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Ask HN: How to self-study mathematics from the undergrad through graduate level?
654 points by hsikka 29 days ago | hide | past | web | favorite | 227 comments
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

Here's my personal opinion about how you should approach this.

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.

My oldest brother had a real love of mathematics and got a masters degree but never even applied to a PhD program. When I asked him about it he just said he was tired of working on other people's problems.

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 don’t know what PhD program your brother was looking at but ideally candidacy committees won’t pass you unless you chose the problem or made it your own. Where your brother is right is that forming such a ideal committee is mostly politics.

I don't know about math, but in CS, a lot of funding comes from specific grants, so you'll be working on something related to what your advisor got grant funding for.

Regardless of grants, etc., your general area of work is going to be based on what the faculty members' focuses are. But in any case, you'll probably still be doing your own sub-piece of some larger effort. You wouldn't choose an advisor at random, so this isn't really relevant.

I mean, if you're interested in astrophysics, you shouldn't go to a school where the Physics department is focused on nuclear physics!

I think you are right on both points. There were more details and his answer was just a quick way to say:

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

Selecting an approachable problem is also a skill that more experienced people have and often the younger people don't. Some problems are just not ready to be tackled because the field hasn't developed for it yet. It is a risk factor between choosing your own problem and floundering through the quagmire or getting help in choosing a problem you have confidence in solving in some years.

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.

Personally, I think this is bad advice, because without an undergrad+ background, the projects above will either be impossibly frustrating or you will make up some crackpot bullshit. Plus, the undergraduate curriculum is its own reward.

Not necessarily. I left undergrad after four terms and self-studied to the point of having published research, giving invited talks, and even being a visiting researcher for three months with my expenses paid.




Look for my name (Thomas / Tom Price) on these pages:





And I think abnry is right that you need some motivating problem / project. In my case it was to prove the Riemann hypothesis. Obviously I have not succeeded but I've learned a lot in the process and it has indirectly led me to some good research questions. I think choosing an outrageously ambitious pie-in-the-sky problem is ok if you are patient and don't try to approach it too directly.

Wow, that's remarkable.

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.

Are you referring to the "An Inequality for Gaussians on Lattices" paper? They cite my paper, but it's to give an example of an application of their result (which I use), not because they built on it. But anyway, I think it's very fascinating that the people who discovered a key result that I needed for that paper, which could probably be best classified as arithmetic geometry, are mainly computer scientists!

Ah, thanks for the clarification. The computer scientists who work on quantum computational complexity and post-quantum cryptography tend to be much more mathematical than the norm :)

Just curious, were you working while you studied? How did you manage this?

Sometimes yes, sometimes no. Of course I can go into much more depth in my studies / research while not working. I work the minimum amount necessary to pay my living expenses so I can devote a maximum amount of time to freely pursuing other interests, which includes pure math among other things.

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”.

Nah, a good way to learn new skills is to pick a destination and then figure out what steps you need to take to get there. This type of “top-down” learning can help one stay motivated through the most frustrating road blocks. This is especially important for self-learning, because unlike an undergrad setting, the person is on their own and can’t rely on peers.

Well, we disagree. Plus, taking on some crazy problem without background saps motivation as well.

I think setting attainable goals and working towards it helps. It is very easy to lose motivation if you don't know what you are going for..

Depending on your background, you probably don't have enough information to pick a long-term goal anyway.

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.

>>Depending on your background, you probably don't have enough information to pick a long-term goal anyway.

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.

The two of you are talking about different things. What forkandwait is talking about is the propensity for people with only an undergraduate education in math (or less) to not actually know what a worthwhile goal is. They usually either lack the mathematical maturity to intuit how difficult a particular problem is (whether it's tractable with available mathematics, whether it's tractable for their ability, etc); or they formulate problems which are "not even wrong."

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.

Not in math. Unless you are a mathematician, I challenge you to pick a math equivalent of "I want to create a website where people can buy and sell yarn." I’ll wait.

People just assume learning math is the same as learning everything else. That is not even remotely true.

Genuinely curious - why do you think that? I have been self studying math for about 4 years now and find it to be the same as everything else that's worth learning: hard! But I haven't found that it's some entirely different realm divorced from all other intellectual pursuits.

I don’t really have time to give a thoughtful answer (it would be quite long), but the exact post you responded to gave an obvious difference. To roughly summarize that difference, producing anything of value in mathematics requires learning a tremendous amount of prior art, and without a tremendous amount of work you won’t even know what’s of value. It’s no wonder that many crackpots choose to work on high profile number theory problems, like Goldbach’s conjecture and previously Fermat’s Last Theorem, since the formulations are simple enough for laypeople to understand, yet the theories behind developed over hundreds of years are incredibly deep.

> 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.

What makes math special, exactly?

This is fair. That's why you need to make the goals your own and make sure they are doable. And sometimes, an unreachable goal will still help you learn and value the fundamental material.

Like, if you are into Rubik's cubes, that's going to make learning group theory a lot more fun and motivating.

I'm with you but I think it depends on the person.

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.

Every example project suggested by the GP can be easily externally verified, so it has built-in protection against crackpotism. Secondly, it may be very difficult, but not impossible.

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).

The OP did specify what he wanted - the basic undergraduate and starting graduate curriculum. That's a pretty well defined area: Algebra, Real Analysis, Geometry and Topology with maybe some complex analysis, number theory, statistics, CS or etc thrown in.

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.

> I personally did work myself up to the graduate in math during the last two years of High School & first year of college.

...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).

How in the world did you manage to do this? Did you actually self-study, or were you placed in a gifted program?

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.


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.

>>> I would guess that each person has a somewhat unique motivation strategy.

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."

I think this is good advice. I tried to read parts of CLRS a few years ago but could not wrap my head around the work. The mathematical prerequisites needed for CLRS gave me an end point to guide my self learning to focus on specific areas: algebra (linear and elementary), calculus 1, combinatorics / probability, discrete math, propositional logic.

This is terrible advice. Apart from the last sentence. A better advice would be to specify which subject to learn.

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.

The advice I gave is not exclusive to working through the typical undergraduate books.

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.

I see. Well, you have a point, but the OP did specifically ask for a plan like that Susan Fowler's blog post. And I am an applied mathematician working mostly on computational physics and I can attest to the requirements I mentioned.

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.

Is this the recommended order to read/learn them in? If I was primarily concerned with getting up to par with math for the sake of being able to actually understand everyone's favorite algorithms textbooks, would you still suggest working through all of these, or could you recommend an abridged list?

If you want to understand algorithms, you need a computer science curriculum, not a mathematical curriculum.

Also, unfortunately, its not the recommended order to learn them in.

This is exactly the kind of advice I give to people who ask me about teaching themselves to "code".

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.

I don't think this is a good comparison. People who are skilled in both areas might not realize that, but learning how to program is a hundred times easier than working through any advanced topic in higher mathematics and coming up with your own proofs.

This thread seems bizarre to me. There's one guy claiming he "worked himself up to graduate level math" in the last two years of high-school. So either he's a literal IMO-gold-medal-level genius, or people here don't quite understand what undergraduate math studies actually involve. Even if he is that intelligent there's no way he actually did the amount of work required on such a broad number of topics.

Can confirm. I’m not an IMO gold medalist, but I come from a very competitive nation that fetch five or more gold medals on most years, and I almost made the national team, twice (failed at TST, which selects 6 out of ~30). And I graduated with a math degree from one of the top institutions. There’s no way I would have completed undergrad plus entry level grad math in the last two years of high school — those took me three years in college (of course I was doing other things, but still).

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.

While I agree with what you've claimed here, in fairness the OP never said anything about proving novel theorems :). Learning advanced mathematics at the higher undergraduate or graduate levels is probably more difficult than programming (at least in my opinion), but it's much closer than proving something original. Likewise programming existing algorithms is significantly easier than designing novel algorithms, which is much closer to proving novel mathematics.

Is this sentence incomplete or am I missing the context of the word "compelling" here?

> be able to write up a compelling

Can someone clarify what this means?

I suspect it was meant to say "be able to write up a compelling proof".

By mistake I swapped lines in my post when composing it. It is fixed now.

This is really good advice, and I thinking about suggesting the same thing. I've accepted that there are some basics that you have to have in place before you can really get traction and learn what you need to learn for specific projects.

There's a set of basics that you will want no matter which direction you go: calculus/real analysis, linear algebra, differential equations/dynamical systems, and sets, groups, rings, and lattices.

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.

Most engineers I know have not learned set theory or groups/rings/lattices. They still seem to be doing pretty well.

in the US, most engineers take the standard two-year lower division sequence (calculus, linear algebra, a bit of diffeqs). for the most part, you learn technique rather than proving things. upper division engineering math courses teach more technique (e.g., more diffeqs).

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

I am a scientist/engineer/mathematician who studied the hell out of abstract mathematics at an extremely rigorous undergrad program. In the 20 years since, not once has that knowledge been useful in my academic or industrial work, not even remotely. I am all for studying theory for its own sake, for the career theoretician and the interested hobbyist, but as an investment I regret those four years of my life as a colossal waste of time.

Math is not just a tool; it is an area of intellectual exploration. By the same logic studying history or philosophy is a colossal waste of time, too.

so, in real world application, what's the most useful/valuable topic in mathematics?

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).

> in real world application, what's the most useful/valuable topic in mathematics?

Basic arithmetic: addition, subtraction, multiplication, division.

It depends on what your relative judgements are for what is deemed 'valuable' or 'useful'. Are the number theory foundations that allow digital transactions to occur more valuable than the subtraction done to figure out how many minutes until the next hour?

From what I've observed, most engineers are glad to be done with math when they finish college. Most engineering is qualitative: Organizing and arranging things, making things fit together, and troubleshooting. Maybe 10% of engineering is quantitative, and that work often goes to the handful of people in the department who have maintained an interest in it.

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 refer to engineers, do you mean actual engineers (BSc in Engineering)? or your web designer with jquery skills who calls himself engineer?

In my experience (as a civil then software engineer with a MS in CS in ML/Big Data) "real" engineering is much more prescriptive than software.

