Self-Taught Mathematician; an Impossibility?(medium.com)
medium.com
Self-Taught Mathematician; an Impossibility?
https://medium.com/@artagnon/self-taught-mathematician-an-impossibility-67d9b6893710
69 comments
I agree that it's necessary to do exercises in order to understand the subject, but I disagree that you need to have them checked by someone else -- in many cases, you should be check it for yourself, honestly.
Beyond the junior undergraduate level, computation-style problems go away, and you're just left with exercises that ask you to prove a certain something. It takes a long time to build up a sufficient degree of rigor, but it happens eventually -- I did go get some of my proofs checked by serious Mathematicians at some point, and it turned out that my intuition wasn't far behind.
Beyond the junior undergraduate level, computation-style problems go away, and you're just left with exercises that ask you to prove a certain something. It takes a long time to build up a sufficient degree of rigor, but it happens eventually -- I did go get some of my proofs checked by serious Mathematicians at some point, and it turned out that my intuition wasn't far behind.
The main reason you'd want proofs to be checked by someone else, at least initially, is to get a feel for writing clear, rigorous and tight proofs. Much as a novice programmer should have their code reviewed, a novice mathematician needs to have their math reviewed for common mistakes and pitfalls.
Weren't there people in history who managed a sound level of self testing ? I understand it's super hard to be non biased and not claiming victory when you think you did a few checks.
Wait, "redefine derivatives to be based on multiplication"? That sounds really interesting, could you please expand on that?
It's a pretty interesting topic: https://en.wikipedia.org/wiki/Multiplicative_calculus
Math tends to bug me because "proving" theorems seems to come from convincing arguments, but we have enough computation power at this point that we should reduce them to mechanical/machine verification and then just run them and know.
With enough rigor, you don't need to "convince" anyone--a machine can verify it.
With enough rigor, you don't need to "convince" anyone--a machine can verify it.
To be rigorous you would need to prove that the machine verified it correctly. Good luck doing that. Also theorem proving is only good for basic rudimentary maths, and not abstract state of the art math research.
An impossibility indeed.
When learning anything you need feedback. In many cases you don't need feedback from humans, and then self-teaching is possible: if you learn chess, you can see if you win more games and if your ELO goes up; if you learn to draw, you can appreciate quite objectively if the apple you drew today looks more realistic than the one you drew one month ago. If you learn programming, you can see if your web page renders or not, or if your sql query returns duplicate records. In machine learning, you can see your AUROC score, is it improving? Is your kaggle ranking breaking the top 100, or you are stuck to 5000 ?
In math, what you deliver is proofs. It's very difficult to "debug" your own proofs, as logical lapses that you might have had when you put together the proof are very easy to persist when you do the self-verification. Sure, mathematicians have developed all sorts of heuristics to flesh out bugs in proofs, but these heuristics come with experience, so for an aspiring self-thought mathematician, this is Catch-22 situation.
What about Ramanujan, or maybe Gauss, or some other historical figures? I don't have a very good explanation, I'm inclined to say they are the exceptions that prove the rule. What I can say for today's day and age is that if someone says they know a self thought mathematician, my reaction would be "extraordinary claims require extraordinary evidence".
When learning anything you need feedback. In many cases you don't need feedback from humans, and then self-teaching is possible: if you learn chess, you can see if you win more games and if your ELO goes up; if you learn to draw, you can appreciate quite objectively if the apple you drew today looks more realistic than the one you drew one month ago. If you learn programming, you can see if your web page renders or not, or if your sql query returns duplicate records. In machine learning, you can see your AUROC score, is it improving? Is your kaggle ranking breaking the top 100, or you are stuck to 5000 ?
In math, what you deliver is proofs. It's very difficult to "debug" your own proofs, as logical lapses that you might have had when you put together the proof are very easy to persist when you do the self-verification. Sure, mathematicians have developed all sorts of heuristics to flesh out bugs in proofs, but these heuristics come with experience, so for an aspiring self-thought mathematician, this is Catch-22 situation.
What about Ramanujan, or maybe Gauss, or some other historical figures? I don't have a very good explanation, I'm inclined to say they are the exceptions that prove the rule. What I can say for today's day and age is that if someone says they know a self thought mathematician, my reaction would be "extraordinary claims require extraordinary evidence".
