I'm surprised at how strongly held most of the opinions are argued in the forum. The reason why is that I know of very little scientific evidence that would lend people such certainty. Firstly, as I understand it, to date, there has been no longitudinal study of mathematics talent in people who develop in interest after adolescence. Secondly, the neurological basis of learning is a rapidly growing field, but the field right now may not have much to say about the development of mathematics talent. The points that have been made: i.e. that you can't do it may be true, it's just that you probably should take what they say with a grain of salt since they are talking about a subject that is not very well understood.
I would wager, though, that going from having a small knowledge of math to going to a level of knowledge of comparable size to a mathematics researcher is going to be very hard. People who focus on math starting freshman year of high school go through 4 years of high school + 4 years of college + 5-7 years of grad school = 13-15 years before they become research mathematicians. So it might take you 10+ years before you know enough to start contributing.
Additionally, if you read psychological research on expert performance (which you probably should, and Eric K. Andersson is one of the experts in that area so perhaps start with an article by him), it normally takes 10 years to develop expert skills in a field, and that expertise comes about from something he terms "deliberate practice". Check out: (http://projects.ict.usc.edu/itw/gel/EricssonDeliberatePracti...) The Role of Deliberate Practice in the Acquisition of Expert Performance.
Most evidence would indicate that the abilities of a Putnam fellow are probably not out of reach to somebody who uses a strict deliberate practice regimen to develop their working memory (google that with 'mathematics') of and the ability to execute the thousands of tricks that mathematicians employ to solve problems. I am not very confident in this particular assertion, and I believe there is only weak evidence for it, however, there is only weaker evidence against it (and most people who argue against it make use of hand-waving has their primary argumentative technique).
On the actual task of becoming a mathematician I suggest you read some essays by Gian-Carlo Rota (http://web.archive.org/web/20070630211817/www.rota.org/hotai...), in particular look at the "10 Lessons I wish I had been Taught" and the reflections on math and mathematicians. He points out that mathematicians seem to have only a couple of tricks up their sleeve which they apply over and over and over again.
The points made by yelsgib are good since mathematics is about a community of researchers, and most problems have been looked at by hundreds of researchers. At the very least you will need to attend conferences at Universities.
This brings me to my final point, which is that if you want to make contributions to the field you'll make it a lot easier on yourself if you attend grad school in mathematics. In general people don't have to pay for math grad school, but it will cost you time. The reason why I say that it will make your life easier is that, firstly, math grad school will induct you into the community of mathematicians. You'll have better guidance than I can give (I am a lowly undergraduate) on how to become a mathematician, and when you get to doing serious research you'll have an advisor who will (hopefully!) guide you through it. One of my friends who is finishing up his P.h.D in analysis points out that he would have no idea whether he was making progress or not if it weren't for his advisor (he's doing research on semiclassical wave functions). Secondly, you'll know whether mathematics is right for you.
And, if you plan on making 'significant' contributions you'll have to do more or less the same preparation that a math grad students goes through and you'll have to put in the same, if not more time as grad students do in preparation to do research.
I admit that I am curious to see what happens if you go through with such a project. It would take a great deal of perseverance on your part as you come across all the barriers I am facing (as a math major) as well as those that come about because of your age and station in life.
My main suggestion is to try to get into correspondence with mathematicians to try to get more guidance. This can be difficult because you are probably just starting out, but you will most likely find one or two gracious ones if you start trying to correspond with mathematics departments.
I lack the experience to be a good person to talk to but I am nearly always willing to discuss this sort of thing so feel free to send me an e-mail to [my sn without the 'm'] [the a with a circle around it] uchicago [period ] edu.