dijkarte said:
Is Academia really necessary to become a mathematician? What about self-taught mathematicians? If someone has the maturity to learn by themselves, why need to go to university and spend time and money to sit in an overcrowded class with who knows what kind of lecture you get...?
The only advantage I see is that someone cannot teach at the university unless they have a related graduate degree. But what about publishing math papers? Do we need to have this graduate academia license to publish something or author a book?
I consider myself a self-taught mathematician. I have a bachelors in musicology and a masters in trumpet performance. My math background in a "formal" academic setting is minimal. I definitely think it's possible to learn on your own. In some cases and at some stages of mathematical development it may be significantly less productive though.
The last time I really had a "formal" course in mathematics was 10th grade when I took calculus. My last two years of high school math were independent studies in linear algebra and differential equations (because my school only offered courses up to AP calculus). I had little interaction with my teacher. My independent study was during her "prep period" where she graded papers for her other classes, made lecture notes, etc... I sat in her room quietly reading and doing problems out of a few books, shoved my work in a folder and that was about it. My independent study "teacher" looked at the work I had in the folder each quarter (didn't grade it or anything), wrote a brief test for me based on how far I had gotten (random problems out of her old college LA/DE book), I took it, she graded it. She gave me the test back, if I had a problem wrong, I was able to correct it and get half credit for it ... I almost always got a 9/10 usually for a stupid mistake, then corrected it for a 95%, which gave me an A for the marking period ... pretty cut and dry.
In college, I majored in music but also enrolled in math and science classes that didn't require attendance (other than lab courses) and made sure to look on ratemyprofessor or talk to students and find sections that didn't have required weekly coursework. I usually scheduled hours of work at my part-time job during class but make sure to take off work the days that were scheduled exam periods. I taught myself the material, often from a few old $5 textbooks rather than the $100+ one that was required for the course. That worked out well for me with all my math/science stuff.
After years of being a professional musician, I eventually found myself doing math again. I am currently doing what is considered "graduate-level" stuff and I haven't had a "formal" class since the late 1990s, so it's definitely possible to teach yourself. Keep in mind it might not be as efficient to do this vs if you were actually in academia due to the support you get from the university system.
I can definitely see this being an issue later on with a PhD. Assume I had to keep up the slower pace I've been working at because I still need to provide for my basic physiological needs. How likely am I to complete something original if it takes me 50-100% longer than the "normal" 3-5 years to crank out my dissertation? Might life happen and I abandon what I'm working on? Might somebody have already solved my problem during the 6-10 years I might be working on it with ~1000 PhDs being granted each year in the US and maybe a 1 in 10 chance they're in the same broad field (excuse that guess if it's totally off, just a round number to serve as an example). How much more time will it take when I don't have an adviser to guide me, tell me what to read, regularly ask me what I've been up to this week, discuss my dissertation problem, etc...
I guess it's possible to eventually work with somebody as an "informal adviser" but good luck finding somebody to volunteer their time taking on a (for lack of better words) "PhD student" who wasn't actually part of their university, and who might only able to devote part of their time to research.
Anyway, my two cents are that: sure it's possible to self study. I think it's much easier / practical to self study all of the topics leading up to PhD qualifier material, especially now when there's: Khan Academy, MIT's OCW, full courses on youtube from Harvey Mudd, Stanford, Princeton, Indian Institute of Technology, USF, Harvard extension school, etc...