I will concede that I have some friends who are Mathematicians who could provide me _some_ feedback. Besides, I did end up contacting a few serious Mathematicians at various points to get feedback on my exercises -- it was just at extraordinarily low levels, compared to coding.
The possibility of making a mistake does not prevent one from learning a subject on one's own as deeply as one wants. In fact, discovering a mistake is an achievement and a great learning experience in its own right.
So many weird jarring points in this article. Self taught, but masters in physics? Takes 10 years to produce original research? Lots of PhDs graduating without publication?
I'd say it might take 10 years to develop a taste for interesting research problems but it isn't at all unheard of for undergrads to get a paper for some research project they did in final year. I think it would be really rare for someone to get a PhD without a publication these days.
Most undergrads who get a paper published are not publishing math papers.
This stackexchange answer is relevant
https://academia.stackexchange.com/questions/48399/how-many-...
Also this quote from a math professor, linked to in that SE link: "Many people get a PhD in mathematics before having a single accepted paper (I did), and if they have an eminent advisor who goes to bat for them, having no papers need not be much of a strike against them in the postdoctoral market."
This stackexchange answer is relevant
https://academia.stackexchange.com/questions/48399/how-many-...
Also this quote from a math professor, linked to in that SE link: "Many people get a PhD in mathematics before having a single accepted paper (I did), and if they have an eminent advisor who goes to bat for them, having no papers need not be much of a strike against them in the postdoctoral market."
"Most undergrads who get a paper published are not publishing math papers."
Err, I am not sure how to respond here. I said "it wasn't unheard of", which means when I went through I heard of a few bright undergrads getting publications in maths. It definitely happens. My personal experience says it would be extremely rare to get a phd now without a publication (again, in maths). It was different in the past. Hard numbers would be better than anecdata I guess.
Err, I am not sure how to respond here. I said "it wasn't unheard of", which means when I went through I heard of a few bright undergrads getting publications in maths. It definitely happens. My personal experience says it would be extremely rare to get a phd now without a publication (again, in maths). It was different in the past. Hard numbers would be better than anecdata I guess.
I enjoy the pure logical aspect of maths, but really dislike the notation/abstraction aspect.
I think that if we could rewind time and humanity had to reinvent math from scratch, it would probably end up very different from what we have today; most of the underlying logic and ideas would be the same, but the abstractions, notations and representations of those ideas would be completely different (and possibly better/simpler and more logical).
When it comes to code; whenever something is kept a certain way because of historical reasons, that's called technical debt and it's a bad thing.
In maths, basically all of the abstractions and notations are the way they are because of complex historical reasons. Math never gets refactored.
I think that if we could rewind time and humanity had to reinvent math from scratch, it would probably end up very different from what we have today; most of the underlying logic and ideas would be the same, but the abstractions, notations and representations of those ideas would be completely different (and possibly better/simpler and more logical).
When it comes to code; whenever something is kept a certain way because of historical reasons, that's called technical debt and it's a bad thing.
In maths, basically all of the abstractions and notations are the way they are because of complex historical reasons. Math never gets refactored.
I disagree, there's been quite a few attempts at refactoring math. Whether it's the Hilbert Program, which attempted to formalize the foundation of mathematics, the Cartesian plane as a way to describe functions in space or the modern reformulation under homotopy type theory, mathematics has gone through some serious refactoring. Even something as simple as algebra has gone through extreme changes in notation and in phrasing. Try comparing a copy of Euclid's elements to your standard geometry textbook. Or a proof of the Galois Correspondence Theorem in an Algebra textbook to the original in French.
Also, I highly doubt that math would significantly simpler or "logical" (a rather peculiar word to use, as math is inherently about logic, so what would more logical logic look like?). Math is inherently about communication and is therefore inherently subjective. People have a million different ways of writing math because they have a million different ways of speaking. And depending on who you ask, one way may be more simple or more logical than another.
