Self-Taught Mathematician; an Impossibility?
medium.com
medium.com
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.
With enough rigor, you don't need to "convince" anyone--a machine can verify it.
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.
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".
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."
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 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.
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.
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.
When you talk about abstractions, what in particular do you mean? Math is in many ways about describing objects through abstraction.
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.
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.
Hollywood isn’t making movies about “tortured geniuses” to pander: they’re doing it to reflect reality.
I hate the "writing math on windows thing".
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 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 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.
Those who come up with new results tend to have had a lot of previous instructed study.
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 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.
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.
That doesn't really tell me much.
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.
Curious, what makes you say this? Not that I strictly disagree.
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.