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.

I'm referring to people with degrees in mechanical, electrical, etc., or programmers with computer science or related degrees. These are all people who got through some level of college math requirement such as calculus.

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.

Depends on what you're doing. My father did a huge amount of mathematical analysis over the years (roofs, bridges, trusses, ship hulls), but, again, he had the mathematical ability to do it. One of my best friends gets called into big institutional HVAC systems when they're misbehaving, and pretty much makes his living from qualitative understanding of dynamical systems that requires this kind of foundation.

Come to think of it, most of the math majors know (who didn't get PhDs and now work as programmers or data scientists) have forgotten the greater part of what they learned about groups/rings/lattices, and they seem to be doing pretty well too! ;)

Most people I know have not bothered to learn any math beyond algebra, and try to forget what algebra they did learn, they still seem to be doing pretty well. You can always limit yourself and pretend that's "enough" but it can very well be self-limiting. You simply don't know what you don't know, and that includes not knowing all of the things you might miss out on by not studying. Set theory and abstract algebra has a huge range of applications, simply being able to earn a living without studying those things is an insufficient reason to avoid studying them.

Another thing most engineering courses skimp upon is variational methods/calculus of variations, and that has effect in how they do classical mechanics. Physics majors and engineers do classical mechanics in very different manner.

Depends what you work on. My suggestions were based on what has served me well professionally. Another way of looking at it is that if you don't have this, you're not going to get to work on things that require them.

Neither did most historians, or most apple farmers. What's your point?

Any suggested reading on rings and lattices? I took a lot of math classes in undergrad but was never exposed to these concepts.

Any standard abstract algebra textbook will cover rings in gross detail. Dummit and Foote is a much beloved standard, but is also used in graduate classes. I've heard good things about Pinter's book (http://www2.math.umd.edu/%7Ejcohen/402/Pinter%20Algebra.pdf).

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.

Really enjoyed Algebra: Category 0. Introduces some basic Category Theory and uses it to develop the standard practice of undergrad algebra. The exercises are medium difficulty (nothing ridiculous like a Knuth or Spivak problem, but nothing breezy either) and there are enough examples in the text of proofs with enough detail to get you working on the exercise problems.

Did you mean Aluffi’s Algebra: Chapter 0? What a wonderful book.

Sorry yes

Herstein Topics in Algebra or Dummit and Foote.

goofy suggestion. those are both either grad or senior undergrad books depending on where you are. much better suggestion is

A Book of Abstract Algebra by Pinter (really gentle)


A First Course in Abstract Algebra by Fraleigh

I used on in high school and the other my second year of undergrad. They have lots of material but they also have good exercises and explanations.

Herstein was my undergrad book back in the early 90's. I remember really liking it. It's been a very long time, but my recollection was that the groups to rings progression felt a lot more natural to me than the rings to groups progression of some of the other books at the time.

Dummit&Foote imho is the best undergrad algebra book.

Lattices (the algebraic structure) seem to exist only on Wikipedia in the sense that the only time I've seen lattices every mentioned in my standard undergrad education is once for a proof of the Stone-Weierstrauss theorem.

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.

There is an interesting rant from Gian-Carlo Rota on why lattices are so rarely taught despite being so ubiquitous. I include them based on the amount of use I have gotten out of them over the years. For example, the entire theoretical structure of eventual consistency is "make your merge operation the meet of a semilattice."

Basic topology is also important.

You're absolutely right, and I kind of quietly slipped it in under "extracting qualitative information about functions" in calculus.

Here is a proven approach for at least the first part, building foundations and being ready for graduate work. Many Berkeley Ph.D. students passed through this route. Get the book "Berkeley Problems in Mathematics." It contains historical problems from the Berkeley math prelim exam, and solutions. Now don't look at any solutions yet.

This is the exam all Berkeley math Ph.D. students must pass within three semesters of arriving to stay in the program, and the fail rate is about 50%.

You will also need reference books, advanced undergraduate and beginning graduate textbooks. Buy, download, or borrow as appropriate.

Pick a problem (start with the older ones, they are easier). Set aside 30-60 mins and try to solve it. No devices, no references at all, go to a library or a coffee shop without your devices. Dont' give up till time is over. If you cannot (usually the case), still don't look at the answer. Hit the reference books (don't look up the problem online either, it will go right to the answer and you won't learn much). Read and try to understand enough so that you can solve the problem. It is ok if you solve it this way (in the course of reading about it).

For bonus points, students studying for the exam will typically take entire old exams (available from the Berkeley website), take that to the library and just sit down for three to six hours and try to solve all the problems correctly. Then self-grade harshly. When you can do that for a recent exam (and get a good score), you will have more or less mastered undergrad math to the point that you could teach it.

Most important: you have to struggle to solve problems. Reading a solution is about as useful as watching someone else lift weights: you get minor tips on form but not any stronger.

Thanks for the excellent analogies!

The hard but maximally useful thing to do, in my opinion, is to regularly meet with at least two other people and a blackboard and beat your heads against it together for about 3 hours at a stretch. Do this at least weekly--and preferably more often. Self study in between meetings is obligatory.

One 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.

Just to add, it's also worth finding someone who might be a bit below you in terms of maths ability. If you can explain the concepts to that person, then you know the concepts.

Yep. You can do this, if you're lucky, with the triad. You and the other person at about your level can trade places depending on subject matter, each explaining to the other and reinforcing their own knowledge.

Like Einstein's Olympia Academy!


Can we start something like this online for Hacker News community who are interested in Mathematics?

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.

I'd be very interested in something like this.

Glad to know you would be interested in this. Please feel free to click on group link I have shared in my previous comment and hit "Apply to join group".

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!

I always got more from working problems on my own with an occasional consult from peers or a teacher. A danger is that you think your group sessions substitute for working problems.

group sessions keep you honest and help you work through high level stumbling blocks. You're expected/obligated to do a fair amount of solo pick-and-shovel work in between.

Well in my experience they are a waste of time, but that's just me.

you have to have the right group. It's nontrivial to manage, unfortunately, and if you do not have the right group, it is indeed a waste of time.

this is an excellent recommendation. it did not work for though. i was not the one that was at the advanced level and the one that was got bored rather quickly with us minions. Happend 5 different times.

if you could find one who willing, this recommendation will def help.

Other people's help is indeed one of the most effective ways to learn. My problem with that is simply finding such people - given my extremely introverted disposition it's much harder for me than for normies - and also what you say, ie. finding people who would not quit after three sessions.

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?

I know people who have done exactly this. You’re looking for grad students - they often know exactly the right amount, and are very willing to work for some extra cash, and some (not all, but some) are happy to teach.

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.

Definitely, plenty of PhD students looking to make some extra money. I saw requests like this on grad student email lists frequently.

As an alternative I would suggest a top-down approach. Start with the theorems/results you truly wish to understand and work backwards.

There was a great quote from an interview of Peter Scholze (one of last year's Fields' Medallists), which has really changed how I view learning:

At 16, Scholze learned that a decade earlier Andrew Wiles had proved the famous 17th-century problem known as Fermat’s Last Theorem, which says that the equation xn + yn = zn has no nonzero whole-number solutions if n is greater than two. Scholze was eager to study the proof, but quickly discovered that despite the problem’s simplicity, its solution uses some of the most cutting-edge mathematics around. “I understood nothing, but it was really fascinating,” he said.

So Scholze worked backward, figuring out what he needed to learn to make sense of the proof. “To this day, that’s to a large extent how I learn,” he said. “I never really learned the basic things like linear algebra, actually — I only assimilated it through learning some other stuff.”


The issue with this approach is that if you learn just "the fun parts" you might be left with some huge gaps in your knowledge, all that in-between stuff - especially if you're not a genius like Scholze. Standard approach is perhaps less motivating and you learn a lot of stuff that frankly you'll never need and you'll probably forget most of it, but it ensures that you've at least heard about all the major ideas. One day when you run into a problem you'll know where to look for more details. This is IMO a common problem with self-thought programmers as well, they often end up inventing a wheel simply because they just never heard that solution to their problems already exist in some 70s CS textbook.

I think fun is an extremely important part of keeping up the pace of learning. I would rank that much higher over "methodically combing through everything", which could cause you to quit from boredom long before you reach your goal.

I'd hypothesize that the problem you describe is due to insufficient curiosity about the way things are already done. Especially with Google, Wikipedia, and well-populated internet forums, it'd be hard not to find "foundational solutions" after a little research.

I had a music teacher who was obsessive over minutiae, and fussing about the exact position of each finger on a piano key was much less fun than loosely noodling around with chords and making things that sounded closer and closer to real music. I made little progress with the lessons and quit, but since I started noodling on my own I've been getting much farther.

I propose that "intelligent" is a synonym for "gets enough pleasure from learning to do a lot of it", so I would overwhelmingly optimize for that.

You're totally right, it's an approach that requires more motivation, but I don't think it's really that hard. University curriculums are made for young people who're not necessarily super motivated to study (many study just to finish the course, not to learn it) and who also have to study a lot of other things in the same time (and to party, and to fall in love and feel miserable and all other things that you do when you're young that are way more important to you than math). Compared to that your starting point is not that bad at all. Being older and more mature, plus genuinely interested in learning that subject you're probably way more motivated, plus you don't need to pass 6 or 10 courses that year, you can concentrate on just that one, at your own pace. Again, it's really up to ones own personality, there's no one-size-fits-all here, but IMO chances are that you'll learn it way better than someone who had that course on Uni.

And regarding wikipedia and google, they're much more effective when you know what you're looking for. If not exact name of theorem or algorithm then at least "there was that thing that we learned related to that other thing". Having at least a faint idea like that can save you tones of time when researching.

I have a lot of sympathy for this approach, and of course you learn more by having fun and continuing, then being meticulous and stopping.

However if at each fork, you choose the fun road, you might get a long way until you run out of options (which may be fine). But the terrible difficulty is necessary knowledge that is not explicitly stated. These gaps ("mathematical maturity") are difficult to even identify, let alone fill. Story time:


In high school, I missed a week or two, and later got stuck on a problem in calculus. Discussed it with the teacher, and he eventually seemed to see my difficulty was, but wouldn't tell me, instead saying "you can work it out". I couldn't.

