The great thing about math research, or really research in any field, is that it's highly parallelizeable. There are a few geniuses like Terry Tao exploring spe

The great thing about math research, or really research in any field, is that it's highly parallelizeable. There are a few geniuses like Terry Tao exploring specific directions of thought, but there are so many things to think about and problems to solve that even the best researchers can only focus on a very small fraction. For every genius there are hundreds of other researchers who are only, say, 10% as productive (this is maybe your typical professor at a top-20 school), but through virtue of numbers they can explore a much wider range of problems and have a large aggregate impact. Fetishizing of genius makes for fun pop-science articles but is not really reflective of how the bulk of research gets done. Of course, even if it doesn't require genius, making serious research contributions in mathematics does typically require many thousands of hours of training and hard work (usually in the form of a PhD) which most people are not going to dedicate their lives to. Given this, I really like Bill Thurston's perspective, from the famous essay "On proof and progress in mathematics" (http://arxiv.org/abs/math/9404236 http://arxiv.org/abs/math/9404236), which is that proving new theorems is only a small part of the project of mathematics. The real goal is to increase humanity's mathematical understanding, which does not necessarily grow monotonically: there are plenty of results proved 100 years ago that are not known or understood by any currently living mathematician, thus have been "lost" to the mathematical community. And of course there are new people born every day who know no math as all, so understanding of mathematics will tend to decrease unless people actively work to sustain it. In this view, understanding and communicating mathematical insights is just as important and valid a part of the intellectual process as proving new theorems, and requires a different set of talents. Plenty of great researchers are not great communicators, and vice versa. (Terry Tao actually is an excellent writer, but again he's only one person and there's a ton of math out there to be shared). You may not know as much math as Terry Tao, but anyone on HN probably knows more than the vast majority of adults (the bar here is really low), and the world at this moment is certainly not suffering from a surplus of mathematical thinking. So whatever your background and abilities, there are almost certainly lots of useful ways to contribute.