All the easy math has been done. People have been working on Mathematics for a long, long time. To "make meaningful and significant contributions to the field" is one of the hardest intellectual tasks a person can do. It is probably impossible for someone at a midpoint in their lives to suddenly develop an interest in math and make a meaningful contribution. Even people who have been immersed in Mathematics their whole lives often cease to make major contributions in middle age.
You do have to be inherently gifted to be a good Mathematician. You even have to be really smart to be a mediocre one. This is hard stuff, the Olympic Marathon of intellectual pursuits. Do you have to be gifted to run in an Olympic Marathon? Someone that was merely determined might be able to qualify for it, but the ones that win are definitely genetic freaks.
You can certainly enjoy math without being a super-genious. If someone were dogged and creative enough and focused on a new enough subfield, they might even be able to contribute a little, provided they've kept their analytical facilities sharp in a field like Physics or Engineering. But then again, anytime a new field opens up there's hundreds of PHD students across the world that jump into it looking for a new bit of math to write their thesis on. It gets fleshed out pretty quick, and the low hanging fruits are the first to go, and the people picking the fruits are PHD math candidates.
Please note that this is just my opinion from being around Mathematicians. I'm not a Mathematician myself, but I was around them when I got an undergraduate degree in Mathematics. That degree taught me that Math was really hard and I would probably have a more significant impact on the world if I pursued something else.
I disagree with most of your points, though you're more qualified to talk on the subject than I am.
| All the easy math has been done.
How easy are we talking about? While I was in high school, I remember a news article about some other high-school-age kids that happened to a solve a long-standing and relatively simple problem in geometry involving triangles. I can't remember the specifics, nor can I find anything about it now, but I think there are still plenty of relatively entry-level problems to work on.
| It is probably impossible for someone at a midpoint in their lives to suddenly develop an interest in math and make a meaningful contribution.
It's a pretty crappy list, and most of the names are from centuries past, but there are a couple of interesting entries there.
| Do you have to be gifted to run in an Olympic Marathon?
Your example of Olympic marathons is specifically a zero-sum game, whereas mathematics is not.
| But then again, anytime a new field opens up there's hundreds of PHD students across the world that jump into it looking for a new bit of math to write their thesis on.
This seems to contradict your previous point that mathematics has become inaccessible. There are a few differences between a grad student and a sufficiently dedicated hobbyist, and none of them are genetic, nor are any of them necessarily bound by a particular age bracket. Indeed, someone in middle age could attend university on a specific curriculum, and in a few years be looking at the same problems as the PHD students. The older person might have some advantages in self discipline or experience in tangential fields.
> There are a few differences between a grad student and a sufficiently dedicated hobbyist, and none of them are genetic, nor are any of them necessarily bound by a particular age bracket.
This raises an interesting empirical question: is there a significant deterioration in brain performance after about 40 years of age or so? Anecdotally it would seem to be the case. The Fields Medal is the highest award in Mathematics and it has an age cap of 40, but the rule hasn't raised controversy because almost all the worthy contenders have been under 40 anyway. However, some people have hypothesized that the dominance of the young is the result of other career and family concerns distracting people in middle age. That could well be the truth, and I hope it is. It would be interesting to know the answer.
You do have to be inherently gifted to be a good Mathematician. You even have to be really smart to be a mediocre one. This is hard stuff, the Olympic Marathon of intellectual pursuits. Do you have to be gifted to run in an Olympic Marathon? Someone that was merely determined might be able to qualify for it, but the ones that win are definitely genetic freaks.
You can certainly enjoy math without being a super-genious. If someone were dogged and creative enough and focused on a new enough subfield, they might even be able to contribute a little, provided they've kept their analytical facilities sharp in a field like Physics or Engineering. But then again, anytime a new field opens up there's hundreds of PHD students across the world that jump into it looking for a new bit of math to write their thesis on. It gets fleshed out pretty quick, and the low hanging fruits are the first to go, and the people picking the fruits are PHD math candidates.
Please note that this is just my opinion from being around Mathematicians. I'm not a Mathematician myself, but I was around them when I got an undergraduate degree in Mathematics. That degree taught me that Math was really hard and I would probably have a more significant impact on the world if I pursued something else.