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.

He had a quiet programming position in a large company that gave him time to work on personal projects. Among other things he volunteered at the local schools teaching "fun" math. I guess I'm just agreeing with you about finding something that inspires you and go with it.

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

He didn't care about the status of having a PhD

He didn't need a PhD to advance his career

He was having fun doing math stuff anyway

While I appreciate your brother's personal choice, to each their own after all. There is quite a lot of merit in your PhD advisor helping you in choosing a problem. That being said, good advisors provide students with an array of good problems out of which the student can choose one they are the most passionate about. This is what happened with me, I was provided with around 7 different choices to make. In the end, I chose 2 of them even though I wanted to chose 3 more but couldn't because of lack of time.

Links:

http://content.algebraicgeometry.nl/2017-2/2017-2-007.pdf

https://projecteuclid.org/download/pdfview_1/euclid.ecp/1508...

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

https://web.archive.org/web/20170121000748/https:/wwwmath.un...

https://www2.math.binghamton.edu/p/seminars/arit/arit_spring...

https://www2.math.binghamton.edu/p/seminars/arit/arit_fall20...

https://www2.math.binghamton.edu/p/seminars/arit/arit_fall20...

As an aside, your work on numerical cohomology appears to have been useful for a new result pertaining to lattices. Given the authors of the followup work it's likely helpful for the study of lattices in post-quantum cryptography.

There are plenty of people who devote several years of their life to studying, and must pay not only their living expenses but tuition fees as well. In my opinion, those are the people who you should be asking “how did you manage this”.

I 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.

Of course this is in the context of choosing research problems to strive towards in math. If you tasked yourself with solving an open problem in math, it's more likely than not that, without any collaboration, you'd have no idea how to even work towards the goal due to all the unknown unknowns. If your goal is something concrete that can be augmented with mathematics, then yes I agree that goal setting can be useful. It doesn't take a volume of missing domain knowledge to develop that kind of goal.

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

> everything else that’s worth learning: hard!

I disagree. I’ve learned many things worth learning that are not hard at all, but to each their own.

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

I 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.

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).

I personally did work myself up to the graduate in math during the last two years of High School & first year of college. I was motivated by exploring ideas and gaining knowledge. I would guess that each person has a somewhat unique motivation strategy. Maybe solving problems gets some people doing stuff. I'm sure simply learning stuff can motivate others. Probably each person has to experiment to discover what works for them - I would pick up a calculus book and read it - well, I'd skim repeatedly and then read in depth, solving a few problems. Math is difficult, of course, so having a bit of patience with your until it gets the ideas on it's own is probably necessary.

...how in the world did you manage to do this? Did you actually self-study, or were you placed in a gifted program? Self-studying all of undergraduate mathematics is more impressive than actually studying all of it in a four year classroom setting. Doing so as a teenager is amazing.

The most gifted person I've ever known in mathematics is actually a physicist who entered Harvard at 16. He was already taking undergraduate math courses at 13/14 and had mastered all the undergraduate material by the first semester of his undergraduate degree. For the remaining years of his undergraduate degree he took graduate math courses.

But in order to do that he needed to not only be gifted; he was placed in a program for extremely gifted kids sponsored by Stanford University from his preteen years. I don't think your anecdote is a great comparison for the OP's expectations (or for calibrating advice they'd benefit from).

I read quite a bit on my own, I took some courses at UCLA through a high school scholars program (including the undergraduate honors seminar). The entirety of the undergraduate program might be a slight exaggeration but I was ready for graduate level courses when I got to Berkeley.

I think going through the material requires determination, not necessarily being extremely gifted. But then, it seems like people at someone's gone through a bunch material say by that fact they're gifted. Thus having done this, one is tautologically gifted.

I've never tested at the extremely gifted level but I'm doubtful of single-measures of intelligence regardless.

Edit:

I don't think your anecdote is a great comparison for the OP's expectations (or for calibrating advice they'd benefit from).

