to its present time.
Those are the basics.
Applications are all over physics, engineering, and the STEM fields.
Linear Algebra: The subject starts with a system of simultaneous linear equation. The property linearity is fundamental, a pillar of math and its applications. The STEM fields are awash in linearity. E.g., a concert hall performs a linear operation on the sound of the orchestra. E.g., in calculus, both differentiation and integration are linear. In the STEM fields, when a system is not linear, often our first step is to make an attack via a linear approximation. E.g., perpendicular projection onto a plane is a linear operator and the main idea in regression analysis curve fitting in statistics.
Most of math can be given simple intuitive explanations such as above.
Susan, I greatly appreciate this list and will definitely come back to use it as a reference if I need a book recommendation. (I don't think I'm the target audience, although who knows what the future brings..)
That being said, I think you are missing out on an opportunity to reach a wider audience. It bugs me a bit that the requirements seem very American-centric. What I mean is the following bit:
> A high school education — which should include pre-algebra, algebra 1, geometry, algebra 2, and trigonometry — is sufficient.
And later the paragraph on "pre-calculus".
I know that many places don't have such names for courses in high school. In fact, often it's just called "Mathematics" and you either take it or you don't (obviously there is a spectrum here).
How is a prospective (non-American) student to know what is covered in Algebra 2 in an American high school?
I'm not asking you to change the article, I just hope I can nudge you into realizing that the text as it is now is more difficult than it needs to be for non-Americans.
If you actually want to study math, you probably shouldn't touch calculus until you've take linear algebra and a fair amount of topology, since these are the two structures on sets that (differential) calculus is founded upon.
For other subjects, you can briefly substitute an intuition for the underlying structures with sufficient finesse in the presentation of the material (see the theory of knots and links, for an example), but calculus is not, in my experience, such a subject, and the early emphasis on it is harmful for the study of mathematics, which is supposedly what your list is for.
For some reason this is heresy, but I have honestly no idea how you are supposed to appreciate calculus from a mathematical perspective without being able to define the large stack of terms that constitute it. The situation is potentially different for a physicist, but if you want to study mathematics, the physical world is not the object of study, rather it is precisely the definitions that we have chosen.
Setting aside the question of motivation, how would a working adult find the time to work through all these books? The author of this clearly leads a busy life outside of reading math textbooks.
Reading a math textbook is time consuming endeavor, regardless of underlying ability. The author herself mentions this in the introduction. I can think of a few factors that might make it possible for a busy person to go through all these books in a few years:
- They include books that were read partially while taking course.
- Consistency: allocating 1 - 2 hours per day for a few years.
- Doing exercises selectively: If you only do a handful of exercises per chapter this would dramatically increase the rate at which you go through a book. This would come at the cost of deeper understanding.
I have a large backlog of math books I'd like to read, but time is a constraining factor. If people have found strategies for reading these types of books, I'd like to hear about them.
I don't agree with this article, it as off-putting as the usual math eduacation it criticizes. I wonder how one can propose a curriculum to study math and not mention Euclid. One learns more mathematics from this article https://mathshistory.st-andrews.ac.uk/Extras/Russell_Euclid/ by B. Russell where he harshly criticizes Euclid than 2 years of calculus. Newton did not know calculus but he knew Euclid's Book 5, the book about ratios and proportions. Euclid's 5th Book must be the starting point for the study of math. When we say "math is the language of nature" we really mean that nature is proportional. Ratios and proportions are fundamental.
Where's statistics? You mean to tell me I could go through all that and come out not knowing statistics?
I really don't.
For many, many years I thought I did. I'd have a brief surge of interest for a few weeks, and then get completely bored of it. I'm not someone who finds it inherently easy, so boredom + difficulty = failure.
When I was foolish enough to do this in university, it meant doing great in the first few assignments, and then abysmally in the exam.
So my policy now is to never study maths for its own sake. Only when there's equations in a computer science paper I don't understand.
How do you like to solve math problems in this day and age?
I'm partial to Jupyter notebooks lately - I run it locally from a docker container, and have a folder of notebooks. Mostly markdown cells, alternating between my narrative thinking and LaTeX math output.
The author doesn't seem to take her own brilliance into account when composing these self-study guides.
For the other 99.99% of us, it would take many lifetimes worth of free time to make a substantial dent in these materials. To me, guides like these are too intimidating to even begin. Maybe what's needed is a meta-guide on structuring one's time and developing the necessary focus to be able to do this within one human lifetime.
If you're interested in both mathematics and physics, does it make sense to learn both concurrently? If yes, what areas complement each other? Or is there no overlap to warrant concurrent study of the essentials? By essentials I mean what a college student must know, or really anyone who pursues self-education without a background in these areas. Beautiful website, by the way!
Wow the philosophy guide is super narrowminded.
IME (as a math-degree-haver) the value of mathematics is in improving one's ability to mentally model and reason about complicated real-world phenomena. A lot of folks lose sight of the reality and get lost in the mysticism, especially within the academic regime.
> [Mathematics] is the purest and most beautiful of all the intellectual disciplines. It is the universal language, both of human beings and of the universe itself. [...] That doesn’t mean it’s easy — no, mathematics is an incredibly challenging discipline, and there is nothing easy or straightforward about it
I am always, always going to condemn this unnecessary mystification and idealization of mathematics. It's exclusive and misleading.
