Ask HN: How to self-study mathematics from the undergrad through graduate level?
Hey HN community,
I've been looking to get deep and build my math skills from the foundation up. I have the time to dedicate to this endeavor and I'd love to hear if you have any specific resources/curriculums you recommend.
Something like https://www.susanjfowler.com/blog/2016/8/13/so-you-want-to-learn-physics would be ideal, but more focused on applied math.
One idea I had was to complete the MIT open courseware courses for the Applied and Pure math fields
230 comments
[ 3.0 ms ] story [ 245 ms ] threadOne of those people should be at about your level. The other should be farther along.
All three of you should trust each other enough that nobody gets caught in a shame/guilt/ego loop.
if you could find one who willing, this recommendation will def help.
Then I thought, why can't I just buy the time of someone competent? There are many students who tutor high school kids, shouldn't students who can teach first/second-year material for pay exist?
I wonder, have anyone tried this? To me, it looks like a perfect solution and I can't think of any downsides. I'm not sure how to approach this exactly - where to find such a tutor in the first place - but it should be possible, right?
I would use such private lessons for getting a summary/overview of the subject first, then after some self-study, I would ask about whatever I couldn't comprehend. It would have minimal impact on my schedule, shouldn't cost that much, and should be quite efficient.
Well, it's just an idea I had some time back, I didn't try it in reality yet, but I think I'll try going this way in the future. I'd like to ask what do you think, is it possible, could that really work out?
One way to find them is to contact your nearest research university’s department of math/physics/whatever and they can help match you with someone, or just post a flyer in their department office.
https://en.wikipedia.org/wiki/Olympia_Academy
I have created a group here if someone is interested to join: https://groups.google.com/d/forum/projectfermat (If you think that there is a better place to have a forum like this that anyone can easily view or participate in, please let us know. I mean we could also create an IRC channel, Slack workspace, etc. but there should be one main starting point and a mailing list/web forum like this seems like a good place for that.)
I am thinking we could also host a web meeting to present, discuss, or share interesting topics and problems regularly. We can form our own mathematics discussion community here.
I have been doing this kind of thing at my workplace as well as outside work and it has been an incredible source of learning. I believe something like this for the Hacker News community would be very helpful and we can learn a lot of mathematics from each other if we can interact with each other on a more topic-focused forum.
There are 12 members in the group so far which I believe is a good size to kick-start the discussions and other activities. More members are welcome!
As a maths undergraduate, I attended around 10 classes per year - you go see who the instructor is and then just read the lecture notes at your own time at home. Then go take the exam at the end and that's it. That sums up my 3 years of undergraduate studies.
Mathematics is well established, essentially has been frozen at undergraduate level for 50 or more years, so there is plenty of material. Also, you don't need any equipment, just your own mind.
In my opinion, it is the easiest major of all if you can follow the logic. No essays to write and no projects to do - just read the material.
What I love about math is that this is completely do-able. Good luck! Also, definitely get a dry erase board. There’s something about having a large space that can be easily erased that helps working out problems.
Edit: deleted an m
A couple resources I would recommend would be:
* A book called "Who Is Fourier? A Mathematical Adventure", which touches on a pretty good variety of math topics. It's aimed at children, but it's probably the best math book I've ever seen. It's the only math book I've ever read cover-to-cover.
* 3Blue1Brown's youtube channel: https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw (as well as other math videos)
His videos, I think, are best consumed twice: once before you explore a topic, so that you go in with a solid overview and know the key things to look out for; and again after you've studied in depth, when you are able to predict what he will say next, which reinforces you learning.
1. Go to Youtube, find the Professor Leonard channel. He teaches math at Merced College and is a very good lecturer. He has recorded himself teaching everything from pre-algebra through differential equations, and a statistics class. The one thing he has not done yet is linear algebra, he he apparently plans to do it.
2. Watch his stuff, and supplement that with the corresponding "Schaum's Outline" or similar book for the topic at hand. Also, if you desire, buy a few editions old used college textbook for the corresponding topic. This gives you more exercises to do and a reference to consult if anything is unclear.
3. As desired, follow the Khan Academy lessons on the topic you're studying. KA has everything from arithmetic / pre-algebra up through at least Calculus and Linear Algebra. I don't remember offhand if they cover Differential Equations or not.
