The real niche that needs to be filled here is not more textbooks, it is a really really good description of how to actually teach yourself this stuff on your own. There are so many ways to get this wrong. It's insanely easy to trick yourself into thinking you understand these things without actually doing so. I've read some of the stuff Thurston and Grothendieck and people on HN have written about this, but it still doesn't feel enough. Just doing exersizes is not enough when you get to the graduate level, at that point introspection and the inner mental game becomes much more significant. And without the collaborative help from a greater wiser community everyone is left to discover this stuff on their own, which means a lot of people are going to fail when other interests take priority.
>The real niche that needs to be filled here is not more textbooks, it is a really really good description of how to actually teach yourself this stuff on your own.
I don't think you can. Sure there are a select few insanely motivated geniuses who could do it. But learning math is just really, really hard. Harder than anything most people ever come in contact with. It's kind of like being able to punch yourself in the face. You can try to, but you'll never be able to do it with the same force as someone else. Or like being on a sports team, where a good coach will push you far beyond the limits of what you could have possibly set for yourself. This is coming from someone who taught themselves to code completely solo from nothing and built a successful career as a software engineer. I've recently gone back to school because I simply could not teach myself even remedial high school math with sufficient rigor to prepare for upper division science/engineering courses.
Learning math isn't "really, really hard", but it is unforgiving in a particular kind of way: you're basically required to internalize a concept fully before going on to later concepts. (This is coming from being mostly self-taught up through calculus-level math)
Students often don't fully grasp one concept before the teacher moves onto the next topic. Once this happens, the student is missing a crucial building block and tends to just get stuck with all the later things that build on it. This is a common experience in schools and I suspect is the origin of the "math is hard" trope.
It can be hard for a student to self-evaluate whether he/she has full mastery of a topic. This is exacerbated by situations where a struggling student will go to office hours or work with classmates and "solve" a problem, without having acquired the skills to solve slightly different permutations of that same problem on their own.
In programming, we don't have this problem. You can write HelloWorld without understanding console IO, terminal protocols, window buffers. You can write a webapp without knowing the OSI stack. Software is built upon abstractions.
This rant got longer than I thought it would. tl;dr I think that "you just have to have the dedication to keep learning more even when the foundation is weak."
I'm not an educator so I can only speak from my experience and the experiences of people that I know.
I've found the opposite around foundational concepts -- very often you simply learn how to use them mechanically with no real understanding, and then move on. Only much later do you get the epiphany that lets you unlock the understanding.
I found this first with trigonometric functions; I had "SOH-CAH-TOA" memorized, and the double-angle and a+b things memorized, and could solve basic problems about triangles. It wasn't until much later on (in calculus) that I broke through and really understood what the hell was going on.
Limits I learned early in calculus, and found them incomprehensible -- I could mechanically complete epsilon-delta proofs and intuitively apply them to things like single-point singularities or rational function evaluation, and even follow the sin(x)/x proof, but it wasn't until much later (after differential calculus, and maybe even later than that) that it just clicked and I had trouble remembering what I found difficult about the concept.
Same with quotient groups in abstract algebra; it wasn't until field theory and Galois theory that I realized why my textbook used the Z/nZ notation instead of just saying Z_n like everyone else did.
Learning about asymptotes in rational functions was something that my high school did in pre-calc; that was a concept that I was perfectly capable of mechanically executing without any understanding. There are a million examples of this kind of thing for me.
And so on -- through differential geometry and topology and algebraic topology, complexity theory (formal computer science notation is made even worse if you've actually used computer programming languages, because it seems like an alien language -- it's always stuff like "take the machine S(sigma, alpha, q, r_alpha, delta) over the language M_gamma" -- what? In what sense is this a machine?). I can even feel echoes of this in terms of learning addition and multiplication vs. understanding what multiplication is -- why is multiplication associative is an intuition that came late to me, after I was able to see the geometric analog of multiplication.
For a long time my thought was that this could be fixed by introducing clarifying concepts earlier -- I feel like the concept of a limit as you go to infinity is easier to understand than local ones (something like, if the limit is infinity, a game, "hey, I say infinity, so you say, 'bet it's not bigger than 1,000!', and I say 'if I plug in 10, then the function evaluates to 2,000', and you say 'bet it's not bigger than a million'" -- if I can prove that I will always win this game, then I've proven the limit).
