I started my Ph.D. at age 27. I would have made a lot more money during my career if I had instead just joined a large company like Microsoft because I was a good programmer, but I really have enjoyed mathematics.
I think that math is a "young person's game" because it is intellectually demanding requiring a fair amount of energy to process what is going on. It takes years to learn the background necessary to do the research, so I think that most discoveries are made by mathematicians in the age range of 30 to 45.
I'm 55 and I have mostly given up on doing research. I just don't have the energy and I've never been a professor, so I just don't have the time to do it for fun anymore. My former advisor continues to publish in his 70s.
to get a PhD is many times easier than to "accomplish groundbreaking work"
Shminux, I want to disagree with this statement. I guess it all depends on what you mean by "groundbreaking work". First, it seems to me that most of the Ph.D. dissertations that I know of entailed discovering new theorems. Second, in my math department, it took 7 years on average to get a Ph.D. It often takes a lot less time to discover and write up a good, popular paper (say one with over 20 citations).
If you don't consider a paper "groundbreaking" unless it has over 100 citations, then maybe you are correct. I just don't know.
I'm not sure what you mean by "a basic formal education". Ramanujan attended Government Arts College, Kumbakonam and Pachaiyappa's College and then he became a mathematical "researcher at the University of Madras". About 3 years later, Ramanujan wrote his first letter to Hardy. It is certainly true that he did not have the normal formal background of a Ph.D. student, but I think he did take a number of undergraduate math classes.