Neither of us know the OP. It's kind of up to them to calibrate what process works for them. Scanning a lot of math until I found good, clear explanations worked well for me.

Indeed, speaking only for myself, I don't think that I had the self discipline to carry through with such a plan as a teenager. I needed the classroom environment to learn math & physics, which became my college majors. At the same time, I was able to teach myself programming and electronics quite easily for some mysterious reason. And I ended up combining all of those things in grad school.

Also, I only had access to the books and resources of a typical suburb with a decidedly anti-intellectual culture. The officials at my high school refused to offer a calculus course because they said it would be "elitist."

For example, (since I don't really have much time)

1. Topology (book by Munkres)

2. Real Analysis and Measure Theory (book series by Stein Shakarchi)

3. Algebra (book by Aluffi)

4. Linear Algebra (book by Friedberg Insel)

5. Measure Theoretic Probability (book by Cinlar)

6. Differential Geometry (book Smooth Manifolds by Lee)

7. Numerical Analysis (book by Quarteroni)

8. Set Theory and Propositional Logic (books by Goldrei)

This is what one will mainly learn in a strong undergrad/grad math program. Once this is done, then there are different tracks to follow.

I was questioning why the OP wants to self-study an undergrad math curriculum to begin with.

It's probably not to become a pure mathematician. So I suggested, instead of creating a massive goal of getting through a collection of books just for the sake of being a completionist, to have a concrete personal goal. Otherwise, people can throw books "you have to read" at you until the cows come home. Especially since this person is talking about applied math.

But your advice has a point, just going through books mindlessly is not motivation enough/ can lead to wandering. And it is always good to have specific tasks at hand. Like, solving a particular ordinary differential equations numerically.

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

1. Find a thing you want to make. 2. Find out how to make it. 3. Try to make it. 4. Learn the skills that previously prevented you from making it.

Unlike programming, mathematics isn’t something you can pick up in a weekend.

EDIT: Now that I think about it, you can probably bang your head against, say, Lang, for two years, “digesting” a big chunk of it, earning you the bragging right of “working yourself up to graduate level math”. That won’t give you the breadth of a good bachelor of mathematics, and it certainly won’t prepare you for quality research.

> be able to write up a compelling

Can someone clarify what this means?

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.

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

Basic arithmetic: addition, subtraction, multiplication, division.

Some of the engineers who attract quantitative work are people who came from outside of the mainstream engineering training, such as scientists and math people.

When you're designing a real-world engineering project, the entire specifications are defined legally (through national, state and local laws) and technically in manuals/books. Many engineering specifications will describe the work done to a T before you even need to think about it i.e. "water main shall be constructed of 12'' coated DIP at depth no less than 2 ft". A lot of the challenges are managerial and logistical.

All of this is on purpose. "Traditional" engineering disciplines are more mature and have the constraint of being safe for the general public. There isn't much room at all to creatively deviate from what's already specified.

I've found software design to be a lot more technically demanding in regards to designing and building things. There's a lot less precedent, more moving parts and many different ways to do one thing.

And I'm not blaming anybody -- for one thing college math is often badly taught, and there's a pervasive message that you won't use any of your math or theory after you finish your degree. And then we get them so busy with CAD and bureaucracy, that they forget a lot of their school stuff.

But anything requiring calculus or above, goes to a handful of "math people" in the department, who accept those tasks in return for avoiding the CAD and organization stuff. (I'm one of those people at my workplace, my degree is in physics).

To make it a bit harder, virtually all math these days is done with computation, which means a person has to be good at both math and programming at some level.

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.

A Book of Abstract Algebra by Pinter (really gentle)

or

A First Course in Abstract Algebra by Fraleigh

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.

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.

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.

https://en.wikipedia.org/wiki/Olympia_Academy

I have created a group here if someone is interested to join: https://groups.google.com/d/forum/projectfermat (If you think that there is a better place to have a forum like this that anyone can easily view or participate in, please let us know. I mean we could also create an IRC channel, Slack workspace, etc. but there should be one main starting point and a mailing list/web forum like this seems like a good place for that.)

