career path - How possible is it to do mathematical research outside academia?

Does anyone know if it is possible for a Ph.D. graduate of mathematics to successfully do research in his/her spare time?

The work environment and conditions in academia seem somewhat deplorable at the moment---no offence intended, but that simply very much appears to be the case from an outsider's perspective---and it just strikes me that mathematics would surely be the subject most suited to Ph.D. graduates who don't want to go there, but still want to research, as "only pen and paper" are required.

If one were really, truly devoted...how much of a disadvantage would they have compared to someone who chose to stick it out?

Are seminars entirely barred?

Is it impossible to publish?

Collaborate?

How much (more) time would someone who stayed in academia have in a day to spend researching?

Also, what about people who have not got PhDs? I would think it's more or less impossible for them to contribute very much of value if they only have a Bachelor's or a Master's degree, but I really don't know.

Answer

Publishing

In mathematics, most people will judge your research on its merits, not based on your credentials or affiliations. So this is not an issue, if you are able to generate significant results and write them up in a clear way.

Seminars

I have seen many members of the public regularly attend seminars. I've never heard of them being kicked out. It's not an issue.

Funding

The primary drawback is that you won't be paid for your research if you are just doing it in your spare time. Unless you are already independently wealthy, that means you will have relatively little time for research since you need a day job.

It is of course possible to get grants to fund your research without being in academia; there are a number of private entities in the US funded entirely by grants, and many of them include mathematicians. But you said spare time only.

Collaboration

Another drawback is that you won't be in close contact with other researchers by default. It wouldn't be impossible to establish that contact, but again you must consider whether you will have the time. And people won't take you seriously unless you have serious results.

Not getting a PhD

In mathematics, even more than in other fields, it is almost impossible to make a contribution without spending several years learning what is already known. In principle you could do that without getting a PhD, but why would you?