4. For Linear Algebra in particular, the Gilbert Strang lectures on Youtube are very highly regarded, and he has a text that was written specifically to accompany those videos. So that's a good resource for Linear Algebra.
5. For "higher" math (real analysis, complex analysis, topology, abstract algebra, etc.) you can almost always find complete lecture series on Youtube / OCW. Depending on the topic, there may also be a "Schaum's Outline" or similar study guide book you can supplement with. And you can always find a used textbook on Amazon, usually for not too much money if you go with an older edition.
You can also find a lot of freely available maths texts online. See, for example: https://math.gatech.edu/~cain/textbooks/onlinebooks.html
If you don't have a background in doing proofs, which is kind of regarded as the dividing line between "simple" math and "higher" math, there are a number of books on that specific topic, including texts written for so-called "transition to higher math" classes. Some of those are freely available online as well. There's also a good class you can find on Youtube, "Math for Computer Scientists" which covers proofs and what-not pretty well. There's a freely available corresponding text as well.
Another thing to do is consult forums where you can ask for help if you get stuck. There is math.stackexchange.com, physicsforums.com, cheatatmathhomework.reddit.com, learnmath.reddit.com, mathhelp.reddit.com, etc.
Somewhere I have a Google doc that lists a lot of the resources I have been using, and have queued up to use in the future. If anybody is interested, I'll clean that up, and make it public and share the link.
One last note: I haven't done it myself, but I've heard that if you live near a University, it's not too hard to find maths students who will tutor you to pick up some extra cash. So that's an option as well.
Edit: somebody else mentioned 3blue1brown on Youtube, and there are a number of other really good Youtube channels, including: Prof RobBob, NancyPi, and Dr. Chris Tisdell.
If you get stuck you could try asking on https://math.stackexchange.com/
Although, if you can, finding someone else to work with, and someone else who already knows some math to ask occasional questions would probably help you a lot.
http://www-history.mcs.st-and.ac.uk/
How much math do you currently have, which courses have you taken and how long ago did you take them?
It's like when you have Calculus and over generalize when entering Analysis or Differential Geometry -- but now I have more structural patterns to extrapolate from :/
You will not improve your math watching youtube videos and reading books. You need to produce stuff that pass the "sniff test" to your colleagues.
That depends on exactly what you mean by "become good at math". If you're talking about "becoming a mathematician" and doing original research in pure math, then you're probably right. But if one means "learning existing math well enough to apply it to a problem", I would argue that one can learn this stuff just using books, videos, etc. At least up through a certain level.
That said, I do encourage the idea of finding peers to work with. I used to coordinate a "math study night" at the local hackerspace for that exact purpose. It fell off because I got busy and couldn't keep committing to it, but generally speaking, it is a good idea to have other people to work with. I may well try to find a math major from UNC to hire as a tutor at some point as I keep working on this stuff.
And if one can't find somebody to work with in person though, and they need, say, a proof evaluated, there is the option of using math.stackexchange.com or the like.
Sure, I'm just saying that it kinda depends on what part of math one is referring to. I think sometimes in these discussions on HN, we overload the term "math" to mean both "calculation" or "applied math", and "pure math" or "math research" and it can be unclear which is being referred to in a given statement.
I believe you can learn the former - "applied math" - (at least up to a certain level) just by reading books, and watching videos (and doing exercises, of course). But for the latter - "pure math" - I agree that you need other people, since you can't easily verify your own proofs.
It you don’t have somebody challenging you and checking your work, you will likely plateau very quickly unless you are especially gifted.
You will Hit walls of concrete and walls of glass.
You will misunderstand concepts and not notice it.
Your proofs will have logical gaps and you won’t notice it.
Although Math 55 is tough, if you are self-paced and have a bit of mathematical maturity I think it is doable. It's also an excellent pure math bootcamp that gives you a solid foundation to branch into any other pure or advanced math topic.
I have gone through Halmos & Rudin myself, and it is a great experience. However, if your end goals are more geared towards pure CS, an alternative route might be much more appropriate. Very interesting and promising parts of CS, such as formal methods, and the foundations of mathematics themselves depend on abstract algebra and logic: https://ncatlab.org/nlab/show/computational+trinitarianism
A minor problem is that beginner literature is not so polished as it is relatively young. But there are some excellent textbooks nonetheless. Some below. Other suggestions welcome:
* http://www.cs.man.ac.uk/~pt/Practical_Foundations/
* https://www.mta.ca/~rrosebru/setsformath/
* https://github.com/ademinn/ttfv/blob/master/2006.%20Sorensen...