Years later, I started asking online, and people responded helpfully, but didn't actually help. I read wikipedia and math websites about it. Watched videos about it, that were interesting and gave a new perspective, but didn't help my particular issue. I looked it up in famous textbooks (Spivak, Hardy), still nothing.

Finally, I did all the problems in the derivatives section of Khan Academy, but when I came to this problem, it still didn't address my specific issue.

Thinking I had gaps in my knowledge, some known to me, and suspecting others unknown, I went back to even earlier material.

Some time later, I reviewed the problem again, and realized that the issue was a completely trivial part of limits - an earlier section. Which was what I missed in those 2 weeks of high school, all those years ago.

A nice maths teacher can save you years.

I've long wished for a series of interactive ebooks or websites based on that type of approach. Each would be devoted to one great theorem or problem.

Each would start with a presentation of the theorem and proof, presented how it would be presented today if it were a newly discovered research result being presented by professionals to professionals in the field.

At each step of the presentation, there would be two expansion options. One is to ask for filling in the details. A detail expansion keeps the presentation at about the same level of required knowledge, but takes smaller steps. You use a detail expansion when you understand where a step starts and end, but you just don't quite see how it made the connection.

The other expansion option is to ask for background or prerequisites. A background expansion is for when you don't have the background to even understand the start and end points of a step. It opens up material to teach you the background necessary to understand what is going on.

A key aspect is that this would all be recursive. You could do a background expansion on a background expansion, and so on, all the way back to common high school math.

The background expansions would just teach enough of their subject to support the step above. So, for example, if you used one of these interactive books to learn an analytic proof of the prime number theorem, and you started knowing nothing beyond high school algebra, you would end up learning all the calculus and complex analysis needed to prove PNT, but only such calculus and complex analysis as are needed.

What I wonder is if you could pick a set of theorems and problems for such books such that (1) someone could go through them all in about the same time as a conventional math degree takes, and (2) combined, the background expansions would have covered as much as a conventional degree.

If so, that could be an interesting way to keep motivation high because everything you are learning has a direct, visible, connection to advancing the proof of the interesting theorem at the top.

This works right up until someone makes you have to pass a qualifying exam. Ask Terry Tao - he failed his first time.

An approach might work well for a Fields medallist but less so for normal people.

I love generative art, so I quite often run into things like Lorenz attractors. I looked up how they are made and that lead me to differential equations which lead me to better understand calculus. So my love of pretty graphics lead me to learning math with a purpose.

I haven't won any awards yet.

I dunno, it's worked for me. I have a problem I want to solve, then learn the tools I need to solve it. Not just math, but just as a software developer.

Look at any undergraduate course curriculum at your university of choice and just follow that. Some of them might even have lecture notes and exercises published.

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.

I was thinking about completing the MIT applied and pure math major course requirements via open courseware

I did this a few years ago, now doing an MS in math, after last formally studying math in high school, more than 20 years previously.

There is lots of advice in this thread, including long lists of books - at your level, I think that’s unhelpful. To get started with the foundations, I think it’s best to have one or two books at most, that you work through completely, then decide what to do next. Otherwise you can waste a lot of time time figuring out “what to do next” instead of actually doing math!

I would personally recommend “Engineering math” and “advanced engineering math” by Stroud. The first covers “foundation” topics like algebra, trigonometry etc, then covers the first year of math for a English engineering degree. The second book covers the second year.

Note that English engineering degrees are 3 years vs 4 in the US, but have no general education requirements, so the math covered in the two books is roughly equivalent to the 1st-3rd year of an engineering degree in the US. Both books are very much focused on calculations, not proofs, but will give you a fluency in handling mathematical calculation that is assumed when doing more proof-based courses later. And they are both designed for self-study, which is important.

The other book that I would get, and do in parallel once you have done the foundation part of the first stroud book, Chartrand, Mathematical proofs. This will teach you how to do proofs, using mainly (high school) algebra and basic number theory to start, but going on to cover some proof techniques in analysis, advanced algebra and others areas.

Once you have done those 3 books, you will have a solid basis for further study. I also agree with the comments about interacting with others studying the same material if possible.

Happy to chat sometime if you want to message me (contact info in profile)

The University of Oxford has all of their notes and exercises available online https://courses.maths.ox.ac.uk/overview/undergraduate

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.

Very nice - I particularly like the links to the MacTutor History of Mathematics archive:


Many of the courses listed there have no course materials available or only very incomplete materials.

Oh really? The ones I cliked through to did!

Hello, I have actually done this. I learned Algebra up to a good amount of Vector Calculus over the course of four years mostly through self-study.

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.

Bet you weren't married at the time. I'm trying to re-learn calculus myself, and I have to hide it from her because she gets mad at me when I try to do calculus problems: "why are you doing this? Are you doing this for work? You don't have to do this. There's no reason for you to be doing this."

I hope you are kidding to some degree. Having to hide things you want to do from your spouse doesn't sound like a good relationship.

"Why are you doing this?"

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.

Gosh, that sounds terrible. Like it sounds like she doesn't even want to understand. Okay, you can study math if you give me three hours of "free time" and then she goes off shopping while you watch the kids after your math break lololol. Sorry man, that sounds like hell.

IMHO the books being advised here are too high level. Its like trying to read about Computer networking basics from RFCs. Some advise are equivalent to searching for a reason to learn "C" by appreciating what can be done with "C" by reading books on Linux Kernel programming.

I will mention a few books which by no means are undergraduate level, but are per-requisite for undergraduate level study. Are you thorough with books given below? These books are available through http://gen.lib.rus.ec.

1. Higher algebra - Hall and Knight 2. Trigonometry I - Loney 3. Coordinate geometry I - Loney 4. Calculus of one variable - Maron

Download the books and check if you already know all those. Once done, then go for the higher level books most the people advising here.

If we are talking about content you can find plenty of advice here or elsewhere. But do yourself a favor and pay a tutor and/or find a study group. As in writing, dancing, etc. you cannot evaluate your own work good enough.

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.

You will not become good at math watching youtube videos and reading books.

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.

I'm sure padthai meant that you need to do math to learn math, not just read or watch videos.

I'm sure padthai meant that you need to do math to learn math, not just read or watch videos.

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.

Let me add to that.

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.

Listen to this wise man, he knows what is speaking about.

I've been following a breadth-first-based-on-my-interests approach for the last several years. It's not been very systematic, but it's been a lot of fun. I think math is essentially clear thinking, and it's best when it's explored from curiosity.

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)

3B1B is amazing at explaining concepts. Lots of commenters in his videos studies math/cs/physics and tell stories about finally understanding a topic they misunderstood in university.

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.

It is really hard to truly self-study in mathematics. Going through the opencourseware and reading textbooks (working as many exercises as you can, of course) will get you only so far. It is important to have someone (ideally with a PhD-level education in mathematics) who you can meet with to guide your study, correct mistakes and answer questions.

I strongly disagree actually. I think mathematics is the best field to self-study. There is nothing in mathematics that cannot be explained on the paper. There are many textbooks that are very good that in most universities professors won't be that good anyway. I went to UC Berkeley to study mathematics (ended up studying CS though) which is supposed to be a top department, but most of my textbooks were better teachers than my professors. I still prefer reading textbook to people explaining me math. I don't even think I understand math when people explain me. I need to first teach it to myself. Then occasionally people offering different perspectives is very beneficial, which, again, can be done on paper.

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.

As someone who did study both math and CS at university I disagree. I think there are numerous courses that can more easily be self-taught[1], mostly what one would encounter in their first ~2 years in a math degree. After that things get conceptually a lot more difficult.

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.

The reality is that having a great mentor is a privilege, not a given. I don't think anyone is arguing that having a genius and brilliant teacher wouldn't help, rather that it is possible to reach an "advanced" (grad/undergrad level) understanding of mathematics without the luxury of having someone who is far more knowledgeable to turn to for help.

Precisely. GP here. I didn't argue against having a great mentor. If you go to a great university you clearly have an advantage. I think it's perfectly doable to teach yourself mathematics if you study intense enough eith correct tools.

I think a big part of "mathematical maturity" is knowing how important it is to have someone else look over your proofs.

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 had the same experience. I tried to self study Math. I hit a wall where it became inefficient at best and beating my head against the wall for the rest. I couldn’t do it alone past material normally covered in the first two years of university.

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 didn't realize the value of this until after I was done with my undergraduate studies.

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 agree, I would not have survived math in school without being able to ask other people.

I learned programming on my own from books but I don't think I could do that with advanced math.


My Background: Current Undergraduate in CS and I recently added Mathematics

The most difficult part for a person who hasn't done a lot of math to become a person who does a lot of math is to read and understand rigorous proofs. You will encounter countless difficult proofs in any mathematical topic you try to study. Read a few books on mathematical thinking and proof techniques before/during/after reading any other dense math book.

Like you, I realize the value of having a mathematical mindset and want to have a deep understanding. When I added math as a major, I had a very hard time jumping from computational courses (typical math courses, geared towards any major) to theoretical and conceptual courses (proof-based courses that use all the fun and interesting math books everyone has linked here). These books helped:




<3 this is a great book, obvi since its george polya

> The most difficult part for a person who hasn't done a lot of math to become a person who does a lot of math is to read and understand rigorous proofs.

I’m afraid you haven’t delved into any advanced topics. That’s actually about the easiest part, and could be mastered by ten year olds (certainly myself when I was ten).

alnar is likely referring to the US system, or something like it: unless you are tutored externally (rich), an autodidact outlier (gifted), or selected for honors courses, you basically take computational courses for 14 years (with one cursory stop for euclidean geometry) and are then thrown into proofs at the age of 19-20, if at all. it's widely recognized as a problem in the math pipeline, which is why many US universities have "transition" courses for non-honors students.

so it's not that the basics are intellectually difficult as much as practically difficult (unfamiliar, disorienting) for many students. many "transition" books talk about the difficulty in adjusting from talent being redefined from perfectionist "plug and chug" (APs, SATs) to reasoning and creativity.

btw, i'm impressed that you could master college-level proofs at 10. i have a kid about that age who is pretty good at logical reasoning, but i'm not sure what topic (at that level) he could do a rigorous proof about; maybe numbers, as in landau? can you say more about the materials you used?