But I think now that the progression from rote and memorization to deep understanding is something that can't be hurried -- as a student, you just have to have the dedication to keep learning more even when the foundation is weak.
I don't buy the "you can't teach yourself this on your own" thing. At least not in the sense that genius is a necessary requirement. The more time I spend on this stuff, the better I seem to become. The pace at which I learn may be different than other people (which might be due to genetics but it could just as well be due to environmental factors such as having fewer distractions than other people or simply having had habits growing up which weren't necessarily mathematical but might have served to grow the components of the brain that made all this stuff easier) but I seem to be able to do it if I just dedicate enough time on it.
In other words how good a person becomes at the task is essentially monotonically increasing (as long as you don't take a long break and forget it all), the difference between individuals is the slope. The curve doesn't seem to have any upper limit, though at times it seems to have a very slow rate of growth. But you'll only reach a desired level if you invest enough time on it. In school you have a fixed time-constraint, which is why, together with the mental model I described, some people fail (as well as not investing enough time on it of course).
Talent or genius only affects how rapidly a person learns, but as long as you're motivated enough to keep going and your time-constraints are wide enough I don't see a rational reason you couldn't do it. But I also think the growth-rate could be bumped up for everyone if we talked more about the ins-and-outs of how to go about teaching this all to ourselves.
It takes special dedication and focus to learn the rigor on your own. However, once you are able to naturally recognize what's rigorous and what's not, I think it's possible to do it on your own.
I think the issue is not how hard it is, it's just that most people have never done anything like learning advanced mathematics before. If you're on your own, you can't learn math by autopilot and reading and regurgitating and allowing the info to naturally propagate through your brain. You have to learn to focus and interactively refine your own version of mathematics.
I'll accept the premise, but I still wonder if there are things that can be done to make it easier for someone. In my case, I've been trying to learn some more mathematics recently, and one of the most annoying things is coming across notation that isn't defined in a paper, presumably because "everyone" who can read the paper is familiar with the context and knows what the "skinny long arrow" means (good luck with that internet search). I wonder if there could be a wiki-like / forum / stackoverflowish site, which people could use to discuss and provide running commentary on a paper/book. Especially useful would be the ability for people to be able to annotate the paper by translating the formulas in to a formal language where you could track down the definition of the various operators, and try to figure out why the author used both of → and ↦ in the paper, when they both appear to be for functions/maps. (Just to preempt the easy objections, I'm not trying to suggest that each paper be formalized and proven in something like Isabelle/Coq).
In the ideal form, this website would allow you to see the paper or book page in question, and then see all the people who commented or had questions on each particular sentence (in the margin?). There could be filtering and voting so that experts could bypass the newbie commentary, etc..
I suppose part of my problem would be solved by getting a book like:
...(which I just came across when composing this message).
Maybe someone has a other suggestions for something like this? Maybe a site similar to this already exists?
And on a slightly related note to making things easier to learn, I think learning programming is much easier than math, because even though both are abstract, at least with programming you get a tangible, concrete thing (the program) that you can run and modify and extend, and the computer will tell you when you went wrong (e.g. won't compile, output result is unexpected, etc.).
Unlike mathoverflow, it is meant for every kind of math question below research level.
(Regarding $\to$ vs $\mapsto$, I think of it as type-level vs lambda expression. I think you can find it in any introductory abstract algebra book that assumes you still need to learn a thing or two about functions.)
In my experience, it seems the usual way people in the math community resolve these issues is to ask an expert, or at least a knowledgeable grad student.
Forgive me if I'm making incorrect assumptions about your background, but usually you learn math from books of varying degrees of difficulty which naturally force you to become accustomed to various kinds of notational conventions.
You wouldn't try to learn math from papers until you've built that foundation (unless you have access to a tutor/mentor), at which point the notation usually shouldn't be an issue.
