This is something I've been wondering about for a while now. I hope I don't offend anyone with this question; but in addition to satisfying my personal curiosity, perhaps an answer to this question might also help to better target the answers we give.
There is a surprising number of questions on advanced subjects asked by people who patently don't have a clue what they're talking about. I'm aware that there are lots of people out there who don't have a clue what they're talking about, and also that there are lots of people studying advanced subjects, but I had no idea how large the intersection of these two sets apparently is.
In my experience (which is limited to a European academic context and might not be terribly relevant to non-European and/or non-academic contexts), people start studying advanced subjects after having studied less advanced subjects. While studying the less advanced subjects, they become familiar with basic aspects of mathematics, such as valid inferences, precise definitions and fundamental concepts. They also learn how to communicate their work to others. If they fail to absorb these basics, they neither get the opportunity, nor develop the motivation to study more advanced subjects. I've never met a single person in my life who tried to understand, say, the representation theory of Lie groups or the difference between a free module and a projective module, while being clearly unable to keep much simpler concepts than that straight in his or her head. Yet such people abound on this site.
So my question is basically: Who are these people? Answers should ideally include insights into and/or reasoned speculation on their background and motivation that might enhance our ability to help them (which might include giving more adequate explanations, or perhaps advice to study other things before returning to the advanced subject, etc.).
Here's one reasoned speculation that I've come up with; I'd be interested in comments on it. There's a conspicuous correlation between this phenomenon and a marked lack of proficiency in English. (This is in no way meant to be disrespectful towards people with low proficiency in English; I think you'll find from my answers that I go to great lengths trying to compensate for language problems.) So one possible explanation might be that universities in countries with generally low levels of proficiency in English tend to have less merit-based admission procedures; e.g., it might be easier to get into a high-level math degree program because one can pay for it and/or has political connections. This could then lead to people trying hard to master a course they've been admitted to "by birth" without first having developed the required skills. Another explanation based on the correlation with lack of proficiency in English might be that it merely seems as if they don't have a clue what they're talking about because they're unable to bring the basic structure of their thoughts across in English. Although this is probably a part of the explanation, I don't think it can be the entire explanation, since the phenomenon includes such things as not defining variables or using them inconsistently that I'd expect to be largely independent of language problems -- a very badly phrased definition would still be identifiable as an attempt at a definition.