This has little to do with education systems, I was talking about objective difficulty. Try to tell any mathematician that the hardest part of math is rigorous proofs; if they don’t laugh in your face, they are just being polite. One may think the transition is “hard”, until one actually gets into more advanced topics. As you said, it’s just unfamiliar to the uninitiated, at best.

Reading and presenting rigorous proofs in elementary number theory, Euclidean geometry, etc. is easy for gifted ten year olds and definitely manageable for a lot of fifteen year olds. You asked about material — it doesn’t actually matter, and I don’t recall specifics; any entry level treatment of elementary number theory should do (really beautiful subject with a very low barrier of entry). For kids who have eyes on IMO, it’s very common to be throwing around perfectly rigorous proofs at young ages.

A nice foundational textbook to start with is Michael Spivak's text "Calculus." This text is fairly conversational, it motivates its concepts well with many examples, and it will help you build a strong foundation in writing proofs and reasoning mathematically.

As others have said, it might be helpful to have a friend or two read with you so you can answer each other's questions. Also, make sure you do a ton of exercises!

Check out Open University (https://www.open.ac.uk), they have a undergraduate (BSc) up to graduate (MSc and PhD) level programmes in Mathematics, and they take distance-learning seriously.

That’s what I would do if I were you. Learning this stuff on your own will be very difficult. I can’t even imagine the amount of discipline needed...

I was a homeschooling parent. I also have a stronger math background than average, though it isn't anything impressive for this crowd.

I will suggest you try to find a written curriculum from a respectable source as your first step so you have some idea what you need to cover to meet the stated goal.

Then you will want a variety of study materials for the subjects in question. You will want to verify that these are respected materials that are not full of errors.

When I brushed up on math to CLEP Alegebra so I could take college statistics, the computer program I used for self study had errors in it. I knew enough math that I knew what was wrong. I was just rusty and in need of a bit of practice. But it was a program I did not use with my sons to homeschool them because of the errors.

You want to know which materials math geeks like. You want them vetted, basically.

Then use whatever quality materials most appeal to you. Different strokes for different folks and self study let's you pursue whichever materials you like the best.

If you want to start with a single self-contained book, then a really good choice is "Modern Mathematical Physics" by Peter Szekeres:


It covers most of the mathematics needed for quantum field theory (which is a big chunk of applied mathematics) starting right from set theory.

I could give you a list of books that are used at the graduate and undergraduate level that are considered good. However, almost none of them really discuss the motivations behind topics. My suggestion is to start with this book by Arnold on Abel’s Theorem. If it interests you then delve deeper into the topics once you are done with it.


Lots of good points here. I can't speak to the graduate level, but for the undergrad level I think the trick is to find textbooks that don't suck (most good textbooks are free, or are old enough no one cares about copyright, and a million pdfs are online, or you could get it used for $15. Most bad textbooks are price gouging undergrads).

Then just put the time in with deliberate practice. I tried self-studying math for a year, and often skipped the harder practice problems, thinking just reading the textbook and examples is sufficient. That's actually entirely useless. You are't learning math (or anything worth learning) unless you spend 6 hours on a Saturday working through some hard problem that makes you so frustrated that you ask God why he made you such an idiot. If you spend enough time doing that you can actually learn something.

The obvious route for self-study is to go through Halmos or Axler and Rudin, plus all additional materials and study aids such as Gelbaum & Olmsted, emulating the most popular Harvard Math 55 incarnation. Another nice Math 55 incarnation covered Hubbard & Hubbard, which is a wonderful book.

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/

This book is fantastic and pretty much takes you through an entire undergrad mathematics course: https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...

The topics covered in that book are undergrad level, but that book is not suitable for learning the topics. It’s more like a high level discussion of the topics looping them together with historical background. It’s more appropriate for people who are already familiar with the material.

I discovered this amazing YouTube channel recently that I wish I had access to back in college. It explains some very difficult math concepts in a very simple and easy to understand way: https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw

It's amazing how seeing something visually or having someone else solve something can aid in your understanding of a topic. I remember having the most difficult time understanding mergesort from my book's pseudocode and my professor's lecture. But seeing mergesort in action on youtube revealed what it truly was and made what seemed impossibly complex, so simple and elegant.

Hey, I am also on the same boat. I have been trying different things lately. I have gone through a lot of courses and a lot of books, but most of these experts have the curse of knowledge. They don't know how it is like to not know. They skip things that seems obvious to them. But, for begineers not having such information can be frustruating and feels like the author/lecturer is moving fast.

I recently stumbled upon this book "The Language of Mathematics: Utilizing Math in Practice" I have been reading this book for a while and the author takes a different approach in teaching maths. I am slowly beginning to learn how to forumulate a real world problem into mathematical model and then solve it. More importantly, I am able to read and understand mathematical expressions better through this book. This book has raised my hopes in learning and apply math. I would recommend you to give it a shot on this before you go through any other courses.

I suggest you get the syllabi from a state college, and then follow that. "work through" the books : copy each page of exposition by hand, try to do the examples and proofs before looking, do all or most of the problems with answers, and get ready for a long journey.

I find a page of math textbook takes about 30 minutes (at least) to really work it over until you understand it.

Personally, I don't much like working in groups or watching video lectures. To learn math, you have to be able work problems by yourself, so you might as well just do that.

David Morin has some great self study books:


I have a math BA from a regional state college, usually scored in the top 5 on the tests, and I am self teaching physics / me /ee about 5 hours per week after work.

Don't study "math." Find a topic or a problem you'd like to understand, look up the prerequisite knowledge for it, and start there.

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.

I had math through some introductory calculus when I was an undergrad 20 years ago, but I let my skills lapse. But I want to understand enough math so that I can do some more complex electronics projects and some statistics / ML / intelligence analysis. This level of math, as I recall, is more or less the math core of the undergrad engineering program I dropped out of in favor of a philosophy degree (cause I am dumb as dirt).

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.

One great thing about Khan Academy is the gameification aspect makes sure that you go over older material in a spaced reptition sort of way which helps really drill in the knowledge so it isn't forgotten

That sounds like an awesome path dude, keep at it.

I'm actually building an app specifically for this use case:


Polar is basically a personal knowledge repository.

The idea is that you use Polar for all of your education. Either official education or continued learning.

Every textbook or technical paper should go into Polar.

You can then annotate the textbooks directly including text highlights, comments, etc.

It supports spaced repetition systems like Anki so you can continually review what you've learned and NEVER forget anything.

It doesn't make sense to pursue a PhD or spend 6-12 months researching a topic to just forget 80% of it due to bit rot.

The reason I named it Polar is that it's meant to 'freeze' all of your knowledge so that you never forget it - ever.

Learning math isn't really about memorizing things.

very true, in the sense that rote memorization is not the point.

but false in the sense that doing math requires fluency - in applying a small amount of technique up to lower division math, and in applying a large number of definitions/results after that. in areas like abstract algebra, failing to memorize will kill you. it is as disfluent as writing text in a foreign language without having memorized the working vocabulary and grammar; even if you can eventually recall, derive or look up what you need, it's just not practical to work like that. (and i say this as someone who prefers to mentally index information rather than memorizing...)

I have to disagree, especially about abstract algebra. Most concepts and theorems feel like abstract nonsense (not specifically talking about category theory here) when you don’t understand them, but should become pretty natural once you do. For true mastery you need to work with the concepts and results on a day to day basis for a while, by applying them; continually reading the text of definitions and theorems hardly helps if at all.

i agree that memorization by internalization (concepts) is different from (and superior to) memorization by rote (text); i was agreeing with the other fellow that rote memorization is not the end goal. however, to me they're both "memorization" because they both represent work to achieve fluency in application.

however, i made my comment because i think i disagree that spaced repetition has no place in learning. i think that if you dig around in the Polar guy's earlier comments, you'll find threads where folks like michael nielsen are talking about using spaced repetition tools for much more than purely textual memorization of theorems - basically, cycles of repetition and (re)synthesis. so i don't feel it's right to completely shut him down about card decks. you may disagree, of course.

When your definition of memorization encompasses all forms of learning, saying memorization is crucial to learning is pretty much a tautology. No one claimed that amnesia sufferers are perfectly good for mathematics. Same goes for repetition. No one expects you to understand and never forget the theory of cohomology by writing down a long exact sequence once.

The thing is, the root comment of this thread specifically talks about "continually review what you've learned and NEVER forget anything" through looking at highlighted notes in the software mentioned (a somewhat out-of-place plug, I'd say). That's not how math works. You refresh your memory by tackling preferably new problems. Reciting proofs is largely pointless (except for certain very elegant proofs, in which case you probably won't need to recite them anyway); reciting definitions and theorems is even less useful.

> it's meant to 'freeze' all of your knowledge so that you never forget it - ever.

Yeah, no, you don't "freeze" your mathematical knowledge.

I think there is absolutely nothing in math one should memorize. "In abstract algebra failing memorize might kill you" is nonsense. Memorize what? Axioms of group? Tactics that can be applied to problems?

please consider the statement from the position of a student just learning the material (like the OP). if you are working through a book with 1000+ problems, like pinter (as some people are suggesting), and you have to keep looking up stuff like how to verify a subgroup, how long is that book going to take you?

to me, memorization means to recall the important parts of something accurately and precisely so that they can be used fluently. that is not to say textually (rote memorization). i do not think it is enough to say "well, that's what learning is" because i doubt most people learn most things to that level.

It's funny that the article (http://augmentingcognition.com/ltm.html) displayed on polars feature page has this:

"I now believe memory of the basics is often the single largest barrier to understanding. If you have a system such as Anki for overcoming that barrier, then you will find it much, much easier to read into new fields."

