Mathematics as a hobby I would like to know if practicing mathematics, constituting a hobby for some of you who are neither academics nor (advanced) mathematics, is an important part of your career. How do you go and learn a new mathematical field on your own?
Do you just pick up a book and go over all proofs and do all exercises on your own? Is there any technique would you like to share?
 A: I do it as a hobby, but then again that is due to my woeful academic record putting academia out of reach.
You don't need to do anything funky to improve your mathematics, you just need time. Look at the average High School student. Their mathematics improves enormously between the ages of 12 and 17. But it isn't because they are especially talented, or that they work especially hard, or that they were especially encouraged, or that their teachers/texts/syllabuses  were especially good; it was simply due to spending time on mathematics. The improvements are so small that they cannot be noticed over a period of days, but the effects are cumulative and over years they become very noticeable.   
A: I described my way here. I guess one should first have an idea why one is curious about it (for example, I like math text's high idea/page ratio).
Then I would say, one should find out by browsing surveys or seminar talks what parts of mathematics one finds thrilling. Conc. sigfpe's "how do you know you understand?": The simplest test it to tutor university students. Imagining how one would explain things to others is useful (and makes fun) too.
Libraries and the Internet provide huge amounts of excellent texts for everyones' taste. Actually there are so many texts available that one risks to drown in the quicksand they can turn into. The nice thing about being interested in mathematics as a hobby is that it liberates one to follow one's nose, be it oldfashioned stuff as 19th century projective geometry, or for example tracking the equally old "playing with infinite series"-mentality from Carr's textbook (which formed Ramanujan's thinking) through Weil's Eisenstein-book to Cartier's Mathemagics, or working through some (surely completely outdated) Bourbaki talks from the early 1960's what I do in the moment.
The back side of it is that one does not know whom to ask for bibliographic hints or about a correction/completition of one's mental image of an issue. For example, I'm just wondering which applications derived algebraic geometry, aside being a kind of natural abstract development out of Grothendieck's work, has in 'normal' algebraic/arithmetic geometry, and about what Grothendieck meant with the "geometric meaning of the biduality theorem", whose discussion is acc. to "Recoltes ets Semailles" missing in SGA5. It is really bad that mathscinet is not openly available.  
A: I'm no longer in academia and while my job has some mathematical challenges, they aren't as interesting as the challenges I find in other branches of mathematics. That means I do mathematics on my own as a hobby.
As I see it, the main challenge is this: how do you know you understand? You can try doing the exercises in textbooks. But how do you know you have the correct solution? And if the book has solutions, how can you trust the similarity metric you use to decide whether your solution is the same one? It seems to me that from time to time you need external calibration to make sure you're on the right track. In a sense you have to always test yourself, in the sense of falsifiability. If you work in academia, your colleagues will keep doing this for you. I know two ways to achieve this on your own:
(1) Blog about what you have learnt. Because of the "someone is wrong on the Internet" syndrome, you're likely to get a response if you say something that is incorrect. You can write page after page of insightful material that will (apparently) be completely ignored, but if you make a mistake you'll get corrected (at least if you can get a following of some sort). As a side effect, communicating stuff to other people, even "rubber ducks", can really deepen your own understanding.
(2) Try to turn what you have learnt into computer programs that solve a problem. Computer programs don't provide proofs (well, that's not always true), but they will give you confidence and sanity checks. This doesn't just limit you to numerical analysis. Whether you're doing group theory, or algebraic geometry, or logic, or algebraic topology, or combinatorics, or even analysis, there are often subdomains of those disciplines for which you can write computer programs that will likely fail if you don't understand the theory.
A: If you want to try to do some research it may be best to pick a field that is not popular with professional mathematicians. You may also want to try to pick something that has not been worked on for some time. I chose convex structures and the result is:
http://www.ams.org/meetings/sectional/1058-52-28.pdf 
A: I think there is a somewhat misleading perception of mathematics as a field with a very high entry barrier, so you need to first spend many years learning and only then can start working, so only professional mathematicians can do this.