Also, I highly doubt that math would significantly simpler or "logical" (a rather peculiar word to use, as math is inherently about logic, so what would more logical logic look like?). Math is inherently about communication and is therefore inherently subjective. People have a million different ways of writing math because they have a million different ways of speaking. And depending on who you ask, one way may be more simple or more logical than another.
Do you have any examples of things due for a refactor? As another commenter mentioned, mathematical notation and language have evolved heavily over time.
When you talk about abstractions, what in particular do you mean? Math is in many ways about describing objects through abstraction.
When you talk about abstractions, what in particular do you mean? Math is in many ways about describing objects through abstraction.
Actually, the view that Mathematics was "discovered" and not "invented" is quite popular among Mathematicians. There is an area of research called the Foundations of Mathematics, which investigates this question.
It's untrue that Mathematics does not undergo any refactoring -- in fact, it does: even in a field as new as Algebraic Geometry, the "classical" textbooks which build up the subject using ideals and varieties are very different from the modern ones which start from Scheme Theory.
Moreover, Category Theory shows that there is a beautiful underlying structure in all fields that can be extracted out into a new field. For example, the concept of adjunctions existed much before Category Theory was invented -- CT came along and showed that similar structures exist in Algebraic Topology, Algebraic Geometry, Differential Geometry, and other fields.
It's untrue that Mathematics does not undergo any refactoring -- in fact, it does: even in a field as new as Algebraic Geometry, the "classical" textbooks which build up the subject using ideals and varieties are very different from the modern ones which start from Scheme Theory.
Moreover, Category Theory shows that there is a beautiful underlying structure in all fields that can be extracted out into a new field. For example, the concept of adjunctions existed much before Category Theory was invented -- CT came along and showed that similar structures exist in Algebraic Topology, Algebraic Geometry, Differential Geometry, and other fields.
Definitions and notations are not discovered, they're invented. I think that's what OP was talking about.
The dy / dx notation has a very deep meaning from differential geometry. With some other notational things, there is a deep reason behind it.
Other times, notation is bad and it is refactored. The thing is, mathematics by its very nature is a lot of highly coupled 'code bases'. So any time you try to refactor, you end up with sprawling rewrites or a boundary with some translation method.
This is different in coding because there is less coupling, and the translation methods (Foreign function interfaces / shims) can be automated.
Other times, notation is bad and it is refactored. The thing is, mathematics by its very nature is a lot of highly coupled 'code bases'. So any time you try to refactor, you end up with sprawling rewrites or a boundary with some translation method.
This is different in coding because there is less coupling, and the translation methods (Foreign function interfaces / shims) can be automated.
Math isn't context free. Neither are most/all human languages. Yet we communicate succesfeully (within some margin of error).
This article brings to mind the movie "Good Will Hunting", specifically the mention of Srinivasa Ramanujan (https://en.wikipedia.org/wiki/Srinivasa_Ramanujan). I disagree that it's difficult to teach yourself mathemitcs. As with all things, success in self-directed teaching depends on the aptitude of the learner combined with their desire to learn.
Can you offer any empirical evidence that it is not difficult? The vast majority of people I've encountered who are entirely self-taught beyond typical high school (or college freshman) material have been quite bad. They repeatedly make fundamental errors of the sort I would expect to be swiftly corrected in undergrad coursework, even in topics they (claim to) have studied for years.
I don't think I've met anyone who learned to write a good proof without significant human instruction. That is probably something to be expected because much of what makes a "good" proof is social expectations, but I see a lot more cases of unjustifiable leaps of logic than excessive, obvious detail.
I don't think I've met anyone who learned to write a good proof without significant human instruction. That is probably something to be expected because much of what makes a "good" proof is social expectations, but I see a lot more cases of unjustifiable leaps of logic than excessive, obvious detail.
What does it mean to ‘learn math’? I feel like there’s a value to learning enough math that you understand the general concepts and vocabulary and having an intelligent conversation with a mathematician about, for example, how group theory might apply to some problem you’re working on, without having an actual working knowledge of group theory yourself.
I feel like it’s relatively easy to learn a lot _about_ math, as a self learner, without knowing a lot about how to _do_ math.
