I think there are two non exclusive ways of doing research successfully. Intense work and collaboration (in particular supervision of students).
PhD students and postdoc fall in the first category. They work full time on their research and do little else.
On the other hand, most successful experienced researchers are busy writing grant proposals, teaching, attending various committees. There's not enough time left to do meaningful research by themselves (plus getting older doesn't help to stay sharp and passionate), so they end up supervising students that to the bulk of the research.
I think someone that would try to do research on its spare time won't have enough time to devote to it (unlike a phd student), and won't be able to supervise students (unlike a professor).
As most on this site can probably attest to, one of the biggest issues with part-time/hobby/side projects is motivation and discipline. That problem seems to be already partially solved by the freedom to pick an area of mathematics one is already interested in.
Not having enough time to devote to a problem part-time doesn't seem like a large problem if somebody wants to work on problems of their own choosing as per the OP. Thus, my confusion.
I can definitely see that collaboration is a tougher nut to crack unless most participation starts shifting to evening group meetings.
The New Yorker had an interesting article about Yitang Zhang and his discovery concerning bounded primes. The discovery itself is interesting for the mathematics involved, but of wider interest is that up until this discovery, he was a relative unknown in the mathematical establishment.
If it were simply a question of motivation and discipline to have math as a hobby, we would expect discoveries from unknown individuals more frequently, but, alas, that's not true, which is what makes this story precious.
It stems from the fact, among other things, that math isn't like programming. It's not like deciding, one day, that you want to write your own operating system, and then the small matter of coding it up. Math requires* those leaps of intuition that come to you after banging your head against the same problem for months. Most employers would start to question a developer's continued employment if it took them five months to resolve a single ticket.
*http://www.newyorker.com/magazine/2015/02/02/pursuit-beauty
Some do actually use supercomputers to do mathematics.
Most however, thrive in an actual academic environment including spending a lot of time interacting with other mathematicians and going to conferences. There's a big difference between living in an academic environment and spending your whole life thinking about math and being around people who do the same... and working a 9-5 and doing a little math on the side. Not to say it isn't possible, but most reasonable jobs don't leave much left over mental energy for doing maths.
I'm sure there is a sizable number of "amateur" mathematicians who work on problems in their free time, but it is hard to know what to work on and where to look.
The "monastery" could also be run as a business, having a for-profit section developing products based on the ideas produced inside. Something like the members of the monastery signing an agreement that they are ok with their results being published with one year delay, time that would give an edge to the monastery over any competitors that would also want to develop products based on them - think of it like a "research monastery self-financed by coupling with a sort of startup incubator" (with some really intransigent and incorruptible people working to keep the two camps of people apart and to stop the "business idea developers" from exploiting the "monks").
And for research fields that require some expensive equipment, like bio-medical research, the "monastery" may make the "monks" sign contracts giving the patents and other right to their discoveries to the monastery, in exchange for having access to a lab where they can work on anything they want. It's not that utopic since you can a pretty usable top notch molecular biology lab plus an animal experiments facility for under $1M (probably not in the US because you have too many regulations and stuff...) if you find alternative channels of buying equipment of user other "hacks" that regular research institutes can't use...
But it's the kind of idea that it's hard to convince other people it would make sense... so for now I'll keep it on my list of to-dos for after I get filthy rich so I can finance this myself.
https://en.wikipedia.org/wiki/Institute_for_Advanced_Study
On the other hand, Richard Hamming thinks this model won't work:
> The great scientists often make this error. They fail to continue to plant the little acorns from which the mighty oak trees grow. They try to get the big thing right off. And that isn't the way things go. So that is another reason why you find that when you get early recognition it seems to sterilize you. In fact I will give you my favorite quotation of many years. The Institute for Advanced Study in Princeton, in my opinion, has ruined more good scientists than any institution has created, judged by what they did before they came and judged by what they did after. Not that they weren't good afterwards, but they were superb before they got there and were only good afterwards.
An HN comment with +20 karma was probably read more times than most research papers in journals!
Even outside academics this is true. A great essay, posted as a comment on HN or reddit, is going to be much less circulated than the same essay posted on a blog. E.g. I'll see a good essay on someone's blog and see that it's posted to 12 different subreddits and HN. If I say a great essay as a comment, even if it's of the same quality, I guarantee it will not be posted anywhere else.
www.amazon.com/Anathem-Neal-Stephenson/dp/006147410X/
You lose maybe 6-12 hours to a job--assuming you never work on research or knowledge expansion during work hours, which is unlikely--but then you don't have to have that stuff hanging over your head.
Bills are getting paid, equipment can be purchased (if you need it), and your social life is happier. So, I think that research can absolutely be pursued outside of academia.
Because it's much much harder to get published when you aren't immediately associated with a university. People just assume "crank".
Does having a full-time industry job allow you to seriously pursue your own research interests on the side?
For mathematics research especially, why don't more research-oriented mathematicians pursue this dual path? I understand part-time research necessarily implies a slower publication rate, but since you're already eschewing this anyway, are there any other drawbacks? Two that spring to mind are a restriction to travel to mathematics conferences and socializing with mathematicians during the day -- are these big show stoppers?