* http://www21.in.tum.de/~nipkow/Concrete-Semantics/
* http://adam.chlipala.net/frap/
* https://softwarefoundations.cis.upenn.edu/
1 Every day on this site https://www.khanacademy.org
2 Use this for graphing desmos.com/calculator
3 Use this to supplement (ie free text) myopenmath.com
I honestly don't think there is better mathematical content than his being made.
Calculus: learn to extract qualitative information about a function (it goes up here, has a maximum there, goes down there, oscillates with an increasing period, goes to this value at infinity...) and to numerically compute quantitative information about it (its value at 3 is blah, its integral over this interval is blah, its maximum value is blah).
Linear algebra: vector spaces and linear operators, and their representation as vectors and matrices. Functions as forming infinite dimensional spaces, and Banach and inner product spaces.
Differential equations and dynamical systems: extending what we did for calculus to differential equations. Phase space, orbits, Fourier and Laplace transforms, sets of linear differential equations, numerical integration, some partial differential equations. You do not need all the little tricks for special kinds of equations that you will find in, say, Boyce and dePrima. They're not helpful.
Sets, groups, rings, and lattices. Mathematics today is written in terms of set theory. You need to understand the basics of manipulating sets and functions between them. Then you should know something about groups, rings, and lattices, which are the most ubiquitously useful algebraic structures besides vector spaces.
After that, where you go is going to vary enormously. Based on what you're aiming to do.
A Book of Abstract Algebra by Pinter (really gentle)
or
A First Course in Abstract Algebra by Fraleigh
For lattices, there's Birkhoff's book on lattice theory, which is where I learned what I know about it. I haven't spent any time with other books.
I know there are textbooks on the topic, and probably lots of people who deal with lattices a lot. But my own experience seems to be that wikipedia puts more emphasis on lattices and things like universal algebra than actually happens in math.
Some of the engineers who attract quantitative work are people who came from outside of the mainstream engineering training, such as scientists and math people.
When you're designing a real-world engineering project, the entire specifications are defined legally (through national, state and local laws) and technically in manuals/books. Many engineering specifications will describe the work done to a T before you even need to think about it i.e. "water main shall be constructed of 12'' coated DIP at depth no less than 2 ft". A lot of the challenges are managerial and logistical.
All of this is on purpose. "Traditional" engineering disciplines are more mature and have the constraint of being safe for the general public. There isn't much room at all to creatively deviate from what's already specified.
I've found software design to be a lot more technically demanding in regards to designing and building things. There's a lot less precedent, more moving parts and many different ways to do one thing.
And I'm not blaming anybody -- for one thing college math is often badly taught, and there's a pervasive message that you won't use any of your math or theory after you finish your degree. And then we get them so busy with CAD and bureaucracy, that they forget a lot of their school stuff.
But anything requiring calculus or above, goes to a handful of "math people" in the department, who accept those tasks in return for avoiding the CAD and organization stuff. (I'm one of those people at my workplace, my degree is in physics).
To make it a bit harder, virtually all math these days is done with computation, which means a person has to be good at both math and programming at some level.
but as madhadron says, you can't read/write proofs of upper division or graduate level math without the "foundations" material, which includes naive set theory.
do you need any of that to do engineering math? well, there are a couple of standard quotes, relating to the fact that the technique taught is brittle, in weird and subtle ways. the claim is that understanding the proofs tells you what the limits of applicability are.
"[F]or more than 40 years I have claimed that if whether an airplane would fly or not depended on whether some function that arose in its design was Lebesgue but not Riemann integrable, then I would not fly in it." - richard hamming, "mathematics on a distant planet"
"It is customary to begin courses in mathematical engineering by explaining that the lecturer would never trust his life to an aeroplane whose behaviour depended on properties of the Lebesgue integral. It might, perhaps, be just as foolhardy to fly in an aeroplane designed by an engineer who believed that cookbook application of the Laplace transform revealed all that was to be known about its stability." - tom korner, fourier analysis
Although I wonder for 90% people of this world that math is just a tool to pass the exam at school, not any real application (or they just can't sense it).