I’m mostly set taught, and I know the notation, I know the vocabulary, and I can follow along with a lot of math papers without too much of looking up terminology and I’ve adapted things I’ve read in math papers into working code, but I’m also well aware of my limitations —- I don’t know how to evaluate whether any of these papers are valid, I don’t have any sort of working knowledge of the tools of proofs to be able to do anything outside of fairly elementary calculus, etc. I know about higher math, but I wouldn’t say I know higher math. But even without that working knowledge, I wouldn’t say that what I know is valueless.
I feel like it’s relatively easy to learn a lot _about_ math, as a self learner, without knowing a lot about how to _do_ math.
I’m mostly set taught, and I know the notation, I know the vocabulary, and I can follow along with a lot of math papers without too much of looking up terminology and I’ve adapted things I’ve read in math papers into working code, but I’m also well aware of my limitations —- I don’t know how to evaluate whether any of these papers are valid, I don’t have any sort of working knowledge of the tools of proofs to be able to do anything outside of fairly elementary calculus, etc. I know about higher math, but I wouldn’t say I know higher math. But even without that working knowledge, I wouldn’t say that what I know is valueless.
Difficult or not, people - programmers, engineers, scientists - keep learning mathematics all the time. True, it may be difficult if you have little background, but for the vast majority of people with a high school diploma it should be no problem, especially given the amount of materials available.
I agree that teaching yourself mathematics is possible, but I don't think Ramanujan is a good example. He was a prodigy with something about his brain that was simply wired for mathematics and patterns. Solutions came to him as divine visions with little formal rationale or proof.
Ramanujan was also (relatively) bad at actually proving theorems, and worked mainly from intuition. He got a bit of culture shock when he moved to Cambridge and was expected to document his logic in the laborious detail that mathematics expects.
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I really wish Hollywood never talked about mathematics ever again.
Oh, come on. Hollywood can tell a compelling story to inspire people to get into fields they wouldn't have considered otherwise, even if only as amateurs and hobbyists. A lot of kids watching Iron Man are probably excited to grow up to get into machine learning and robotics.
They can tell a compelling story but I hate how they distort the reality. I feel like a lot of the plots are altered to fit this Hollywood format. Sometimes the antagonists feel forced. Or the friendships feel forced. Or the back story fells forced. I don't want a movie where they just take someone's name and likelihood and plaster the standard Hollywood tropes on it.
I hate the "writing math on windows thing".
I hate the "writing math on windows thing".
But... people actually do write on windows and glass like that. It happened at the University I worked at, it happens at my current job. It's just a thing.
Why is that?
I agree its somewhat disingenuous, and not representative of how mathematics is really done (most of which regular people would probably find pretty un-exciting and mundane). But it does increase visibility about those things. Instead of John Nash being someone who only mathematicians/academics knew of, his name is now (somewhat) more better known, isn't it? A book and a movie.
To be honest, I grew up in a pre-internet, pre-wikipedia world, and without movies like those I maybe would have never heard/known about these folks and what they did.
I agree its somewhat disingenuous, and not representative of how mathematics is really done (most of which regular people would probably find pretty un-exciting and mundane). But it does increase visibility about those things. Instead of John Nash being someone who only mathematicians/academics knew of, his name is now (somewhat) more better known, isn't it? A book and a movie.
To be honest, I grew up in a pre-internet, pre-wikipedia world, and without movies like those I maybe would have never heard/known about these folks and what they did.
I don't have a strong opinion one way or the other, but I do want to point out that they didn't make a movie about John Nash because he was a great mathematician, they made a movie because he was a great mathematician who had schizophrenia. It's not obvious to me that this image of mathematicians in popular culture is better than no image at all.
Yeah, for some reason people love this idea of a tortured "genius". I think that it makes the audience feel better. The idea that people can be smart and also relatively ok would make people jealous.
Name one genius who ain’t crazy. Seriously: ONE.
Hollywood isn’t making movies about “tortured geniuses” to pander: they’re doing it to reflect reality.
Hollywood isn’t making movies about “tortured geniuses” to pander: they’re doing it to reflect reality.
Don't confuse eccentricity and actual mental illness. Schizophrenia is something else than say OCD or anxiety.
