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I will soon finish my undergraduate degree in math, and I want to study mathematics as a hobby after I graduate.

Online, I see a general trend of people saying

You can't learn real math without collaboration and feedback. Go to graduate school!

But I don't want to go to graduate school, and I have found that I learn better through self-study than by taking notes in class. That being said, it always does help to get my questions answered by professors, and collaboration with other students is sometimes helpful as well.

Question: Is it possible to have this kind of mathematical community without actually being in academia?

What I fear is studying for years by myself obsessively on a subject only to find that I had a horrible misconception the whole time that could have been avoided if only a professor had looked at my work.

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  • You know, you do not have to graduate if you do not want that degree, you can hang around while learning some material, and connecting with people. – onurcanbektas Feb 16 '18 at 13:50
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    I would still seriously consider graduate school if you really want to learn mathematics. Sure you will have to take some classes, and pass some tests including qualifying exams, but the bulk of your work would be self-directed research. There's also no reason you couldn't self-study the subjects that you are learning in class. In fact, you are encouraged to do so! – rviertel Feb 16 '18 at 18:36
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    @rviertel I have many reasons for not wanting to go to graduate school. If math was my life goal (like it used to be) then graduate school would be for me. But I realized that I wasn't interested in doing math research as a career, and I wasn't really into the idea of teaching. Math for me is a hobby, and I mainly wanted a PhD for the status it would give me. I decided to go into computer science instead -- I think I will have a lot more fun being a programmer or software engineer, doing math on the side as a hobby. – Malachi Holden Feb 16 '18 at 19:03
  • For what it's worth you shouldn't really be afraid of mistakes. They teach you a lot, and usually you gain something useful from failed attempts which you would not have gained by following a path pointed out to you from the start. At least that's my experience, including my math habit. – Yuriy S Feb 17 '18 at 10:37
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I’ll give an answer based on personal experience. My situation is not exactly like yours, but may be similar enough to provide some insight.

I recently completed my undergrad in physics and applied math. Upon finishing this I decided that I was more interested in pure math, and wanted to pursue this in grad school. I ended up not getting into any of the Ph.D programs I wanted (probably because of a lack of pure math experience), so I was left in limbo, so to speak, still wanting to study math but not yet able to enroll on a graduate program.

My solution was to email some professors at a local university who I had found doing things somewhat related to what I was interested in. I told them my situation and explained that I was just looking to learn more about their research and math in general. This ended up working out and now I attend regular seminars at the university and I meet up with them every once in awhile to just discuss math.

I have found it very helpful to be able to have face to face conversations with other mathematicians, even though I do most studying on my own at home. Though Stack Exchange is a valuable resource, I don’t think it can adequately replace this interaction.

I’d like to point out that I am not sure if my experience is typical. After all, this is a somewhat small university that I am talking about, and it may be harder to find professors at larger universities who are able to take extra time to talk to you (I certainly found this to be the case at UC Berkeley, which is where I did my undergrad). Still, I think that this is an option you may be interested in considering.

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  • How big is your university? I don't think it's true that bigger school implies busier professors. I go to a small liberal arts college (1600 students) and the professors are always super busy. – Malachi Holden Feb 17 '18 at 15:03
  • The place I’m at now is less than 20,000 students. So smallish but not tiny. And you are certainly right that small doesn’t imply not busy professors, and vice versa. I do think there is a correlation there though, in general. If nothing else, a place with larger class sizes means more students filling up office hours, emailing with problems and questions, etc. Not to mention large high profile places might put more emphasis and pressure on their professors to research and publish papers (it seemed profs at Berkeley we’re always extremely busy with research related work). – wgrenard Feb 17 '18 at 15:39
  • But again, I don’t pretend that my experiences on this front are going to be typical of everyone. It’s possible that I just got lucky. One of the profs here even told me that he enjoys working here so much because they put a big emphasis on teaching and interacting with students, and the pressure to publish papers is not as great as at other places he has worked. Still, I think you should definitely give it a shot because you never know if something really worthwhile will come of it. – wgrenard Feb 17 '18 at 15:42
  • I definitely will. There are some medium-sized universities in the area -- I'll see if they have seminars in math that I'm interested in. – Malachi Holden Feb 17 '18 at 16:03
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Nowadays, you can get feedback and answers to questions from places like MathStackExchange, and MathOverflow. Most of the responses are (in my observation) either entirely competent or quickly corrected by others. Good quality control. No, that wouldn't be the same as having a good advisor (ideally a world-class scholar on your subject) that you talk to in person every week. Still, these sites can give guidance.

Also, quite a few of the more-senior people who like to contribute to such sites may be amenable to direct email contact, etc.

(Grad school is not about "taking notes in lectures" per se, but is about having access to the informal observations of very good mathematicians, and being able to ask them questions immediately, in the moment.)