While this is certainly true of some fields, it is definitely not true of others.  In fact, in the 19th century there was a large number of amateur mathematicians (there is even a list of them). I recommend finding a nice elegant but accessible open problem and working on it (take books listed on this page as a starting point). There are many nice lectures by excellent mathematicians with a variety of open problems, which are available online in PowerPoint format and sometimes in video format (as well as numerous blog posts, etc.) Later on, you might have to read some books and papers to understand the problem better, but I think the problem should come first.  
To see an example of non-professional work, I recommend reading an article by Doris Schattschneider, "In Praise of Amateurs" (The Mathematical Gardner, ed. David A. Klarner, pp. 140-166). This article shows how modern day amateurs with no mathematical background, influenced by Martin Gardner's Scientific American column, disproved a number of early discrete geometry results, paving a way to a (correct) general result. Of course, the more advanced math you know, the more serious problem you can study.  
In conclusion, since Schattschneider's article does not seem to be available online, let me mention Freeman Dyson's article with the same title and a similar theme (amateurs in astronomy).  
A: I try to learn and understand as many facts as I can. Of course, many people would like to benefit from the opposite, that is, digging into a certain branch as deep as they can.
I try to do the opposite, which I see as my main advantage, as the opposite to professional mathematicians. This is because they have their own careers and has their professional criteria to fulfill (writing articles in journals, gaining citation points, etc.).
As an amateur I am not obliged to do so, and this is a great freedom. If you want to be creative, you may try to dig here and there, and probably you will be lucky to find certain problems which are not penetrated, or you may find just something interesting enough (for example, your own point of view on a well-known area, maybe you find a surprising connection and, even if it is well known, it is funny to discover it once more, etc.) to write it somewhere, maybe on a blog. 
In summary: I read as much as I can, I learn as much as I can, and I ask as much as I can. 
As regards to low-level entry (you need of course to be genius to discover it, but nothing more;-), an example is Feigenbaum's famous discovery about chaos, etc. As far as I know, he used only a programmable calculator to discover it. He was just inquisitive, nothing more, nothing less. 
A: One learns best by explaining things, and, when learning mathematics outside of academia, the main obstacle is that there is no one to talk to, no class to teach, etc. Therefore I use every opportunity to communicate online, and if I can't, I try to ask myself "dumb" questions and pretend that I'm both the lecturer that tries to explain the answer, and the student that does not get it.
Everything else, like getting the books and papers you need, is basically solved by knowing Google and Wikipedia :-)
A: I do mathematics as a hobby and have found various blogs and MathOverflow and Stack Exchange site Mathematics very helpful in continuing my education and revealing my limitations.
I have always enjoyed maths and have done well, but in the end that wasn't a good enough reason for me to make a career. I became interested in too many other things as well.
So what do I do?
First, I do puzzles and problems - I enjoy solving things and always have done it.
Second, I read textbooks - the demise of bookshops in favour of online resources is a bit of a menace here, because in a bookshop I could browse more easily for something interesting which appeared to be within my range and opened up an area I might be a little unfamiliar with.
Third, I do intentional study - for example to understand the classification of finite simple groups, or (as far as possible) the proof of Fermat's Last Theorem, or the Riemann hypothesis, or PvNP. But (to give a benefit of online resources) I do download a number of the papers linked in posts on this site. I did some algebraic geometry when I was younger and am now trying out Ravi Vakil's notes to get myself up to speed - but as a hobbyist I don't always have the time to consolidate what I've read.
What I do find is that I miss some of the informal knowledge (the stuff people talk about, but don't write down), so I don't always link things together as quickly as I might. And also I find that my intuition is not what it was - I think the exercises in good textbooks feed intuition by giving a sense of what is possible and what is not, and help to direct imagination in fruitful ways. I am not doing enough work to sustain my intuition at a high level.
However posting answers on sites like this does force me to commit myself in public, and I find that a learning experience, with lots of helpful (if occasionally sharp) comments and feedback. Since such self-learning is not in line with the research goal on MathOverflow, I indulge myself rather more on Stack Exchange, and tend to scan here for insights on things I might be reading at the moment.
A: Mathematical activity is driven primarily by intellectual challenge.  The infamous  mountaineering apology, "because it's there," applies to mathematical problems just as well as it does to mountains.  No wonder so many take up mathematics as a hobby.
A: Here's a couple of interesting articles about math as a hobby:


*

*Mathematical learning (and math as a hobby) by Rishidev Chaudhuri

*Mathematics as a hobby (math as a hobby) ? Yes – Really!