Peter Scholze
$1000000000000000 says on a long enough timeline he will be proven crazy
A recent HN discussion got into some of the difficulty of self-taught mathematics, with emphasis on the Good Will Hunting level:
"P = NP Proofs: Advice to claimers (rjlipton.wordpress.com)"
https://news.ycombinator.com/item?id=19716303
Let's start with Boolean algebra and throw in same basic set theory. Surely that is not too hard to handle for someone able to program. Indeed it is not hard to imagine that our programmer could write some program that can systematically spit out Boolean formulas of set relationships that are tautologies, a.k.a. math theorems. In theory her program could conceivably produce all possible theorems. The problem is that only vanishingly small portion of the produced theorems are of any interest to another person. So the hardest part is actually to know/find what is interesting and relevant.
> Let's start with Boolean algebra and throw in same basic set theory.
I know where you are coming from, but yawn. Learning something that has practical applications is much more enlightening and fun (besides being useful). Linear algebra is one example.
I know where you are coming from, but yawn. Learning something that has practical applications is much more enlightening and fun (besides being useful). Linear algebra is one example.
I think it's incredibly disingenuous to call yourself a self-taught mathematician when you already have a masters in physics.
Undergraduate physics gets by with surprisingly little, if any, advanced math, so it is only natural when one realizes how inadequate one's knowledge of math actually is.
Have to disagree with this. I'm a math undergrad but a good chunk of my algebra class this semester was physics students. Additionally, the author said they have a graduate degree. As far as I know (which is not very far) modern physics makes considerable use of algebraic constructions such as tensors and lie algebras, so I'd imagine the author would be familiar with these concepts, which puts them lightyears ahead of any self-taught mathematician.
Math is being able to do logic with numbers.
I've tried creating my own math as an engineer, it's possible, but I also know every attempt I've made failed at finding something new.
My point is, we learned a lot about numbers in engineering school.
I've tried creating my own math as an engineer, it's possible, but I also know every attempt I've made failed at finding something new.
My point is, we learned a lot about numbers in engineering school.
Yes but physics != engineering. You really need very minimal higher math to do engineering work. As far as I know, at my school the furthest engineers go is a course in vector calculus, ODES, baby linear algebra and simple probability. None of which would be really be considered higher math. In contrast, physics needs complex variables, Pdes and most students take at least a group theory class.
Is there something about those topics that don't follow logic?
It's not mathematics degree, fair enough.
Math,queen and servant of science..
Spoiler alert: As with almost all headlines of this form, the answer to the question is "no". The author seems to be telling us that something they did is hard, but they did it anyway... Because.... Humble genius?
I've found math to be one of the easiest subjects to self learn (though I'm not saying that I've learned it or anything else to any noteworthy degree). The body of inexpensive literature is vast, and education in the field requires less practical support than, say, chemistry, physics, or engineering. I haven't gotten around to laser-induced plasma metal crystal deposition in my garage, but I do have bookcases of read and pending math books in the living room.
I've found math to be one of the easiest subjects to self learn (though I'm not saying that I've learned it or anything else to any noteworthy degree). The body of inexpensive literature is vast, and education in the field requires less practical support than, say, chemistry, physics, or engineering. I haven't gotten around to laser-induced plasma metal crystal deposition in my garage, but I do have bookcases of read and pending math books in the living room.
This is a great point, and I think it shows why people don't study math in their spare time. It's the same reason why most people aren't body builders with 5% body fat: it takes a tremendous amount of hard work.
Not only that, but the work isn't glamorous or particularly creative. In my teens, I always though I was "bad" at math -- I never did the homework, didn't do the supplementary exercises, had an "intuitive" understanding that wouldn't translate when exam time rolled around, and I consistently got C's and F's.
In college, I decided to take things more seriously and I remember in Calc I-III doing literally every single problem in the books. If I missed any problem, I would do it over and over again until I got it right. "Miraculously," I started getting A's in math classes.
Not only that, but the work isn't glamorous or particularly creative. In my teens, I always though I was "bad" at math -- I never did the homework, didn't do the supplementary exercises, had an "intuitive" understanding that wouldn't translate when exam time rolled around, and I consistently got C's and F's.