Basic arithmetic: addition, subtraction, multiplication, division.
You need to know what you want to do before you go looking for resources. There is no set agreement on what should constitute an undergraduate applied mathematics curriculum, and you are likely to get lost in the deluge of conflicting information. On the other hand, the undergraduate pure mathematics curriculum has been more or less stable for half a century. Any college curriculum will do at this point, and many are freely accessible online.
Either way, there is no shortage of information and resources available. Any topic you'd choose as a layperson likely already has a course or a seminar covering it, and the corresponding syllabus should give you what you need.
Gerhard t'Hooft has a page on how to become of a good physicist. http://www.goodtheorist.science/ Not math though.
I'm not necessarily suggesting that a mountain of books is the best way to go about it. I think I respond best to video lectures. I'm not sure where you're starting from or how applied/pure you want. Here's a quick mind dump some of my favorites and some that I haven't watched. Roughly in order of how much I liked them.
Gilbert Strang's Computational Methods for engineers was life changing for me. It is a two part MIT opencourse. https://ocw.mit.edu/courses/mathematics/18-085-computational...
A Stanford course on the Fourier transform https://see.stanford.edu/Course/EE261
Bartosz Milewski's Category theory for programmer's https://www.youtube.com/watch?v=I8LbkfSSR58
Stephen Boyd's courses are online. http://web.stanford.edu/~boyd/ Linear Systems, convex optimization. Useful stuff.
Francis Su's Real Analysis is very good https://www.youtube.com/watch?v=sqEyWLGvvdw
Indian universities have an astounding collection of videos https://nptel.ac.in/ I have a tough time with the accents, which is a bummer.
UCCS MathOnline has quite a haul https://www.uccs.edu/math/vidarchive
I've been enjoying this Visual Group Theory course lately https://www.youtube.com/playlist?list=PLwV-9DG53NDxU337smpTw...
Math Doctor Bob https://www.youtube.com/user/MathDoctorBob/playlists
Wildberger has some interesting takes on elementary and non elementary topics https://www.youtube.com/user/njwildberger
https://www.perimeterinstitute.ca/training/perimeter-scholar... Perimeter scholars lectures. Physics not math. Good stuff.
Federico Ardila has a number of combinatorics courses. https://www.youtube.com/channel/UCWwECTsgjp_S-c73pO2c4gQ
Nonlinear algebra course https://www.youtube.com/playlist?list=PLRy_Pn1LtSpejKLClqbrW...
Also of course there is Coursera and edX stuff.
Godspeed.
In March of '18 I started doing lessons on Khan and just played around until I couldn't do the problems easily, and for me that was, like literally adding fractions and using exponents. So I had pretty basic skills at that point.
At thus point, I'm finishing the unit on using derivatives to optimize functions around min/max. Not a big deal, but a long way from where I started 10 months ago.
I've had a lot of luck with:
a) khan academy
the lessons are very simple and well broken up, the teaching is interesting, and the site is gameified in a way that is rewarding
b) doing it as close to literally every single day as I can manage
Math fluidity feels (at least to me) very much like my fluidity with music theory or programming. As such I need to do it regularly. Even if I don't get all into a flow state about it (which, I think is necessary on the scale of any given week), I do need do it a little bit every day... that both keeps me doing it and keeps it in the forefront of my mind.
c) keeping a notebook
I do all my work in a single notebook and use it to both track my progress and as a reference. It's also been neat to see how far I've come.
-=-=-=
I dunno if any of that is useful to anyone else. But I feel like I've had a lot of luck educating myself in math. At this rate, I should be through integral calculus by the end of the spring and through the linear algebra class by fall, and then I will move my deep, long-term learning projects over to something else, hopefully a deeper dive into electronics design.
Textbooks and lectures will teach you what math is. The concepts, the different proof methods can all come from a book.
The value from an instructor is that they'll give you feedback on the _how_ of math. A halfway decent professor will edit your proof just like an English professor will -- from the level of word choice all the way to the method you constructed and presented your argument. And, just as importantly, they'll tell you when you fucked up and didn't notice.
I don't think you need a PhD level educator. You need mathematical maturity. Mathematics follows a very specific logical structure that needs you to shift the way you think. Human brain simply doesn't work the way mathematics needs it to work. But this is a constant time overhead. Once you understand how to approach mathematical problems, I can't see why you cannot learn everything from a textbook.