Also I just made an account to say thank you - polar looks awesome. I just wanted to start using anki and stumbled upon it.

Khan Academy has multivariable calculus, differential equations and linear algebra --- with many exercises, solutions, and worked solutions (under the "hints" link after a problem), and gamification as a motivation aid.

The videos are above average, but seem about 5x too slow on the steps, then rushing through the interesting stuff almost too quick to catch: boredom punctuated by rewinding. Probably because it targets highschool kids.

BTW You may find you need to cover even earlier material anyway (I did), to refresh and fill gaps. You can cope with some gaps and hazy recall, until there are too many... and some gaps cannot be filled piecemeal (because you don't know you have a gap).

I don’t follow curricula online. But you should be able to get student and teacher copies of textbooks. Hold yourself firm to attempting the problems before looking at solutions to simulate homework. To simulate lecture, pick 1-2 of the tougher problems and work through the solution for them. Then try your homework out.

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

I have a Masters degree in mathematics and struggle so much to continue learning in my (relatively ample) free time. I was never good at self evaluating my proofs so often I solve a problem and have to scrutinize it for nontrivial steps made "intuitively".

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 :/

First I think you need to think about a direction. "Learn Math" is a very broad and vague statement. You will never learn all of math. So you really need to think about why you're wanting to learn it, and pick a direction that suits that.

If you're wanting to learn math just because you like the subject, then it's wise to get a sampling of a wide array of different areas. Then you have a better chance of finding an area that you like best.

If you're goal is to learn it so you can apply it in some other area, you should focus mostly on learning what math applies in that area. Then find the prerequisites and start studying.

Studying alone in your spare time will be nothing at all like attending a university. The full immersion you get to focusing primarily on one thing, and the ability to engage others at will also focusing on the same thing is a great help in learning. So set your expectations right. You won't be able to reach the level of understanding a 60 year old math professor has. Since he/she has been fully immersed in the topic for probably 40-50 years. So set your expectations accordingly.

There are some awesome resources on github on many topics including math. Just search for "github awesome-X" like https://github.com/rossant/awesome-math

And here is a root project for all resources https://github.com/sindresorhus/awesome

Personally I'd like to have something like a learning path. Not only a list of resources/topics but also some guide how to approach the learning process. O'Reilly has something like this but it's not mature yet https://www.safaribooksonline.com/learning-paths/ in my opinion.

MIT OCW has some guide for prequesities https://ocw.mit.edu/courses/mit-curriculum-guide/#map

I hope that helps you

I actually started from the very beginning of mathematics on Khan Academy, thats starting from pre-school going through every video and exercise, taking the quizzes and each end test for each subject. I am currently almost done with Algebra 1 and has taken longer than expected. I highly recommend! The exercises are key and do everyone until you get 100% then move on to the next topic/subject.

This is really great.

One specific point:

> One idea I had was to complete the MIT open courseware courses for the Applied and Pure math fields

A note about Open Courseware: they tend to be much of or all of the material handed out in a course (lecture notes, problem sets, etc). They aren't a "course" in the sense of Coursera, Khan Academy and the like.

A few years ago I needed to brush up on my thermodynamics and was able to read the material for the same course I'd taken as an undergraduate (well, same course number; I took thermo in 1983, and the prof and some of the material had changed). This was merely an undergraduate thermo class that I had already taken 30 years before and it was still quite hard.

I don't mean in any way to discourage you!!! This is an excellent idea. But OPenCourseware itself is more like a box of legos for someone who wants to teach a class in the subject. You may find a different source better, especially for topics that are new to you.

Ah very good points, thanks for the heads up! I’ll think about this a bit more

One thing that worked for me was to have a community of fellow mathematicians to study with. There's a lot of cultural stuff that is kind of hard to osmose from books, such as for example, how to pronounce things that you read.

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.

That is also part of the loosely defined term, “mathematical maturity.” When you’ve reached it, you can generally grasp the basics of unfamiliar math quickly. Failing that, you can find out what you need to study to understand it and do so on your own.

You don’t typically obtain that kind of maturity until graduate school, maybe upper undergrad if you’re quite good.

Yeah! There's that whole thing where you understand how a work is structured, you know what part to pay attention to, what part to skim or read later, where you're required to do your own calculation to fill in something...

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.

I disagree with the common advice that before you study math you should try to develop love for it or find an application you care about. If you don’t know any math, how can you know what you like or want?

What I think one should do, initially, is try to be better than other people in some way. That’s a motivation everyone has accessible. One way is what you’re doing already — trying to learn math on your own. That’s pretty cool.

From there I’d use Reddit and Quora to build yourself a curriculum. Most people start with calculus, linear algebra, differential equations, and real analysis. Look for textbooks that are known to be pedagogically well-written for self-study (rather than intended to be used as a manual in conjunction with lectures). Skim the table of contents and the first chapter of several books to get a feel for which ones you’ll be able to stand going through. I like Spivak, Arnol’d, Axler, and Pugh.

In each subject, find a book full of problems, and do the hardest ones in each section.

I started doing this, and the first stumbling block I hit was simply assessing at what level my math education falls apart and figuring out what courses I needed to refresh with, what materials I needed etc.

There's a market opportunity for building a math placement test product, or even better a fully automated math education platform with examinations at each stage, that takes you from K-12 refreshers all the way up to grad school. Something a bit more advanced than the existing online learning programs that are still very much based on the college lecture model.

There's a lot of products out there that do this for software development, but not for other subjects. I have noticed there are bits and pieces I can download as mobile apps, but they are designed to be supplements to certain college curriculum rather than an end-to-end education.

Obviously math is a subject where software is an optimal instructional tool, so much more could be done here

These concept maps might be helpful as a general overview of the basics: https://minireference.com/static/tutorials/conceptmap.pdf

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.

When you first start out in math you won't even know the basics like e.g. "proof by contrapositive" but a basic "proof course" perhaps accompanied by "Linear Algebra by Hoffman and Kunze" will get you pretty far.

You can never ever ever ever learn too much linear algebra. It is pivotal in the development of e.g. "field extensions" and "differential geometry" to name two random items.

That said, partway through Hoffman you should start an algebra book like Dummit and Foote so you can see groups and rings.

*After you know how to prove things you could read an analysis text instead of Linear Algebra

Lastly, any serious math requires some topology, but it's less about what a topology is and more about knowing how to quickly apply the basics so certain statements are easy to state/prove/think about

Lastly, the further you go the more you'll see that there are no islands in math. Everything uses everything else. (For example, you can go back and forth between groups and topological spaces via e.g. the fundamental group)

I think it is pretty much impossible to learn mathematics without a teacher beyond a certain point, multiple teachers are ideal. The reason for that is in order to do mathematics you need to solve problems and develop intuition. It will be very hard for you to solve problems (write proofs) without someone that evaluates the quality of your solutions, and it is very easy to deceive yourself that your reasoning is without gaps without experience. The same is true for intuition, without a master at whatever field you want to study it will be hard to work through most books (Hartshornes, Algebraic Geometry comes to mind. His seminars and lectures contained plenty of simple examples. The book itself is famously an uneasy compromise, which no one was really happy with but also with few better alternatives)

Math is a huge field and I'm not sure any can truly have both a deep and wide understanding of everything in math.

But if you want to give it a go, why not start with the tried and tested "Euclid's Elements" and a book on the history and philosophy of mathematics. If you are still interested, then tackle a course plan laid out by MIT or any other reputable university.

Also, I suggest getting an exercises/solutions book because math is doing. You can read the theorems, corollaries, etc and think you understand it but until you try to solve a problem yourself, you really won't know if you really understood the topic. And in your down time, you can check out math related channels/videos on youtube or elsewhere. It's going to take a lot grit and determination. Best of luck to you.

Just take community college courses. Work lots of problems. Ignore the grand theorems unless you're going to be a math major (in which case you should go to college).

Applied math; if you deal with matter it will generally be linear algebra and differential equations. Vector math important also, and Calc-3 is inadequate; junior level classical physics mechanics book is how I learned stuff like action angle and rotating frames of reference.

If you deal with electronics/signal processing you'll need some kind of Hilbert space course to get you through Greens functions, Laplace transforms and so on.

And for computer science/machine learning/OR: just be really good at linear algebra.

(Applied obv) linear algebra is the highest leverage thing you can do.

I don't at all recommend limiting yourself to self-study when you start this endeavor, it's too daunting given all the unknown unknowns. To start with it helps a lot to take courses, even if it means sitting in/enrolling in night courses at the local community college. It provides a reference point to what others understand, allows real-time querying with an instructor, and is more motivating than self-study. Once you have completed and fully grasped a difficult proof-based course (real-analysis, algebra, etc), then you can possibly embark on a successful self-study. I'm not saying it can't be done, but you will get to mathematical maturity faster this way and there will be less risk of early abandonment.

We need more info:

How much math do you currently have, which courses have you taken and how long ago did you take them?

Be a social human and find a local math prof or someone in industry to mentor you.

That aside: Haskell Road to Math and Programming. Concrete Mathematics. Pick your favorite computer graphics book to learn working with vectors and physics equations. From there chase your passions.

Just follow MIT Open Courseware for curriculum as for how to learn either the lectures or textbooks, lectures are good for initial understanding but then you'll need a textbook to go over the definitions,theorems and very important EXERCICES . you might want to take it easy don't try to put everything in some sort of timeline like (I am going to do all undergraduate in 1 year) that's really stupid as may be advertised on the internet. Your real goal ,as you're taking this endeavor ,is to understand it by making it your own, get a feel for it.I cannot describe it but you'll know . Best of luck in your quests .

Leaning math is primarily a solitary endeavor that takes time to learn properly. If you are an adult starting from basic algebra be prepared to spend at least 2 years of your life in order to get to where you need to be in order to begin physics. (Ie Allegra, function, trig, calculus) You have lots of options but assuming you don't want to go back to college these are 3 tools that are 100% free and as good if not better than any of the paid options.

