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This question might not make much sense or be a big deal for someone but I genuinely would like to know the opinions.

I am currently a software developer in a small start-up. I had to drop out of college and do not have a degree. I have always been interested in mathematics and research in mathematics, I would like to know if there is a way I could be involved in this or take up mathematics. I am planning to get a degree in distance mode and start from scratch but meanwhile, I wanted to know what can I do to have a career in the field.

Right now I have started with this to brush up what I had learned back in college. I would like to know if there is any way I can start working, It need not pay me, I would like to work on something that will help me with my career in Mathematics.

*If this question does not belong here please do not downvote or close the question, Do let me know and I'll remove it

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    $\begingroup$ Research is a conversation among a community of researchers. To join the conversation, you need to catch up on what has been said, what people are interested in, and have something worth saying. Otherwise people will simply ignore you and keep talking to each other. Undergrad curriculum catches you up to the conversation c. 1900, then you study a field in depth, read papers until you're caught up to the state of the conversation... and then you answer a question that the researchers are interested in to start joining the conversation. $\endgroup$
    – Neal
    Commented Aug 25, 2023 at 0:44
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    $\begingroup$ See also these: mathoverflow.net/q/422196/454 mathoverflow.net/q/132237/454 mathoverflow.net/q/44244/454 $\endgroup$ Commented Aug 25, 2023 at 1:28
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    $\begingroup$ @Neal I think you've summarized this beautifully. I think it would be worthwhile to turn it into an answer. $\endgroup$
    – Wojowu
    Commented Aug 25, 2023 at 6:51
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    $\begingroup$ If you want a career in mathematics, then you most likely have to follow the standard path of spending a lot of time in school, if not formally pursuing a degree then at least auditing classes and talking with professors a lot. On the other hand, it is possible (though not easy) to make significant research contributions to mathematics without making a career out of it. See the MO question, What recent discoveries have amateur mathematicians made? $\endgroup$ Commented Aug 25, 2023 at 13:52

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Research is a conversation among a community of researchers. To join the conversation, you need to catch up on what has been said, what people are interested in, and have something worth saying. Otherwise people will simply ignore you and keep talking to each other.

The undergrad curriculum catches you up to the conversation c. 1900. You start to speak the language that the researchers are using.

A graduate curriculum gives you the basics of what the researchers are talking about in many different conversations, and often brings you up to speed to the conversation c. mid/late 20thc - when many of the senior people in the field were just starting.

You pick a conversation. You study a field in depth and read papers and preprints until you're caught up to the conversation in broad strokes. Ideally you have a mentor or advisor, who helps teach you what is interesting and introduces you to other researchers and can vouch for your potential.

You meet many of those people in person. You hear how they talk about their subjects among themselves, you learn the tricks they use to think about the subject efficiently.

As you immerse yourself in the conversation, you become aware of the problems that are interesting to other researchers, and how you might take your knowledge to solve them. After working hard, you answer one of these questions, and you start talking about your answer by email - in person - in talks - via the arxiv - eventually publishing.

Because you have said something interesting, the other researchers become interested in what you have to say, and you are on your way to joining the conversation.

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    $\begingroup$ This is a great answer. But I would push back on the notion that "The undergrad curriculum catches you up to the conversation c. 1900." This makes it sound like even after you've completed a full undergrad course, you're still a hundred years away from doing research. That's simply not true in general. Much of what I learned as an undergrad, I never use in my research; I'd be doing fine had I never taken those courses. Most crucial for research is developing problem-solving ability, and finding the right problem to work on. I elaborate on this point here. $\endgroup$ Commented Aug 28, 2023 at 13:16
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If you want a career in research mathematics, I think you basically need to get a PhD. There are some people who get hired as researchers without having a PhD, but I don't think it is common. That said, you can still do mathematics without a degree; either for fun or to help get your career started. One thing I will say is that it is very difficult for amateurs to know what kinds of questions current professional research mathematicians are interested in, and harder still to make meaningful progress. This is an important role of supervisors in graduate school. They are well-read of the literature in their niche field and so they can point their students towards questions that are not too difficult to solve, but also will be publishable in good journals.

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    $\begingroup$ If you mean a career in academia, your first sentence is probably right, although the PhD would not always have to be in mathematics. A person with a PhD is chemistry or astronomy might at some point become a professor of mathematics. $\endgroup$ Commented Aug 26, 2023 at 0:33
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    $\begingroup$ @MichaelHardy Even in industry and government, it's becoming increasingly rare to hire a non-PhD to do research in mathematics, unless "research in mathematics" is defined very broadly. $\endgroup$ Commented Aug 28, 2023 at 22:08
  • $\begingroup$ @TimothyChow : But my comment was about people whose Ph.D. is in some other field, rather than about people without a Ph.D. $\endgroup$ Commented Aug 28, 2023 at 22:51
  • $\begingroup$ @MichaelHardy Oh yes, I agree with your comment. My remark was just that your hypothesis "if you mean a career in academia" can be weakened. $\endgroup$ Commented Aug 29, 2023 at 12:03
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Some concrete advice for anyone attempting to pursue research mathematics without the infrastructure of a PhD:

Use math.stackexchange for the wonderful resource it is.

I cringe thinking about trying to "solo" research mathematics pre-internet; I am reminded of Feynman's quote on the topic. MSE has successfully become a repository for excellent explanations on basically all common undergraduate topics, and if you have any specific questions over the first few years of your journey you'll be able to search them specifically, and if nobody has asked them before you have a rotating cadre of experts on standby waiting to elucidate things for you. Who could ask for more?

Now, suppose you begin to ask questions which go unanswered on MSE; this is the juncture at which you should begin to

Use mathoverflow for the wonderful resource it is.

It is difficult for me to articulate how valuable of a resource MO is for an independent researcher without understating things. Suffice to say that I don't think it would really be possible to do 'independent' research without it, in the sense that otherwise one would need to maintain personal correspondence with some professional mathematicians in order to participate meaningfully in the 'conversation' we're all having. Of course, 'independent' anything is really an illusion; any meaningful and worthwhile task is one we do shoulder to shoulder, standing on the shoulders of those who came before and preparing to bear those who come after.

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