In college, I decided to take things more seriously and I remember in Calc I-III doing literally every single problem in the books. If I missed any problem, I would do it over and over again until I got it right. "Miraculously," I started getting A's in math classes.
Sadly Calc iii is not “real” math, just an applied version of what newton created 300 years ago. Real math starts at analysis and abstract algebra. It is extremely inefficient to learn this kind of math completely by yourself since there is a specific ‘math intuition’ that is hard to gain without a great teacher.
That seems a little harsh as a reply to the previous comment - calc 3 is math, but sure, upper division or graduate math also requires the ability to prove things, which is why people who did will in calc-3 might not necessarily do well in more advanced courses until they learn the additional skills. But it's not like the ability to apply theorems and calculate accurately is orthogonal to the ability to do proofs - the attention to detail and ability to keep track of the steps is certainly similar, I would argue.
You dont need to be a graduate in math to prove things. Check out the IMO and tell me it isn't real math, and it's only highschool level.
I'd guess Khan Academy was among the first to provide such teaching. Is it "self-taught", I don't know (I think it qualifies, though). It can be done.
Reading the article, the issue doesn't seem to be that math is particularly hard compared to other STEM fields.
Rather, to be productive in maths, you need a really high level. Before your PhD, chances is that you can't do something meaningful in the field. It means that your degree is the only proof of your level.
So the idea is : you can teach yourself maths but in the order to become a working mathematician, you need to do it all the way up until the end, with little help besides books.
Contrast with programming, after a few weeks, you will be able to write working software, maybe get an internship or even a low level job. You can then work your way up. You don't have to be at the top just to start working or make meaningful contributions.
Rather, to be productive in maths, you need a really high level. Before your PhD, chances is that you can't do something meaningful in the field. It means that your degree is the only proof of your level.
So the idea is : you can teach yourself maths but in the order to become a working mathematician, you need to do it all the way up until the end, with little help besides books.
Contrast with programming, after a few weeks, you will be able to write working software, maybe get an internship or even a low level job. You can then work your way up. You don't have to be at the top just to start working or make meaningful contributions.
Even a monkey can write a simple program, that's easy. However, to come up with something unique in computer science field, can easily take decades of study and research.
Math is the purest form of art humans can produce.I'd say one can learn a fair bit of it on his own and be relatively knowledgeable,but..There are many many things in math that are not just mind boggling,but also so difficult that unless you can singlehandedly crack problems that haven't been solved for centuries, you'd need a support from someone more senior. Those more seniors most likely will be found at good universities teaching and researching the subject.
> but I do have bookcases of read and pending math books in the living room.
That doesn't really tell me much.
That doesn't really tell me much.
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All mathematicians who get very far after formal schooling, especially in research, are essentially necessarily "self-taught".
At least at one time the math department at Princeton stated that graduate courses were introductions to research by experts in their fields, that no courses were given for preparation for the qualifying exams, and students were expected to prepare for the qualifying exams on their own.
A standard remark is -- "Learning mathematics is not a spectator sport.".
For learning math, teachers, courses, recommended text books, in high school and in good math departments in colleges and universities are necessary at least for a good start; else students will too often drift off into nonsense. So, such guidance, direction, motivation, feedback, environment, explanations, seminars, etc. are from helpful, ..., to crucial.
Still, in the end, especially after formal classes, mathematicians are essentially always "self-taught".
In K-8, the teachers were all females, really liked how the girls worked and behaved, and treated me like dirt. I learned enough anyway -- Dad really good at education monitored my progress and was satisfied.
But I was not a usual good student. Ninth grade was algebra, and it was a dream for me. Still in most ways I was still not a usual good student, but I learned the material, mostly on my own, well and got sent to a math tournament. The 10th grade with plane geometry was a big turning point: The teacher was the most offensive person I ever knew, and the subject was a total dream for me. So, no way did I want the teacher to have any credit for my learning and just ignored her, slept in class, refused to admit doing any homework, etc. In fact, I was likely the best math student she ever had: I solved a few of the hardest problems in the main part of the book then turned to the more difficult supplementary problems in the back and solved them ALL, never once missing one. I started college at a cheap place I could walk to. The math class they had me in was beneath what I'd done in high school, so a girl told me when the tests were and I showed up for those. The prof said I was the best math student he'd ever had. But starting in my sophomore year I was going to a good college with a quite good math department and didn't want to fall behind. So, I got a calculus book, taught myself, and started on sophomore calculus at in the good department. So, I've studied freshman calculus, taught it, applied it, learned much more in advanced calculus, ..., and published research, but never really took freshman calculus!
