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!
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.
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.
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 :)
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.
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.
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.
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.
>>> 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."
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?
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.
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.
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.
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.