thank you so much i WILL use them to scribble diagrams like a maniac

pwease mutuals answer the poll i need feedback

"learning languages is easy, all you have to do is practice, put in the time and effort"

"math is easy, all you have to do is practice, put in the time and effort"

i think you may be missing the entire fucking point of saying that something is "easy"

how to prove proposition 2.1

step 1: say it's easy step 2: give some geometric construction that i think may have an (inconsequential but still) error in the formula to prove (a) step 3: vaguely reference the literature to prove (b) step 3: straight up says a previous theorem shows a bit of (c) (it probably does) then vaguely reference the literature to prove the rest (the reference is 55 pages long and no specific theorem number is given) step 4: forget to prove (d)

99% of mathematicians quit staring intensely at the problem right before it solves itself

thanks for the compliment!

first of all it's not a dumb question. trust me i'm the algebraic-dumbass I know what I'm talking about. okay so uh. how does one learn math without a class? it's already hard to learn math WITH a class, so uhhh expect to need motivation. i would recommend making friends with people who know more math than you so you have like, a bit more motivation, and also because math gets much easier if you have people you can ask questions to. Also, learning math can be kind of isolating - most people have no clue what we do.

That said, how does one learn more advanced math?

Well i'm gonna give my opinion, but if anyone has more advice to give, feel free to reblog and share. I suppose the best way to learn math on your own would be through books. You can complement them with video lectures if you want, a lot of them are freely available on the internet. In all cases, it is very important you do exercises when learning: it helps, but it's also the fun part (math is not a spectator sport!). I will say that if you're like me, working on your own can be quite hard. But I will say this: it is a skill, and learning it as early as possible will help you tremendously (I'm still learning it and i'm struggling. if anyone has advice reblog and share it for me actually i need it please)

Unfortunately, for ""basic"" (I'm not saying this to say it's easy but because factually I'm going to talk about the first topics you learn in math after highschool) math topics, I can't really give that much informed book recommendations as I learned through classes. So if anyone has book recommandations, do reblog with them. Anyways. In my opinion the most important skill you need to go further right now is your ability to do proofs!

That's right, proofs! Reasoning and stuff. All the math after highschool is more-or-less based on explaining why something is true, and it's really awesome. For instance, you might know that you can't write the square root of 2 as a fraction of two integers (it's irrational). But do you know why? Would you be able to explain why? Yes you would, or at least, you will! For proof-writing, I have heard good things about The Book of Proof. I've also heard good things about "The Art of Problem Solving", though I think this one is maybe a bit more competition-math oriented. Once you have a grasp on proofs, you will be ready to tackle the first two big topics one learns in math: real analysis, and linear algebra.

Real analysis is about sequences of real numbers, functions on the real numbers and what you can do with them. You will learn about limits, continuity, derivatives, integrals, series, all sorts of stuff you have already seen in calculus, except this time it will be much more proof-oriented (if you want an example of an actual problem, here's one: let (p_n) and (q_n) be two sequences of nonzero integers such that p_n/q_n converges to an irrational number x. Show that |p_n| and |q_n| both diverge to infinity). For this I have heard good things about Terence Tao's Analysis I (pdf link).

Linear algebra is a part of abstract algebra. Abstract algebra is about looking at structures. For instance, you might notice similarities between different situations: if you have two real numbers, you can add them together and get a third real number. Same for functions. Same for vectors. Same for polynomials... and so on. Linear algebra is specifically the study of structures called vector spaces, and maps that preserve that structure (linear maps). Don't worry if you don't get what I mean right away - you'll get it once you learn all the words. Linear algebra shows up everywhere, it is very fundamental. Also, if you know how to multiply matrices, but you've never been told why the way we do it is a bit weird, the answer is in linear algebra. I have heard good things about Sheldon Axler's Linear Algebra Done RIght.

After these two, you can learn various topics. Group theory, point-set topology, measure theory, ring theory, more and more stuff opens up to you. As for category theory, it is (from my pov) a useful tool to unify a lot of things in math, and a convenient language to use in various contexts. That said, I think you need to know the "lots of things" and "various contexts" to appreciate it (in math at least - I can't speak for computer scientists, I just know they also do category theory, for other purposes). So I don't know if jumping into it straight away would be very fun. But once you know a bit more math, sure, go ahead. I have heard a lot of good things about Paolo Aluffi's Algebra: Chapter 0 (pdf link). It's an abstract algebra book (it does a lot: group theory, ring theory, field theory, and even homological algebra!), and it also introduces category theory extremely early, to ease the reader into using it. In fact the book has very little prerequisites - if I'm not mistaken, you could start reading it once you know how to do proofs. it even does linear algebra! But it does so with an extremely algebraic perspective, which might be a bit non-standard. Still, if you feel like it, you could read it.