So, from the ninth grade on, I've heavily taught myself. For my Ph.D. dissertation, I identified the problem in industry, was chatting with a math prof about something else, mentioned my problem, got three words of advice, and when my plane landed I had an intuitive start on my dissertation. In my first year I took a terrific course in pure math, and independently in the following summer turned my intuitive solution into a solid one. I got essentially no direction on the dissertation research -- was "self-taught".
So, being "self-taught" is important. Still, it is crucial to have the guidance, feedback, etc. of a good college math department for at least some of the time.
And if want to have a good career in academic math research, then likely it is really important to learn from some of the best research profs, listen carefully for hints of good directions at research seminars, etc.
At least at one time the math department at Princeton stated that graduate courses were introductions to research by experts in their fields, that no courses were given for preparation for the qualifying exams, and students were expected to prepare for the qualifying exams on their own.
A standard remark is -- "Learning mathematics is not a spectator sport.".
For learning math, teachers, courses, recommended text books, in high school and in good math departments in colleges and universities are necessary at least for a good start; else students will too often drift off into nonsense. So, such guidance, direction, motivation, feedback, environment, explanations, seminars, etc. are from helpful, ..., to crucial.
Still, in the end, especially after formal classes, mathematicians are essentially always "self-taught".
In K-8, the teachers were all females, really liked how the girls worked and behaved, and treated me like dirt. I learned enough anyway -- Dad really good at education monitored my progress and was satisfied.
But I was not a usual good student. Ninth grade was algebra, and it was a dream for me. Still in most ways I was still not a usual good student, but I learned the material, mostly on my own, well and got sent to a math tournament. The 10th grade with plane geometry was a big turning point: The teacher was the most offensive person I ever knew, and the subject was a total dream for me. So, no way did I want the teacher to have any credit for my learning and just ignored her, slept in class, refused to admit doing any homework, etc. In fact, I was likely the best math student she ever had: I solved a few of the hardest problems in the main part of the book then turned to the more difficult supplementary problems in the back and solved them ALL, never once missing one. I started college at a cheap place I could walk to. The math class they had me in was beneath what I'd done in high school, so a girl told me when the tests were and I showed up for those. The prof said I was the best math student he'd ever had. But starting in my sophomore year I was going to a good college with a quite good math department and didn't want to fall behind. So, I got a calculus book, taught myself, and started on sophomore calculus at in the good department. So, I've studied freshman calculus, taught it, applied it, learned much more in advanced calculus, ..., and published research, but never really took freshman calculus!
So, from the ninth grade on, I've heavily taught myself. For my Ph.D. dissertation, I identified the problem in industry, was chatting with a math prof about something else, mentioned my problem, got three words of advice, and when my plane landed I had an intuitive start on my dissertation. In my first year I took a terrific course in pure math, and independently in the following summer turned my intuitive solution into a solid one. I got essentially no direction on the dissertation research -- was "self-taught".
So, being "self-taught" is important. Still, it is crucial to have the guidance, feedback, etc. of a good college math department for at least some of the time.
And if want to have a good career in academic math research, then likely it is really important to learn from some of the best research profs, listen carefully for hints of good directions at research seminars, etc.
If you went through a math undergrad and some grad courses, you are not self-taught.....
Second, it's quite easy to convince yourself that you understand a topic. I don't know what it is about math, but there's so many people, myself included, who fall into the trap of thinking they understand a topic, when in fact they don't at all. For instance, there's a multitude of people who can recite verbatim the quadratic formula, or the power rule for differentiation. But ask them to explain why this is true, or give them variants on the power rule (what if we redefine derivatives to be based on multiplication? How does the power rule change?), and they won't be able to give you satisfactory answers. This goes back to my first point; the only way to truly understand math is to do problems and have them checked.