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'd very strongly recommend https://en.wikipedia.org/wiki/How_to_Solve_It to familiarize yourself with some fundamental processes and https://www.amazon.co.uk/Playing-Infinity-Mathematical-Explo... before you lose yourself in a highly symbolic and abstract world, appreciate the beauty of it first, it'll make the long journey much easier.

I went back and learned a lot of math recently. I found the 3Blue1Brown youtube channel to be hands down the most useful for developing intuition.

I honestly don't think there is better mathematical content than his being made.

I know this is not what you're asking for, but I'd recommend stopping at the end of the sophomore level classes in the open courseware material.

I finished an undergrad degree in math from a school that had a large PHD program, and once I got to the junior level it was a lot of proving obscure facts about complicated objects (say, Lie Algebras) that have zero applications outside of very speculative theoretical physics. Proving theorems was fun sometimes, so if you're interested in math as entertainment, it's worth trying the more advanced material.

Here's a reading recommendation list I put together over the years: http://math.mit.edu/~notzeb/rec.html

Two caveats: first, it is obviously biased to the sort of math I find enjoyable, and second, many of the suggestions require the reader to have that elusive quality called "mathematical maturity". (I have no idea how one goes about gaining mathematical maturity, despite having gone through this process myself.)

For those like me who learned programming first and want to improve their math, but struggle with the crazy notation inconsistencies and the "pure math" approach to things in general, I highly recommend this book:

A Programmer's Introduction to Mathematics by Dr. Jeremy Kun

I've only just started it, but so far it does a great job of introducing math while assuming just the right amount of programming knowledge to make math approachable.

Is there a particular reason why you want to learn upper level mathematics? I have a pretty decent math background and I personally found majority of math boring (particularly linear algebra). The only reason why I did so well in Math is because I know I needed a strong math foundation to do the fun stuff (like fluid dynamics). Without this purpose, I would of passed my math courses in college but definitely would not of excelled in it.

I've been catching up with things I've missed, already forgot or never learned in school with Paul's Online Notes: http://tutorial.math.lamar.edu/

It's an incredible resource with great introductions to each topic and examples that are easy to follow and try for yourself.

Not sure if its great for your purposes, but I like the 'No Bullshit guide to Math and Physics' at least as a reference. It might be a good place to brush up on lower level stuff before advancing to harder topics.


Can anyone share some insight on if there has been any improvements on this since the last time it was posted here [1]? From that thread it appears that it has few misinformation there.


This might also help a similar thread was posted a while back

"Ask HN: How to self-learn math?" https://news.ycombinator.com/item?id=16562173

John Baez (physicist / category theorist / smort dude) has a list of books to learn math and physics. Is an interesting source to crib from. http://math.ucr.edu/home/baez/books.html

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.


Good advice here.

If you choose the project/problem based approach I recommend https://projecteuler.net

You will learn a ton solving these problems (the programming aspect is the trivial piece in most cases).

This is absolutely the right approach!

I'll add that, as a working faculty member, most people do not take naturally to writing proofs correctly. Even quite good students have to work with someone, at some length, before they get good instincts at it.

I have found that talking to mathematians is the best way to learn mathematics.

I think this is something like what you're asking for

How to Become a Pure Mathematician (or Statistician) http://hbpms.blogspot.com/

This might be a good roadmap for pure math, but don't use this for guidance on how to be a statistician.

Start small. I would look at Stat 110 from Harvard, Strang's OCW lectures, and the intro calculus MIT course.

When you're done with that, it'll be easy to see where you want to go next.

Use community college as a non-degree student to do the introductory undergrad classes. It's cheap, you are goal oriented and can get a lot of face time with your professor.

My $0.02: Make a goal to fully understand General Relativity as Einstein published it. Then work through the evolution of some of its mathematics.

Why set an arbitrary goal like that? Why not understanding Curry-Howard correspondence or Godel's Incompleteness or Hilbert's 17 instead.

That was my $0.02 as someone who majored in Physics. You just gave $0.04. Good suggestions.

Part I

I have some opinions on the question in the OP: I'm heavily self-taught from independent study in math; that study helped me with, and at times was part of my good career in, applied math and computing before my Ph.D. in pure/applied math and helped a lot for my Ph.D.

For a curriculum:

(1) Do the standard high school math, Algebra I, Plane Geometry (based on proofs), Algebra II, Trigonometry, and Solid Geometry (based on proofs). In each of these, it is sufficient (A) to work most of the more challenging exercises and (B) try to think a little about what is going on. For (C), it would be good occasionally, say, 1-4 times a year, to chat about your progress with a good mathematician, Ph.D., likely a college prof. To find a Ph.D. who will give you an hour or so per visit, maybe do some networking. If you don't like the first choice, e.g., if they are not encouraging and helpful, then try a different Ph.D. mathematician.

(2) Do Analytic Geometry and Calculus. Use a good college textbook. Since highly polished texts have been widely available for decades, there's no need to pay big bucks for new copies of the latest. So get a good used text. Better yet, get 1-4 good texts, use your favorite one as your primary source and the others for more content, alternate descriptions, evidence of what is more/less important, etc.

Sets: Now in an important sense, essentially all of math is heavily about sets. For first calculus, you don't need to know much about sets; the following should be nearly sufficient:

By 1900 or so, a lot of math was known, but there was a nagging, somewhat philosophical, question of what math really is. The answer of pure math was to construct a foundation in, say, the deep basement, that would answer the question but not much change the rest. The answer was to start with sets and then define everything else in terms of sets. In particular do some somewhat tricky and obscure work to define some sets that look, work, walk, talk, etc. like the numbers on the line, the real numbers. We don't think of the real numbers this way, but the effort addressed the philosophical question -- or, if I can get you to believe in sets, then I've got you for the rest of math. Much of the early work on sets was from G. Cantor. Yes, soon there were some issues from Bertrand Russell and then Kurt Gödel and, later, Paul Cohen.

For a long time in math, you can regard a set as a conceptual (imaginary) collection of some kind, e.g., all the peas in a little jar in the back of the refrigerator, all the people with an iPhone, all the real numbers on the line, all the water molecules in the earth's oceans, all the parabolas, all the trials you might do in a lab experiment, and much more -- really, from the philosophical approach to what math is for all of math. Or intuitively sets are the containers for what we are thinking about.

Functions: Much of math is about functions. E.g.,

     f(x) = 2x^2 + 3x - 4
So, in pure math, the function is f. The x is the argument of the function, i.e., a variable which is essentially always in math from,

     "Think of something; call it x."
where at least in calculus usually the thing are thinking about is a number. Or, to "solve for x", think

     "I suspect there is a number such
     that these conditions hold; I want to
     find if there is such a number and if
     so what its actual numerical value
Now you understand what math means when it talks about variable x and function f. Then, given variable (e.g., number) x and function f, f(x) is the value of f at x.

E.g., for this set stuff, a function gets defined in terms of sets, e.g., the set of all ordered pairs

     (x, f(x))
for x in some set called the domain of the function. In calculus, the domain is usually the set of real numbers, that is, the points on the number line (yes, that's not the tricky set theory definition of the real numbers).

So, with the ordered pairs

     (x, f(x))
we have a very precise definition of a function in terms of sets (ordered pairs can also be defined in terms of sets).

Of course, in actual work, especially in applied math, nearly no one actually thinks of a function as a set of ordered pairs.

Completeness: There is an old joke, partly appropriate, that

     "Calculus is the elementary
     consequences of the completeness
     property of the real number system.".
Intuitively completeness is, if you are converging to something by more and more accurate approximations, then there really is something there for you to converge to. This property does NOT hold for the rational numbers -- the rational numbers are all the numbers of the form (that can be written as) p/q for p and q integers (whole numbers, positive, negative, and zero) with q not zero. Why? E.g., can use the rational numbers to approximate as closely as we please the square root of 2, but the square root of 2 can't be a rational number. Why not? Because if the square root of 2 were rational, then we would also have for integers p and q, q not zero

     (p/q)(p/q) = 2

     (p)(p) = 2(q)(q)
So the left side has some even number (possibly 0) factors of 2 while the right side has an odd number of factors of 2, and this violates the fundamental theorem of arithmetic that each integer can be factored in only one way as a product of prime numbers.

Well, a big deal about the real numbers is that they are complete; i.e., intuitively, if something converges, then there is something to converge to. Completeness is a big deal because it generalizes, especially to Hilbert space that you may come to.

So, in calculus often we make approximations that can become as accurate as we please, and we want the completeness property to know that what these approximations approach, limits, can exist.

E.g., with the real numbers, if we make more and more accurate approximations to the square root of 2, then we know that our approximations will converge to the actual square root of 2 which DOES exist as a real number.

Sets get to be quite important in math in the last years of college math and beyond. But for calculus, nearly all of that subject, especially what learn in a first calculus course, was quite solid well before G. Cantor. So, in practice, when studying first calculus, we don't see much about sets. So, for first calculus, sets remain mostly a topic in the deep basement that addresses an old philosophical issue. So, for first calculus, sets are no big issue.

Analytic Geometry: Here you study the conic sections. They are a bit amazing and important in physics, mechanical engineering, and more and something definite to work with while learning calculus. So, imagine a cone, say, an ice cream cone. Take two of them that are the same and put the points together so that they are both on the same axis through the points (I'll let you get a precise definition of axis). Now imagine a sharp sword, say, as in the old John Belushi Samurai Tailor skit and use the sword to slice the cones and look at the cut edges. Right, we assume that the sword moves in a plane. Depending on where you slice, you will get (A) just a point (where the two cones touch together), (B) two lines crossed as in an X (the cross in the X is at the point where the two cones come together), (C) a circle (we are awash in circles), (D) a parabola (to a fairly good approximation, a grand slam baseball follows a parabola), (E) an ellipse (to a quite good approximation, each planet goes around the sun in an ellipse), (E) the two parts of a hyperbola (an electron shot at a negative charge will follow one of the halves of a hyperbola as it avoids the negative charge).