That sounds like the traditional method of learning math. I was wondering if we could leverage technology and our experiences with teaching/learning the formal systems of programming languages to make more math more accessable. For instance, I'm thinking this little instance of geometric algebra:
...might be easier for me to understand if I could use Haskell to implement the wedge and geometric product operators on an algebraic data type describing the scalar/vector/bi-vector thingy. There is probably an applied vs. pure thing here as well. My motivations for investigating geometric algebra is to see if geometric algebra makes synthesizing mechanical linkages easier, whereas maybe most expositions on geometric algebra are focused on teaching geometric algebra to advance the state of geometric algebra. That's probably a long winded way of saying that mathematicans are writing for mathematicians (whether by design or accident). I suppose I should re-read Mindstorms again, but this time in the context of adult learning.
I'm not sure if this is what you're looking for, but I've had this book on my wishlist for a quite a while and it seems to fit: http://www.geometricalgebra.net/
I also what a running commentary would do for authors. Would they get ideas for improving their next paper, by looking at what had people confused? Surprised by who is reading their papers (especially those outside of their field)? Would they merely be horrified by YouTube style commenters?
i really feel the same way. i would identify another niche, limited university-like courses. Something like google's 20%, but instead of persuing your side-project, you just take one lectures, tutorial and a seminar every friday morning (may be a high workload, you could also switch between seminars and lectes every semester). While not much, i would would guess that it really accumulates over the years, motivates you to keep on learning and deepening your knowledge while being high quality and rigorous (in math i think this is really important).
I am interested in Machine Learning and there is just so. much. to. learn. I am currently a student, but i really don't know how to master the life-long learning in these more theoretic aspects. Universities are just made for this. From analysis, probability theory, statistics to information theory there is an abundance in things that are relevant to this field.
This is true and becomes more so as the subject gets more abstract. It is not hard to understand and learn calculus. All you need is motivation, attention, and - yes - plenty of practice. Learning most of the "college math" is like that. Even linear algebra is "easy" in that sense. The problem arises as you move into the domain of abstract algebra, topology, category theory, etc. All of a sudden you need, in addition, a lot of concrete examples and purely mental practice just so you could get an intuition about even the basic ideas. The process of learning slows down to the point that it becomes easy to lose the much needed motivation and/or attention; or you could just realize that the level of the required dedication is such that no matter how interesting the subject is for you it may no longer be worth your while (you weren't going to be a full-time mathematician, were you?). The degree of difficulty varies with the person, of course, but the fact remains that as math becomes more abstract, there will always be a point where one has to decide on the balance of "life and work".
You would also likely benefit from scheduling review of stuff you already understand with tools like Anki.
Beyond how to study any one thing, the most important thing is studying the right things in the right order starting at a level that is perhaps too easy so as to make sure your foundation is strong and not the source of your difficulties. You can more or less recreate this on your own through online resources and following along typical undergraduate mathematics curricula. There are often dedicated problem books with solutions for most common subjects even at the advanced undergraduate level.
I support the sentiment. I'm not sure anyone has all that much insight though, especially if you don't consider the things Thurston and Grothendieck have written to be adequate. I'm got a PhD in mathematics and have interacted with a lot of mathematicians, some of whom are up there with Milne, and I could pretty much say everything I know about how to learn mathematics in a paragraph or two.
I've found his notes most useful when, because of the medium, he is able to provide more background, detail and examples compared to the standard texts.
I read the first chapter on algebraic group theory, and was shocked at how unhelpful it would be to an undergrad attempting to learn the subject. Then I realized I was a dolt, and read some of the Group theory chapter which did, in fact, effectively explain the terms and concepts thrown about in the AGT chapter. Fair enough, but I prefer my college textbook which merged the two.
Most of these texts were originally intended for graduate courses at Michigan. In any event, I think it's helpful to note that algebraic groups are to algebraic geometry as Lie groups are to differential geometry; it's not a primary subject, even if the name does not emphasize this.
Ooh, nice a discussion of complex multiplication. I first heard about this a few weeks ago during the Mathematical Congress of the Americas, and I was a bit embarrassed to admit (to a group of number theorists) that I had never even heard about this. They were happy to explain the basic idea, although I still need a lot of time to digest the mechanics of it.
It's a third year course! If you want a more leisurely treatment I hear good things about Cox's book "Primes of the Form x²+ny²". Even if you don't reach the CM part you will have gained a lot of motivation for it, all the way back to Disquisitiones.