For me personally I didn't really need an instructor for most of my calculus courses, or ordinary differential equations, or most of the linear algebra stuff. It was a bit more difficult around real/complex analysis, non-linear dynamics, and courses of that nature. The classes that taught me the value of having an instructor were abstract algebra and topology. Those were such a massive shift away from what I had perceived math to be that an instructor being able to impart intuition, correct my own incomplete or incorrect assumptions, and generally just help guide me to a different mode of thinking was invaluable.
The problem with books/texts in this instance is they are not reactive, they have no idea what you're thinking and can't steer you in the right direction. Worse is that as the person trying to learn the subject matter you don't know where to look to get on the right track and correct your own assumptions because you don't know enough yet.
Now I'm not saying you need an instructor per se, but having some place to ask questions where someone far more knowledgeable than you can help might be a good substitute. I'm sure there are some websites like this, although I don't know of any since I graduated a long time ago.
[1]This does somewhat depend on a person's skillset going into this.
Of course you can learn a ton on your own by reading and working exercises and doing research, but there is no substitute for collaboration.
Mathematics is inherently a social activity, even if the bulk of it can (counter-intuitively) be done in relative solitude.
I eventually did get to university and get a Math degree. It was much more rewarding and fun to do it with professors and other students.
I learned programming on my own from books but I don't think I could do that with advanced math.
I would leave for work an hour early and either sit in my car or go into a Starbucks and do math. Doing time before work is important. That's when you are at your best. Then after work I would sit in my car and do math for an hour. Then on the weekends, in the morning, I would do three hours of math straight, sitting in my car. Then sometimes in the evening I would do a first pass over a section that I knew I would have to really think hard about the next day that way I had sort of taken the first layer of difficulty off.
Why in my car? Because I simply cannot sit at home and focus enough to study math. I've tried the library but it's too restrictive on what you're allowed to do "oh f*! that's how you do that" doesn't generally go over well in public places. Plus I feel very safe in my car. I can be relax which makes learning a lot easier.
I always take one day off a week (for me that's Tuesday) but aside from that I don't skip.
Get your books at half price books. Math book reviews on Amazon are almost always wrong. I usually have to try two or four books before I find one I can understand. I would just get a book and read it in your car. You don't need the pressure of following along with a schedule where you fall behind and miss out on a topic. You also don't want to miss a topic just because the teacher chose not to include it. Some of the coolest topics/examples don't get covered when you take a course because they don't have the time.
Also, when you get done with a book go through it again and take good notes on note cards that you can review on your drive to work or while you are waiting around. You'll find yourself going back to them over and over again.
I'm trying to figure out where I went wrong with my life by retracing my steps. Starting with when we met in freshman calculus.
It's all well and good to want to cover undergraduate math courses. When you are actually enrolled in a university, you will have enough inertia and motivation to complete the courses.
However, when you are self-studying you are doing it all on your own. It's hard to be as thorough and cover everything.
And so I ask, what really is your goal here? You don't have to learn everything about mathematics, because that is in fact impossible.
My advice is to FIRST construct a bunch of projects, tasks or goals that require knowledge.
It could be something like (a) implement a machine learning algorithm to do X from scratch (b) implement a simple physics engine (c) try to verify a number theory conjecture (d) be able to solve all the exercises in a book (e) be able to write up a compelling description/theorem/problem in math (d) numerically solve the quantum mechanics equations of a certain system
Spend some time on material that will inspire you first to help get these goals. Numberphile on YouTube, or any of Brady Haran's videos, is a good place to start. But make the goals your own and make them personal.
Math is not a spectator's sport. Make sure to DO mathematics, not just LEARN mathematics.
> be able to write up a compelling
Can someone clarify what this means?
I am afraid that sounds curmudgeonly, but I have also seen students shoot themselves in the foot because they decided they didn't need a class for their not very well informed goals.
Nah, it's totally possible for newbies to pick high-level long-term goals.
This can be something like "I want to teach my computer to tell apart dogs and cats", or "I want to create a website where people can buy and sell yarn." From there, Google searches can direct someone towards concepts and various methods of learning them.
I mean, you can disagree all you want, but this is in fact how many people learn things.