> My goal here is to provide a roadmap for anyone interested in understanding mathematics at an advanced level. Anyone that follows and completes this curriculum will walk away with the knowledge equivalent to an undergraduate degree in mathematics.
NO, NO, NO.
There is no real way to go up to the real deal without having understood elementary Functional Analysis, which the article doesn't even mention. FA is roughly what Linear Algebra would look like if instead of finite dimensional vector spaces we considered infinite dimensional vector spaces. It opens the rigorous path to non-linear optimization, analysis of pdes, numerical analysis, control theory, an so on. What this article mentions is a way to work around things, but nowhere near an undergraduate degree in mathematics.
I'm astonished that the PDE section has such books, they look like the calculus aspect of partial differential equations. A more appropriate book would be L. C. Evans' Partial Differential Equations. Same with ODEs, no mention of Barreira's or Coddington & Levinson's books.
Here is the best mathematics course on YouTube:
Introduction to Higher Mathematics (Bill Shillito) - https://youtube.com/playlist?list=PLZzHxk_TPOStgPtqRZ6KzmkUQ...
No recommendation on probability. Thats strange given that the author is a physicist and fundamentals of modern physics rests on probability. My recommendation is the classic "Probability Theory, The logic of Science by E.T.Jaynes" which is a Bayesian formulation.
While I understand that the author has good intentions, I strongly disagree with the general idea of this post, which is that anyone can learn math through an almost entirely analysis-focused curriculum while other topics like topology, game theory, set theory, etc. are presented as advanced and graduate-level. This is practically equivalent to saying that anyone can learn history, and they should learn all about British history in undergrad, and then graduate-level courses might teach you more about the history of South America.
Some of my thoughts (mostly drawn from personal experience, feel free to disagree):
1. IMO "learning math" is really about learning how to recognize patterns and how to generalize those patterns into useful abstractions (sometimes an infinite tower of such abstractions!). So it really doesn't matter if one does abstract algebra or linear algebra or combinatorics or number theory or 2D geometry or whatnot at the beginning. Any foundational course in any branch of mathematics, or any book on proofs, will fulfill this need. People learn in different ways and have affinities for different topics, so some subjects will be easier and/or more interesting for them, so aspiring mathematicians should start with a topic they're at least initially entertained by. If you don't know where to start, one fun (for me) topic is the game of Nim; other combinatorics topics are also elementary and entertaining to think about. I'm fairly sure that if I had to take this suggested curriculum as an undergraduate, I would have picked a different major entirely, I personally find analysis quite difficult :(
2. One's first foray into a topic should be a one-semester course, not a textbook. Lecture notes for many courses are freely available online also, so you don't have to pirate the books you want if you aren't willing to pay $100 :P The reason is this: courses are curated by a mathematician to teach students the basics of a topic in one semester, so they will better highlight what you need to know, like important theorems, and have a more careful selection of problems. If you're confused, you can read the relevant textbook chapters. On the other hand textbooks are more like comprehensive references - reading a textbook through and doing all the problems will make you an expert at the material, but it's not as time-efficient (or interesting) as a course.
3. There are benefits to diving very deeply into a topic, but IMO one's mathematical experience is much richer if there's more consideration for breadth, especially when you're starting out. A student learning basic real analysis would benefit from understanding some point-set topology (not just the metric topology that usually begins these courses) and seeing how (some of the) pathologies of topological spaces disappear when you impose a metric and then you get things like being Hausdorff or having many different definitions of compactness coincide. After learning real and complex, of course one could move onto differential equations, but there are so many other ways to branch out, like exploring differential topology or learning about measures & other forms of integration, which also meshes very nicely with statistics. Exploring different branches emphasizes that there are so many directions you can go with math, even when you're just starting out, and gives you a better feel about how "math" is done, as opposed to just the techniques for a specific topic.
This is my first comment on HN, so please let me know how I can improve this comment!
Wrong: math is neither hard nor difficult. It is this single belief that deters many people from learning it. Math is all about logic. It is nothing but how to go from A to B. Any person who can reason should be able to be good at math.
You can get good or better at something with effort, but few will ever make to leap to being great or world class at it, no matter how hard they try.
"... but make sure you get the paperback or hardcover version for readability purposes."
As opposed to... the ebook?
I don't think most math undergrads (as opposed to engineering or physics students) take a PDE course.
It's a crime not to mention Gödel, Escher, Bach.
whats the point of learning them, i studied them in undergrad and i dont even use them anymore
Unfortunately math wont make you smarter in other fields
Only reading does
Are there any "math for people who just want to use it" tracks in math pedagogy? I don't care a bit about proving any of it's true, or even reading others proofs of same. "Recognize which tool to apply, then apply tool", all focused on real-world use (so, yes, it wouldn't be "real" mathematics). That's the math education I'd like—try as I might, I just can't make myself care even a little about math for math's sake.
I've got Mathematics for the Nonmathematician by Kline and that's kinda heading the right way, but what about whole courses of study? More books? It's more of an introduction than a thorough resource or course, and feels like it needs another four or five volumes and a lot more exercises to be really useful.
I want a mathematics education designed for all those kids (likely a large majority?) who spent math from about junior high on wondering, aloud or to themselves, why the hell they were spending so much time learning all this. One that puts that question front and center and doesn't teach a single thing without answering it really well, first.