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    How obnoxious would it be for me to post a proof for review on stack exchange, say, every week? Do you people get tired of looking at people's homework? Or is that what you're here for? – Malachi Holden Feb 16 '18 at 12:26
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    @user87627 as long as you show that you make efforts you can get positive responses. If you only ask questions and don't seem to work on it then you may get negative responses. At least on math.SE. Mathoverflow I don't know because I have not been on there so much. – mathreadler Feb 16 '18 at 12:56
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    Proofs for review are considered off-topic on Mathoverflow, and so is any kind of homework. – Emil Jeřábek supports Monica Feb 16 '18 at 15:01
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    @user87627 Grad school isn't about homework any more than it is about taking notes in class. There will probably be a bit of classwork at the very beginning, but that's not the main focus. For most of grad school, you'll be finding interesting problems you want to work on and then working on them, while having more senior mathematicians around whose job responsibilities include giving advice to the apprentice mathematicians (which is what grad students are; they're there to do research, not take classes). Disclaimer: I'm CS, not math, so there might be some field-related differences. – Ray Feb 16 '18 at 21:32
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    @user87627: I'd rather Math SE's front page be covered by all your proof attempts than by the Please-Solve-Questions that we see more and more of nowadays from students who ask others to do their homework for them! – user21820 Feb 18 '18 at 9:24
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Disclaimer: I am a mathematician, but I do research in CS.

My observation is that all mathematics, but especially research-grade mathematics has a lot of features of a folklore. It is much easier when someone explains it to you, even more if informally. Reading books helps, of course, but for me a quick informal explanation done in five minutes could replace a tedious drilling through formalism for understanding for few days. Of course, you'd still need to be able to get through the formal and completely correct description. But it's much easier, when you already have an informal understanding.

To give an example, the determinant is actually an oriented volume of the n-dimensional parallelotope spanned with matrix component vectors. Compare this with a definition from almost any undergrad book on linear algebra.

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    "the determinant is actually an oriented volume of the n-dimensional parallelotope spanned with matrix component vectors." ohhhh... you should write a textbook like that, I'd buy it. Hell, I'd buy two! – DonQuiKong Feb 16 '18 at 9:11
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    I've read it in a critical paper on teaching by V. Arnold and it blew my mind. – Oleg Lobachev Feb 16 '18 at 10:23
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    +1 for "[Mathematics] is much easier when someone explains it to you, even more if informally.", and I'd like to add that in my experience this also holds for other fields. – Pont Feb 16 '18 at 10:49
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    It seems there is a clear demand for textbooks including informal understandings. – Trilarion Feb 16 '18 at 11:47
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    This is true. I do think part of this is because textbooks are so poorly written, unfortunately. In general I learn better from reading than from listening -- however, I think one on one explanation is the best, especially when it's combined with the book. – Malachi Holden Feb 16 '18 at 12:16
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You have a couple of good answers about what you could do. You should also be aware of what you shouldn't do. Don't send unsolicited e-mails to random mathematicians in which you launch into long mathematical discussions. Unfair as it might be, there is a good chance that your message will be quickly deleted as the work of a probable crank. Most professional mathematicians receive numerous such e-mails (or in the old days, physical letters) over their careers and have learned not to pay too much attention to them. If you have a legitimate mathematical signal, don't broadcast it over a channel which has been drowned out by the noise of cranks.

The good thing about the suggestion of being active on Math Overflow is that this would allow you to develop a relationship with professional mathematicians. I have had a number of unsolicited e-mails from people that I have encountered on Stack Overflow, and I have almost always made an effort to respond to them. The virtual rep that you earn on Math Overflow entails a certain amount of real-world rep with the mathematicians who participate on that forum.

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    Regardless of whether or not you look like a crank, there are very few people who are prepared to spend an hour or more helping a total stranger to no obvious personal benefit. Heck, even if a friend or colleague asks for that, there are plenty of circumstances in which one would say no. – David Richerby Feb 16 '18 at 15:53
  • @DavidRicherby, nevertheless, there are some people who view select responses of this sort as part of their professional/societal role, if not "obligation". To be a bit rhetorical, if experts don't try to help "cranks" improve their viewpoints, who will? – paul garrett Feb 20 '18 at 22:08
  • @paulgarrett Sure -- there are a lot of people in the world and some of them are really nice. I was just trying to convey the scale of what's being asked for. – David Richerby Feb 20 '18 at 22:45
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    @DavidRicherby, indeed, it's a non-trivial favor to ask, so appropriate perception (on the part of the asker) of the response (from the asked-of) is a bit subtle. On one hand, if one doesn't ask, one gets nothing, but the mere "asking" itself does not immediately/truly entitle one to a response. All that kind of thing... – paul garrett Feb 20 '18 at 22:52
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One good option is to hire mathematicians to help you when you have questions. Many graduate students and some folk with Ph.D.'s work as tutors. If you're going into programming in the US, you should have the budget. You might not connect with the first person you contact, but if you live near a research university there should be some opportunity for this.

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    This is right, if you want someone to do work for you that's not part of their job, then you should pay them. You're not likely to pay enough for advice from a professor, but grad students are much cheaper. – Noah Snyder Feb 18 '18 at 2:43

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