To conclude I'd say I don't really belive there's a "correct" way to learn math. Sure, if you pursue pure math, at some point, you're going to need to be able to read books, and that point has come for me, but like I'm doing a master's, you can get through your bachelor's without really touching a book. I believe everyone works differently - some people love seminars, some don't. Some people love working with other people, some prefer to focus on math by themselves. Some like algebra, some like analysis. The only true opinion I have on doing math is that I fully believe the only reason you should do it is for fun.

Hope I was at least of some help <3

haha, well, I don't know that much algebra to be honest (me using a fancy word in a joke means i have heard of it before, not that I actually know how to work with it!)

But yknow I could give out some resources, so here they are (so far I have mostly learned from classes but yknow i'm at that point where i'm starting to need to transition from listening to someone ramble to reading someone's ramblings and then rambling myself)

For basic linear algebra I didn't learn through a textbook, but I have heard good things about Sheldon Axler's Linear Algebra Done Right and it seems similar to what the classes I had did (besides the whole hating on determinants part, though I kinda get it).

For some introductory group theory, I also had a class on it, but the lecture notes are wonderful. I would happily give the link to them here but since they're specifically the lecture notes of the class from my uni I would be kinda doxxing myself. Also they're in French. I will give out some of the references my prof gave in the bibliography of the lecture notes (I have not read them, pardon me if they're actually terrible and shot your dog): FInite Groups, an Introduction by Serre (pdf link), Linear Representations of Finite Groups also by Serre (pdf link), Algebra by Serge Lang (pdf link). Since our prof is a number theorist he sometimes went on number theory tangents and for that there's Serre's A Course in Arithmetic (pdf link). I'm starting to think our prof likes how Serre writes.

For pure category theory and homological algebra I have read part of these lecture notes. I think a good book for category theory is Emily Riehl's Category Theory in Context (pdf link). For homological algebra, a famous book that I have read some parts of is Weibel's An Introduction to Homological Algebra (pdf link). Warning: all pdfs I found of it on the internet all have some typographygore going on. If anyone knows of a good pdf please tell me.

For commutative algebra, A Term of Commutative Algebra by Altman and Kleinman (pdf link). I haven't read all of it (I intend to read more as I need more CA) but the parts of it I read are good. It also has solutions to the exercises which is neat.

For algebraic geometry (admittedly not fully algebra), I am currently reading Ravi Vakil's The Rising Sea, and I intend on getting a physical copy when it gets published because I like it. It tries to have few prerequisites, so for instance it has chapters on category theory and sheaf theory (though I don't claim it is the best place to learn category theory).

For algebraic topology (even less fully algebra, but yknow), I have learned singular cohomology and some other stuff using Hatcher. I know some people despise the book (and I get where they're coming from). For "basic" algebraic topology i.e. the fundamental group and singular homology I have learned through a class and by reading Topologie Algébrique by Félix and Tanré (pdf link). The book is very good but only in French AFAIK.

For (basic) homotopy theory (does it count as algebra? not fully but what you gonna do this is my post) I have read the first part of Bruno Vallette's lecture notes. I don't know if they're that good. Now I'm reading a bit of obstruction theory from Davis and Kirk's Lecture Notes in Algebraic Topology (pdf link) and I like it a lot! The only frustrating part is when you want to learn one specific thing and find they left it as a "Project", but apart from that I like how they write. It also has exercises within the text which I appreciate.

For pure sheaf theory, a friend recommended me Torsten Wedhorn's Manifolds, Sheaves and Cohomology, specifically chapter 3 (which is, you guessed it, the chapter on sheaves). I only read chapter 3, and I think it was alright (maybe a bit dry). I also gave up at the inverse image sheaf because I can only tolerate so much pure sheaf theory. I will come back to it when I need it. The whole book itself actually does differential geometry, but using the language of modern geometry i.e. locally ringed spaces. I have no idea how good it is at that or how good this POV is in general, read at your own risk.

Also please note I have not fully read through any of these references, but I don't think you're supposed to read every math book you ever touch cover to cover.

thanks for the kind comments, and I hope at least one of the things above may be helpful to you!

sometimes i'm like "dang i wanna reblog this" but this is a math blog and i don't wanna subject my dear followers to whatever sadpost came across my dash

Shit man, this algebra war is fucked. I just saw a guy clap his hands together and say "the six functors" or some similar shit, and every chain complex around him got put into a short exact sequence, had their long exact sequence taken out and then got their homology calculated. The camera didn't even go onto him, that's how common shit like this is. My ass is casting lagrange's theorem and degree 2 equations. I think I just heard "power word: operad" two groups over. I gotta get the fuck outta here.

I am normal and can be trusted with the power of spectral sequences