First Calculus in a Nutshell: Then for first calculus, there are two big topics, differentiation and integration. Differentiation is finding rate of change, intuitively the slope (as in high school algebra) of a tangent line to the graph of a curve (defined in terms of a function with domain some or all of the real numbers). E.g., for time t, let d(t) be the function that gives us our distance from home at time t. Then the derivative of function d at time t is the velocity, say, function v, at time t. So, in a car, the odometer gives d and the speedometer gives v. That is, at time t we have traveled distance d(t) and are moving a velocity v(t). Here to be simple, I do not make a careful distinction between what physics calls speed versus velocity.

What differentiation does, integration reverses, undoes. So, we can use integration on function v to recover function d. We have now introduced the fundamental theorem of calculus.

Right, v is the function, and v(t) is its value at time t, but in first calculus it is common to drop the distinction between v and v(t) and just say "the function v(t)" and mostly avoid mentioning v by itself.

Part II

Again, for variable t, think

     "There is a number; call it t".
That approach to the meaning of a variable works well enough for essentially all of math.

Calculus was invented mostly by I. Newton, mostly for physics, especially for explaining the motions of the planets. Since then calculus has become a pillar of civilization, especially Western Civilization.

Mostly in calculus, integration is finding the area (or more carefully defining the area) under a curve, maybe a parabola.

But can also have a line integral, say, the work do when carrying 100 pounds of hay to the top of the hay loft of the barn. Here we integrate from beginning to end of the work. If we let the hay fall, then neglecting friction, etc. get the work back as energy. So, have to pay attention to the direction of the integration, from the ground up to the hay loft or from the hay loft back to the ground. The two values have opposite signs.

You can teach yourself calculus: Just get 1-4, at least one good calculus texts and dig in -- read the chapters, follow the material, work nearly all the more difficult exercises, and maybe check 1-4 times a year with a good mathematician. I did that for freshman college calculus (but didn't check with anyone). I never took the course and, instead, started on sophomore calculus and made As. No problem.

(3) Linear Algebra. Likely the next course would be linear algebra. It would be good to do this subject 2-4 times from more elementary treatments to some of the more advanced material. Some of the more advanced material can cover linear programming optimization, classic two person game theory, the proof of the Nash result in game theory, some treatments of the fast Fourier transform in digital signal processing, group representations in the quantum mechanics of molecular spectroscopy, error correcting codes in algebraic coding theory, numerical methods, and more.

The start of linear algebra is just one or several linear equations in one or several variables. E.g., for two equations in three variables, unknowns:

     2x + 3y - 6z = 0

     5x - 2y + 7z = 3
For positive integers (whole numbers) m and n, given m such linear equations in n variables (unknowns), the number of solutions, depending also on the numerical constants in the equations, is none, one, or infinitely many.

The main way to find all the solutions is just Gauss elimination.

That covers a good chunk of a first course in linear algebra.

The notation

     2x + 3y - 6z = 0

     5x - 2y + 7z = 3
gets to be clumsy so we write instead

     2 + 3 - 6

     5 - 2 + 7
write that with big, square brackets, and call it a matrix with two rows and three columns. The individual numbers are components. Of course, a matrix is conceptually close to an array with two subscripts common in programming languages. Maybe we call the matrix

     2 + 3 - 6

     5 - 2 + 7
A. Since matrix A has 2 rows and 3 columns, we say that it is 2 x 3 ("2 by 3").

Then we write the x, y, z in a column



apply the big, square brackets, and call it a matrix with 3 rows and 1 column and maybe call it v. We write the right side


also with brackets, and maybe call it b.

Then we define a product, a matrix product Av so that

     Av = b
means just the same as

     2x + 3y - 6z = 0

     5x - 2y + 7z = 3
From then on we work with matrices and try to avoid notation like

     2x + 3y - 6z = 0

     5x - 2y + 7z = 3
In this way, a course in linear algebra is commonly called a course, Linear Algebra and Matrix Theory.

A matrix with just one column is a column vector; with just one row, a row vector. In short, either is called a vector.

Linearity: Given m x n matrices A and B, m x 1 matrices u, v, and real numbers c, d, we have that

     A(cu + dv) = cAu + dAv
Here we regard Au as case of function A with argument u. That is, we could have said that matrix A defines function f so that

     f(u) = Au

     cA + dA
we form products cA and dA and sums

     cA + dA
So, we have to define both of those: The definitions are close to just obvious.

The equation

     A(cu + dv) = cAu + dAv
means that matrix A acts like a linear function. Big deal! The real world is awash in linear behavior, and linearity is an enormously powerful property mathematically. The rest of linear algebra and matrix theory is nearly all about the consequences of such linearity.

The fundamental theorem of algebra is that each (high school style) polynomial in variable x can be uniquely factored into a product of terms of the form (ax + b) where the a and b are possibly complex numbers. Essentially from this fact, the more advanced parts of linear algebra use the complex numbers and not just the real numbers.

At the end of a second course in linear algebra, will pay attention to the essentially geometrical notions of length and angle and, in particular, to orthogonality, that is, perpendicular. Big topics are the Gram-Schmidt process where find some orthogonal vectors and the polar decomposition.

The polar decomposition result says that a matrix acting on a circle will yield an ellipse, that is, for matrix A and vector u, if we let u take on the values of all the points on a circle, then the Au will generate all the values of all the points on an ellipse. The ellipse has two axes, and they are mutually perpendicular. So, if have the vectors of the two axes, then have enough to construct the whole ellipse.

In linear algebra, this situation generalizes to any finite dimension and yields the singular value decomposition, principle components, factor analysis, and more.

With more advanced work, this situation becomes the spectral theory of self-adjoint linear operatiors central to quantum mechanics.

(4) The Four Main Parts of Math

There are four main parts of math:

(A) Foundations, that is, deep in the basement with set theory.

(B) Algebra, as in high school, the integers, prime numbers, the fundamental theorem of arithmetic, greatest common divisor and least common multiple, the fundamental theorem of algebra, linear algebra and matrix theory, the generalizations of number systems such as groups, rings, fields (the rational, real, and complex numbers are all fields but so is the set of integers modulo a prime number), etc. Fermat's last theorem, settled by A. Wiles, is part of algebra; so are the deep, difficult questions about prime numbers, etc. There are applications in error correcting codes and cryptography.

(C) Geometry, e.g., high school plane geometry, analytic geometry, and differential geometry, a deep subject important for relativity.

(D) Analysis, as in calculus, differential equations, functional analysis, e.g., Banach and Hilbert spaces, partial differential equations, e.g., as in Maxwell's equations, fluid flow, and relativity, probability, statistics, stochastic processes, optimization, and more.

In differential equations we are given an equation with the derivatives of some function and want to find the function.

E.g., for real valued function y(t) and constants k and b, we might have

     y'(t) = k y(t) (b - y(t))
where y'(t) = d/dt y(t), the first derivative of function y(t). At one time, that little differential equation saved FedEx from going out of business. The solution is a lazy S curve and a first cut at viral growth.

There is topology which is partly in geometry and analysis. The main idea of topology is continuity, that is, changing without sudden jumps or some cases of wildly fast oscillations.

There is algebraic geometry partly in algebra and geometry.

Much of number theory has deep connections with analysis.

Much of linear algebra is an introduction to functional analysis in analysis.

The older applied math is mostly analysis.

(5) Analysis. Usually after linear algebra will study advanced calculus and analysis. Broadly there are two approaches, (A) theory and (B) applications.

The theory is mostly to give fully careful proofs of the results and first generalizations of what you saw in calculus. For the theory maybe the most respected text is

R. Rudin, Principles of Mathematical Analysis, Third Edition.

For this, get the third edition and not either of the two earlier editions.

So, will discover that the integral learned in calculus is called the Riemann integral because B. Riemann made the theorems solid. Rudin also does the easy generalization to the Riemann-Stieltjes integral.

Part III

Rudin has a nice chapter on Fourier series, that is, representing a periodic function f(t) with a sum of e^(iwt) for overtones at frequencies w and where each frequency w is a whole number multiple of a fundamental frequency. The linear algebra and geometry here are that the e^(iwt) are perpendicular projections of the f(t). So, the e^(iwt) are orthogonal axes.

The line integral generalizes to the exterior calculus of differential forms (keep track of signs from the direction do the integrations, i.e., as in line integrals) and the fundamental theorem of calculus generalizes to Stokes theorem crucial in Maxwell's equations, fluid flow, and partial differential equations. Now we are close to differential geometry, and Rudin also does the inverse and implicit function theorems important in differential geometry and parts of nonlinear optimization, e.g., Lagrange multipliers.

I warmly suggest that take a really fun, one weekend, pass through Stokes theorem parts of

Tom M. Apostol, Mathematical Analysis: A Modern Approach to Advanced Calculus, Addison-Wesley, Reading, Massachusetts, 1957.

Get it used, and pay whatever you have to. This way get to see the 2-3 dimensional versions and, really, mostly enough for what physics and engineering do with vector analysis and Stokes theorem.

In H. Royden, Real Analysis and the first half of W. Rudin, Real and Complex Analysis, can see the Lebesgue integral, due to H. Lebesgue near 1900. In essentially all cases where the Riemann integral is defined, the Lebesgue integral is also defined and gives the same numerical value. But the Lebesgue integral has more powerful theorems and is defined in more general situations.

As in a 1933 paper of A. Kolmogorov, the Lebesgue integral gives a solid foundation to probability, statistics, and stochastic processes.

In W. Rudin, Functional Analysis get a treatment of distributions that cleans up what physics tries to do with the Dirac delta function and also covers spectral theory.

From there, can go for a Ph.D. For that will need mostly (A) pass the qualifying exams and (B) do some original research. The standards are commonly something like "an original contribution to knowledge worthy of publication" and for publication, "new, correct, and significant".

For the qualifying exams, first pick a department, pure/applied math or some math area in engineering, optimization, probability, economics, computer science, etc., get their description of their qualifying exam topics and references, study, and take and pass the exams. For the research, do some.

One suggestion: For the research, pick a real problem and use some math, at least in part new, to get a good solution. Can get "significant" from the importance of the real problem. Can get much of "new" if have the first or better attack on the real problem. Can get "correct" if do the math with careful theorems and proofs. Can publish if pick an appropriate journal. Might get to regard the math as significant based on what it does for the real problem and not just its contribution to pure math.