I am thinking we could also host a web meeting to present, discuss, or share interesting topics and problems regularly. We can form our own mathematics discussion community here.

I have been doing this kind of thing at my workplace as well as outside work and it has been an incredible source of learning. I believe something like this for the Hacker News community would be very helpful and we can learn a lot of mathematics from each other if we can interact with each other on a more topic-focused forum.

There are 12 members in the group so far which I believe is a good size to kick-start the discussions and other activities. More members are welcome!

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

Then I thought, why can't I just buy the time of someone competent? There are many students who tutor high school kids, shouldn't students who can teach first/second-year material for pay exist?

I wonder, have anyone tried this? To me, it looks like a perfect solution and I can't think of any downsides. I'm not sure how to approach this exactly - where to find such a tutor in the first place - but it should be possible, right?

I would use such private lessons for getting a summary/overview of the subject first, then after some self-study, I would ask about whatever I couldn't comprehend. It would have minimal impact on my schedule, shouldn't cost that much, and should be quite efficient.

Well, it's just an idea I had some time back, I didn't try it in reality yet, but I think I'll try going this way in the future. I'd like to ask what do you think, is it possible, could that really work out?

One way to find them is to contact your nearest research university’s department of math/physics/whatever and they can help match you with someone, or just post a flyer in their department office.

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.”

[1]https://www.quantamagazine.org/peter-scholze-and-the-future-...

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.

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.

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.

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.

I haven't won any awards yet.

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.

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)

If you get stuck you could try asking on https://math.stackexchange.com/

Although, if you can, finding someone else to work with, and someone else who already knows some math to ask occasional questions would probably help you a lot.

http://www-history.mcs.st-and.ac.uk/

I would leave for work an hour early and either sit in my car or go into a Starbucks and do math. Doing time before work is important. That's when you are at your best. Then after work I would sit in my car and do math for an hour. Then on the weekends, in the morning, I would do three hours of math straight, sitting in my car. Then sometimes in the evening I would do a first pass over a section that I knew I would have to really think hard about the next day that way I had sort of taken the first layer of difficulty off.

Why in my car? Because I simply cannot sit at home and focus enough to study math. I've tried the library but it's too restrictive on what you're allowed to do "oh f*! that's how you do that" doesn't generally go over well in public places. Plus I feel very safe in my car. I can be relax which makes learning a lot easier.

I always take one day off a week (for me that's Tuesday) but aside from that I don't skip.

Get your books at half price books. Math book reviews on Amazon are almost always wrong. I usually have to try two or four books before I find one I can understand. I would just get a book and read it in your car. You don't need the pressure of following along with a schedule where you fall behind and miss out on a topic. You also don't want to miss a topic just because the teacher chose not to include it. Some of the coolest topics/examples don't get covered when you take a course because they don't have the time.

Also, when you get done with a book go through it again and take good notes on note cards that you can review on your drive to work or while you are waiting around. You'll find yourself going back to them over and over again.

I'm trying to figure out where I went wrong with my life by retracing my steps. Starting with when we met in freshman calculus.

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:

https://www.amazon.com/How-Read-Proofs-Introduction-Mathemat...

https://www.amazon.com/How-Think-Like-Mathematician-Undergra...

https://press.princeton.edu/titles/669.html

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

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).

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?

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.

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.

You will not improve your math watching youtube videos and reading books. You need to produce stuff that pass the "sniff test" to your colleagues.

That depends on exactly what you mean by "become good at math". If you're talking about "becoming a mathematician" and doing original research in pure math, then you're probably right. But if one means "learning existing math well enough to apply it to a problem", I would argue that one can learn this stuff just using books, videos, etc. At least up through a certain level.

That said, I do encourage the idea of finding peers to work with. I used to coordinate a "math study night" at the local hackerspace for that exact purpose. It fell off because I got busy and couldn't keep committing to it, but generally speaking, it is a good idea to have other people to work with. I may well try to find a math major from UNC to hire as a tutor at some point as I keep working on this stuff.

