I am a research mathematician, a professor at a US university. Anyone can do research. It is much harder to do research that other researchers will read and ta

I am a research mathematician, a professor at a US university. Anyone can do research. It is much harder to do research that other researchers will read and take an interest in. "The strict formats, all the requirements before you can publish, the references you have to use, the nitty-gritty of writing essays, and the conditions that seem to be part of the status quo and hidden secrets between universities": If you want to write a paper which others will read, then you have to learn what topics others are interested in reading about, what other work has been done in the field, and how to write up your results so others will find them interesting. You would learn all of this (and much more!) in a good PhD program. But if you don't care about recognition? Then all doors are open to you. Just do it! I'm quite serious. Never mind what others have done; maybe you're duplicating something that someone else did seventy years ago, but if you're not gunning for an academic job do you really care? You'll need to accept that no one might ever take a serious interest in what you've done, but the joy of discovery and problem-solving is available to you, as it is to everyone.