Of course this is in the context of choosing research problems to strive towards in math. If you tasked yourself with solving an open problem in math, it's more likely than not that, without any collaboration, you'd have no idea how to even work towards the goal due to all the unknown unknowns. If your goal is something concrete that can be augmented with mathematics, then yes I agree that goal setting can be useful. It doesn't take a volume of missing domain knowledge to develop that kind of goal.
People just assume learning math is the same as learning everything else. That is not even remotely true.
> everything else that’s worth learning: hard!
I disagree. I’ve learned many things worth learning that are not hard at all, but to each their own.
Like, if you are into Rubik's cubes, that's going to make learning group theory a lot more fun and motivating.
I think maybe the most likely outcome, if they were really motivated and somewhat capable, is that they learn lots of mathematics well—but maybe not quite to, "... through graduate level" (that phrasing is a bit ambiguous though; self-teaching 'up to' graduate level is definitely doable).
Links:
http://content.algebraicgeometry.nl/2017-2/2017-2-007.pdf
https://projecteuclid.org/download/pdfview_1/euclid.ecp/1508...
Look for my name (Thomas / Tom Price) on these pages:
https://web.archive.org/web/20170121000748/https:/wwwmath.un...
https://www2.math.binghamton.edu/p/seminars/arit/arit_spring...
https://www2.math.binghamton.edu/p/seminars/arit/arit_fall20...
https://www2.math.binghamton.edu/p/seminars/arit/arit_fall20...
As an aside, your work on numerical cohomology appears to have been useful for a new result pertaining to lattices. Given the authors of the followup work it's likely helpful for the study of lattices in post-quantum cryptography.
There are plenty of people who devote several years of their life to studying, and must pay not only their living expenses but tuition fees as well. In my opinion, those are the people who you should be asking “how did you manage this”.
I have actually found people to be very different in this regard.
For instance, I work in data science and a lot of my peers like to learn about new techniques by applying them to real problems or working with datasets and exploring.
I don't like that approach. I always like to learn the theory of something before using it.
Similarly, I had the goal of learning math for statistics and general relativity among other subjects.
But my desire to have a deep understanding, ended up with me essentially learning the equivalent an undergraduate math degree and selected graduate level topics.
In my case, just learning in a bottom up way with maybe a slight direction would have been sufficient.
He had a quiet programming position in a large company that gave him time to work on personal projects. Among other things he volunteered at the local schools teaching "fun" math. I guess I'm just agreeing with you about finding something that inspires you and go with it.
I mean, if you're interested in astrophysics, you shouldn't go to a school where the Physics department is focused on nuclear physics!
He didn't care about the status of having a PhD
He didn't need a PhD to advance his career
He was having fun doing math stuff anyway
While I appreciate your brother's personal choice, to each their own after all. There is quite a lot of merit in your PhD advisor helping you in choosing a problem. That being said, good advisors provide students with an array of good problems out of which the student can choose one they are the most passionate about. This is what happened with me, I was provided with around 7 different choices to make. In the end, I chose 2 of them even though I wanted to chose 3 more but couldn't because of lack of time.
1. Find a thing you want to make. 2. Find out how to make it. 3. Try to make it. 4. Learn the skills that previously prevented you from making it.
Unlike programming, mathematics isn’t something you can pick up in a weekend.
EDIT: Now that I think about it, you can probably bang your head against, say, Lang, for two years, “digesting” a big chunk of it, earning you the bragging right of “working yourself up to graduate level math”. That won’t give you the breadth of a good bachelor of mathematics, and it certainly won’t prepare you for quality research.
I personally did work myself up to the graduate in math during the last two years of High School & first year of college. I was motivated by exploring ideas and gaining knowledge. I would guess that each person has a somewhat unique motivation strategy. Maybe solving problems gets some people doing stuff. I'm sure simply learning stuff can motivate others. Probably each person has to experiment to discover what works for them - I would pick up a calculus book and read it - well, I'd skim repeatedly and then read in depth, solving a few problems. Math is difficult, of course, so having a bit of patience with your until it gets the ideas on it's own is probably necessary.
...how in the world did you manage to do this? Did you actually self-study, or were you placed in a gifted program? Self-studying all of undergraduate mathematics is more impressive than actually studying all of it in a four year classroom setting. Doing so as a teenager is amazing.