And if one can't find somebody to work with in person though, and they need, say, a proof evaluated, there is the option of using math.stackexchange.com or the like.

Sure, I'm just saying that it kinda depends on what part of math one is referring to. I think sometimes in these discussions on HN, we overload the term "math" to mean both "calculation" or "applied math", and "pure math" or "math research" and it can be unclear which is being referred to in a given statement.

I believe you can learn the former - "applied math" - (at least up to a certain level) just by reading books, and watching videos (and doing exercises, of course). But for the latter - "pure math" - I agree that you need other people, since you can't easily verify your own proofs.

It you don’t have somebody challenging you and checking your work, you will likely plateau very quickly unless you are especially gifted.

You will Hit walls of concrete and walls of glass.

You will misunderstand concepts and not notice it.

Your proofs will have logical gaps and you won’t notice it.

A couple resources I would recommend would be:

* A book called "Who Is Fourier? A Mathematical Adventure", which touches on a pretty good variety of math topics. It's aimed at children, but it's probably the best math book I've ever seen. It's the only math book I've ever read cover-to-cover.

* 3Blue1Brown's youtube channel: https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw (as well as other math videos)

His videos, I think, are best consumed twice: once before you explore a topic, so that you go in with a solid overview and know the key things to look out for; and again after you've studied in depth, when you are able to predict what he will say next, which reinforces you learning.

I don't think you need a PhD level educator. You need mathematical maturity. Mathematics follows a very specific logical structure that needs you to shift the way you think. Human brain simply doesn't work the way mathematics needs it to work. But this is a constant time overhead. Once you understand how to approach mathematical problems, I can't see why you cannot learn everything from a textbook.

For me personally I didn't really need an instructor for most of my calculus courses, or ordinary differential equations, or most of the linear algebra stuff. It was a bit more difficult around real/complex analysis, non-linear dynamics, and courses of that nature. The classes that taught me the value of having an instructor were abstract algebra and topology. Those were such a massive shift away from what I had perceived math to be that an instructor being able to impart intuition, correct my own incomplete or incorrect assumptions, and generally just help guide me to a different mode of thinking was invaluable.

The problem with books/texts in this instance is they are not reactive, they have no idea what you're thinking and can't steer you in the right direction. Worse is that as the person trying to learn the subject matter you don't know where to look to get on the right track and correct your own assumptions because you don't know enough yet.

Now I'm not saying you need an instructor per se, but having some place to ask questions where someone far more knowledgeable than you can help might be a good substitute. I'm sure there are some websites like this, although I don't know of any since I graduated a long time ago.

[1]This does somewhat depend on a person's skillset going into this.

Of course you can learn a ton on your own by reading and working exercises and doing research, but there is no substitute for collaboration.

Mathematics is inherently a social activity, even if the bulk of it can (counter-intuitively) be done in relative solitude.

I eventually did get to university and get a Math degree. It was much more rewarding and fun to do it with professors and other students.

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

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!

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 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.

Shaum's Outlines has lots of cheap math books with exercises and solutions. You can use them to complement other books

https://www.mhprofessional.com/9780071623667-usa-schaums-out...

https://www.mhprofessional.com/9780071635349-usa-schaums-300...

https://www.mhprofessional.com/9780071383929-usa-schaums-out...

https://www.mhprofessional.com/science-math/math/topology

https://www.cambridge.org/core/books/course-in-modern-mathem...

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

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.

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/

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.

https://www.theatlantic.com/education/archive/2014/03/5-year...

Math is not linear, So why do we teach math in hierarchical steps? - https://prezi.com/aww2hjfyil0u/math-is-not-linear/

http://ocw.mit.edu/resources/res-18-001-calculus-online-text... highly recommended for Calculus

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 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:

http://www.people.fas.harvard.edu/%7Edjmorin/book.html

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.

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.

https://getpolarized.io/

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.