Even the best US grad schools are hungry for good students. With a good ugrad math major and good work on independent study, should get a good reception at grad schools and at least a tuition scholarship. In that way I got accepted to grad school at math departments at Cornell, Brown, Princeton, and more.

In a sense, independent study is recommended: (A) That is basically what research profs have to do for all their careers. (B) At least at one time the math department at Princeton stated that no courses were offered for preparation for the qualifying exams, students were expected to prepare for the exams on their own, courses were introductions to research by experts in their fields, and students should have some research underway in their first year.

For a little on how to do the original parts of math research, there is the A. Wiles comment:

"Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. You go into the first room and it's dark, completely dark. You stumble around, bumping into the furniture. Gradually, you learn where each piece of furniture is. And finally, after six months or so, you find the light switch and turn it on. Suddenly it's all illuminated and you can see exactly where you were. Then you go into the next dark room ...."

That's some of how to learn some math, largely with independent study, and maybe to get a Ph.D. But you might want more: You might want your Ph.D. work to get you a good start on a good career in pure/applied math research/applications. For that, look around, pick up what you can in seminars and conferences, and get what your profs can explain to you.

Broad point: Long the best opportunities in applied math, even with some advanced pure math prerequisites, have been in US national security, especially within 100 miles of the Washington Monument.

I'm working on this myself and while I'm not very far along, I can share my approach.

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.

Get Thomas 13th or 14th edition and solve all the odd problems. That takes care of calculus.

I need this too, thanks for asking what I was too afraid to ask

Check brilliant.org. Got everything there, worth the money.

What is the purpose of a proof? You may think that it's intended to solve a problem, but that's really only half the point. A good proof is one that communicates your solution to other people.

Mathematical writing is hard to learn well under the best of circumstances, but if you don't have someone else giving you feedback on whether they understand what you're writing, it's absolutely impossible. You need to have a mentor or at least an editor at some point. That's not impossible to find outside of the university system, but it's very difficult.

(This is the single biggest reason why MOOCs for higher math haven't taken off. There are a lot of people who'd love to communicate something about the field that they've dedicated their lives to, but the feedback system just doesn't scale. If anyone can figure out how to fix that, it'll be a game changer.)

So the first thing you have to do is to figure out what you can reasonably expect to get out of this process. You can learn the definitions and theorems of higher math, and that might be enough, but you're never going to develop an intuition for them without understanding how to produce proofs on your own. And don't fool yourself into thinking that you can evaluate your own proofs. It just doesn't work.

If all of that doesn't have you turned off, then here are some ideas on what to do.

A university level math curriculum is split into roughly three components: * Lower level classes that focus on basic definitions and calculations; * Mid level classes that teach some basic theorem-proving skills in subjects that are useful for people in other quantitative fields; * Upper level classes that offer serious practice in theorem-proving as well as the core ideas of mathematics. You don't generally have to do classes in any particular order, but you do have to master the skills of each level before you go on to the next one.

To begin, you must be very comfortable with the contents of a high school math curriculum. Serge Lang's book on basic mathematics is a great refresher if you're not, or you can use any of the various popular study guides (Schaum's, Barron's, etc.).

At the first level, you have calculus. This is generally split into three semesters, with the first dedicated to limits and derivatives of single-variable functions, one dedicated to integrals of single-variable functions as well as sequences and series, and the last dedicated to derivatives and integrals of multivariate functions. There are plenty of very expensive books with glossy page and many color pictures and few ideas, but if you want a serious introduction, look at Peter Lax's books on calculus.

At the second level you'll almost always find introductions to differential equations and linear algebra. Differential equations have historically been the workhorse of applied mathematics and you really need to have some familiarity with them, but I've never seen a book on the topic that I liked. I probably won't be satisfied by anything at this level, though, so look around and see if you can at least find something inexpensive.

Linear algebra is a more recent topic (with many of its key ideas actually originating in the 20th century), but it's probably actually more important now. Gilbert Strang's books are popular and are worth reading for a first look, but you really can't regard them as a serious introduction to the mathematical side of the topic. Axler is probably the best book in that regard, but it's best taken on a second pass.

I think that probability should be regarded as a core class at this level. I don't think that's a fringe view, but it's not as universal as I'd like. I learned from Pitman's book, and I think it's as good as any to start with.

You can also take classes on complex variables or "discrete math" here. I don't know what a good textbook for complex variables is--maybe Saff & Snider?--but I'm sure there are recommendations out there. Needham's "Visual Complex Analysis" is a fantastic book, but maybe not really suitable for a very first introduction. As for "discrete math" (a jumble of topics from logic, combinatorics and number theory), find the cheapest book you can get that has decent reviews on Amazon.

At the third level, there are three main topics: analysis, algebra and topology/geometry. You can think of these as the three main viewpoints in higher math, and other topics being populated by people who primarily look at things with the tools of one of those three topics.

Analysis starts out as the theory behind calculus. In a first course, you'll revisit a lot of what you saw in single-variable calculus, but you'll learn why it's true rather than just how to use it. For a single semester undergraduate course, Ken Binmore's book is probably the gentlest introduction.

Modern geometry is related to what you studied in high school, but with a few more centuries of development. It also doesn't get a lot of coverage at the undergraduate level, which is highly unfortunate. Stillwell's "The Four Pillars of Geometry" is a wonderful book and completely accessible.

Algebra is a bit difficult to explain without getting into the weeds. Pinter's "A Book of Abstract Algebra" is very good at motivating the topic and explaining the basics, which is the best you can hope for in an introductory textbook.

Beyond that but still at the undergraduate level, you can get electives in combinatorics (use Brualdi), number theory (?), logic (?) and some applied topics as well. Looking through the course offerings of various math departments will help you to fill in what the other possibilities are.

Check out mathvault.ca.

You first need to define your boundaries and limits. Don't get me wrong, but mathematics is grinding, and if you think you have an idea where you're getting yourself into because you've studied calculus and algebra as an undergrad CS, you will fail miserably. Don't make the same mistake I did :)

Undergrad math is tooling; I like to use the analogy that it's like entering a workshop where you have all the tools available, but you're blindfolded, and you have no idea what each stuff is used for. So you'll have to touch everything. Look down for the tooling advice.

When venturing in Mathematics, for your own sanity, please have an objective in mind. I'm down serious! Understand what you want to research and go in that direction.

One last piece, Find a mentor, share, talk to people. You won't advance from yourself.

As a piece of tooling advice, I have the following I've stolen from Reddit a few years ago (sorry I haven't found the source to it) __ TOOLING __ First of all, most important, GO LEARN ALGEBRA. Seriously, I know you think its bullshit but its the most basic skill in some ways that any mathematician should know. Second learn Calculus: Single and Multivariable. If you are still interested here are some things to go onto next:

Discrete mathematics: This includes equivalence relations (probably one of the most important things for you understand ever), propositional calculus (logic) proof techniques (induction) and some basic combinatorics (Pigeonhole principle). You can literally find any textbook and start reading. The theory is kinda a hodgepodge, but those are the major themes.

Linear Algebra: Again, one of the most important subjects you will ever study. Once you understand this, you are really on your way, and this stuff comes up everywhere. Many mathematicians have said many of the biggest proofs in the world come down to "just some linear algebra". The major point here is to understand that there is only one vector space for each dimension over a field and understand how a linear transformation becomes a matrix only after a choice of basis. Here equivalence relations come up again!

Differential equations: Unless you're focused on engineering math or serious applied stuff, don't worry too much about this. Seriously, it's not that integral (haha get it!).

Complex Analysis: Yes, mathematicians and Engr. Actually, do study "imaginary" numbers, but there is nothing imaginary here. This is serious stuff, do it.

Okay, so now you're about as a sophomore/junior level place in mathematics. How to finish it off? It's not that unclear:

Abstract Algebra -- Grab any book read about groups, rings, fields, vector spaces, and modules. Proofs will be difficult here but work through it. There are so many books here, avoid Lang (good book but not for starting out), Dummit/Foote is okay. As a undergrad I had a good time with Rotman's An introduction to abstract algebra.

Analysis -- Grab Baby Rudin. No seriously, Grab this book, sit in a room for a semester and just fuckn' read it. You will basically be "redoing" calculus. This is a trial by fire, go!

Topology -- Grab Introduction to Topology by J. Munkries. Its so well written it might as well be a coffee table book.

There now, you have done everything a math major would. Yes, there are lots of things that are missing, arguably the most important things depending on what your goals are. Typically one studies Number Theory along with Abstract Algebra, or studies Analysis and Differential Equations together or Analysis and Topology. Seeing the links across different topics is essential, but I'm just giving the overview here.

Not every mathematician studies logic, and there are LOT of fringe topics that I'm omitting (including some of my favs: Projective Geometry, Varieties, Lattice/Order theory, Combinatorics, Elliptic Curves, Coding theory, Harmonic Analysis, etc.). However, none of these are required courses at more than a 1% of programs

Go to the library and read their math books. I don’t know why the internet is filled with these questions that are always “how do I start doing x?” Just f* do it dude. Nobody is going to be able to or bother to answer anyway. The answers wouldn’t make deep sense to you because they rely on having done it before. Once you’ve started and you’re really stuck and everything you read doesn’t seem to illuminate it, post a question. People will notice you’ve put some work and time in and will be eager to help you.


You want to learn more about both pure and applied mathematics. Pure math is very proof heavy which is excellent for self learning because you never have to take anything on faith. Proofs are meant to be very clear explanations.

One of the best resources for learning in this way is to find books that are structured to have you develop the theory via the problem sets. Of the top of my head, Tao's Analysis books and Atiyah & MacDonald's Commutative Algebra book are exemplars of this style. These books often show the basic definitions, some results, and then break up remaining results into step by step problems.

Proofs will be the hardest part to wrap your head around. Once you understand proofs, can read them, the remaining is going to be a question of time. Think of it like programming: you can't learn by reading, lectures, etc. You need to learn by doing. In this case, the doing is doing proofs.

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