The most gifted person I've ever known in mathematics is actually a physicist who entered Harvard at 16. He was already taking undergraduate math courses at 13/14 and had mastered all the undergraduate material by the first semester of his undergraduate degree. For the remaining years of his undergraduate degree he took graduate math courses.
But in order to do that he needed to not only be gifted; he was placed in a program for extremely gifted kids sponsored by Stanford University from his preteen years. I don't think your anecdote is a great comparison for the OP's expectations (or for calibrating advice they'd benefit from).
I read quite a bit on my own, I took some courses at UCLA through a high school scholars program (including the undergraduate honors seminar). The entirety of the undergraduate program might be a slight exaggeration but I was ready for graduate level courses when I got to Berkeley.
I think going through the material requires determination, not necessarily being extremely gifted. But then, it seems like people at someone's gone through a bunch material say by that fact they're gifted. Thus having done this, one is tautologically gifted.
I've never tested at the extremely gifted level but I'm doubtful of single-measures of intelligence regardless.
Edit:
I don't think your anecdote is a great comparison for the OP's expectations (or for calibrating advice they'd benefit from).
Neither of us know the OP. It's kind of up to them to calibrate what process works for them. Scanning a lot of math until I found good, clear explanations worked well for me.
Indeed, speaking only for myself, I don't think that I had the self discipline to carry through with such a plan as a teenager. I needed the classroom environment to learn math & physics, which became my college majors. At the same time, I was able to teach myself programming and electronics quite easily for some mysterious reason. And I ended up combining all of those things in grad school.
Also, I only had access to the books and resources of a typical suburb with a decidedly anti-intellectual culture. The officials at my high school refused to offer a calculus course because they said it would be "elitist."
For example, (since I don't really have much time)
1. Topology (book by Munkres)
2. Real Analysis and Measure Theory (book series by Stein Shakarchi)
3. Algebra (book by Aluffi)
4. Linear Algebra (book by Friedberg Insel)
5. Measure Theoretic Probability (book by Cinlar)
6. Differential Geometry (book Smooth Manifolds by Lee)
7. Numerical Analysis (book by Quarteroni)
8. Set Theory and Propositional Logic (books by Goldrei)
This is what one will mainly learn in a strong undergrad/grad math program. Once this is done, then there are different tracks to follow.
I was questioning why the OP wants to self-study an undergrad math curriculum to begin with.
It's probably not to become a pure mathematician. So I suggested, instead of creating a massive goal of getting through a collection of books just for the sake of being a completionist, to have a concrete personal goal. Otherwise, people can throw books "you have to read" at you until the cows come home. Especially since this person is talking about applied math.
But your advice has a point, just going through books mindlessly is not motivation enough/ can lead to wandering. And it is always good to have specific tasks at hand. Like, solving a particular ordinary differential equations numerically.
Also, unfortunately, its not the recommended order to learn them in.
They are extracted from my concise books for adult learners on MATH & PHYSICS https://minireference.com/static/excerpts/noBSguide_v5_previ... and LINEAR ALGEBRA https://minireference.com/static/excerpts/noBSguide2LA_previ...
I don't want to self-promote too much here, but maybe HN users who have read the books can add a comment to say what they thought.
Mathematical notation is meant to be read out loud. It's shorthand for English, or whatever your natural language is. You should be comfortable seeing a sigma sign and thinking in your head, "the sum as ehn goes from one to infinity of eff of ecks squared times ..."
That's just the basic part of it. There's lots of cultural shorthands that are just kind of osmosed, seldom overtly acknowledged (for example, the precise meaning of "without loss of generality"). Maybe you can acquire them by watching videos, but if you can find people that you can hang out with that can teach you the cultural aspects of mathematics, that would help a lot. Picking up a book and reading and working through it and understanding it gets a lot easier once you have the culture down.
You don’t typically obtain that kind of maturity until graduate school, maybe upper undergrad if you’re quite good.
People complain a lot that books and presentations don't do everything and that mathematics has a bad "user experience". They may have a point, but the complaining alone won't fix it and will leave most people still feeling like frustrated outsiders.
https://www.cambridge.org/core/books/course-in-modern-mathem...
It covers most of the mathematics needed for quantum field theory (which is a big chunk of applied mathematics) starting right from set theory.