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...)

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.

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.

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.

"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.

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).

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

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 :/

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.

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

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.

Mathematical notation is meant to be read out loud. It's shorthand for English, or whatever your natural language is. You should be comfortable seeing a sigma sign and thinking in your head, "the sum as ehn goes from one to infinity of eff of ecks squared times ..."

That's just the basic part of it. There's lots of cultural shorthands that are just kind of osmosed, seldom overtly acknowledged (for example, the precise meaning of "without loss of generality"). Maybe you can acquire them by watching videos, but if you can find people that you can hang out with that can teach you the cultural aspects of mathematics, that would help a lot. Picking up a book and reading and working through it and understanding it gets a lot easier once you have the culture down.

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

People complain a lot that books and presentations don't do everything and that mathematics has a bad "user experience". They may have a point, but the complaining alone won't fix it and will leave most people still feeling like frustrated outsiders.

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.

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

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.

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

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

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.

1 Every day on this site https://www.khanacademy.org

2 Use this for graphing desmos.com/calculator

3 Use this to supplement (ie free text) myopenmath.com

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

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.

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.)

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.

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

https://minireference.com/

[1]https://news.ycombinator.com/item?id=4994367

Gerhard t'Hooft has a page on how to become of a good physicist. http://www.goodtheorist.science/ Not math though.

I'm not necessarily suggesting that a mountain of books is the best way to go about it. I think I respond best to video lectures. I'm not sure where you're starting from or how applied/pure you want. Here's a quick mind dump some of my favorites and some that I haven't watched. Roughly in order of how much I liked them.

Gilbert Strang's Computational Methods for engineers was life changing for me. It is a two part MIT opencourse. https://ocw.mit.edu/courses/mathematics/18-085-computational...

A Stanford course on the Fourier transform https://see.stanford.edu/Course/EE261

Bartosz Milewski's Category theory for programmer's https://www.youtube.com/watch?v=I8LbkfSSR58

Stephen Boyd's courses are online. http://web.stanford.edu/~boyd/ Linear Systems, convex optimization. Useful stuff.

Francis Su's Real Analysis is very good https://www.youtube.com/watch?v=sqEyWLGvvdw

Indian universities have an astounding collection of videos https://nptel.ac.in/ I have a tough time with the accents, which is a bummer.

UCCS MathOnline has quite a haul https://www.uccs.edu/math/vidarchive

I've been enjoying this Visual Group Theory course lately https://www.youtube.com/playlist?list=PLwV-9DG53NDxU337smpTw...

Math Doctor Bob https://www.youtube.com/user/MathDoctorBob/playlists

Wildberger has some interesting takes on elementary and non elementary topics https://www.youtube.com/user/njwildberger

https://www.perimeterinstitute.ca/training/perimeter-scholar... Perimeter scholars lectures. Physics not math. Good stuff.

Federico Ardila has a number of combinatorics courses. https://www.youtube.com/channel/UCWwECTsgjp_S-c73pO2c4gQ

Nonlinear algebra course https://www.youtube.com/playlist?list=PLRy_Pn1LtSpejKLClqbrW...

Also of course there is Coursera and edX stuff.

Godspeed.

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).

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

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

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

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

     "Think of something; call it x."

     "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
     is."

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))

So, with the ordered pairs

     (x, f(x))

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.".

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

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

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.

Again, for variable t, think

     "There is a number; call it t".

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

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

     2 + 3 - 6

     5 - 2 + 7

     2 + 3 - 6

     5 - 2 + 7

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

     x

     y

     z

     0

     3

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

     Av = b

     2x + 3y - 6z = 0

     5x - 2y + 7z = 3

     2x + 3y - 6z = 0

     5x - 2y + 7z = 3

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

     f(u) = Au

     cA + dA

     cA + dA

The equation

     A(cu + dv) = cAu + dAv

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))

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.

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.

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.

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.

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

https://www.mathcamp.org/2015/abel/abel.pdf

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.

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.

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.