[Edit by Anton: see the comment thread to this question for background]

@Pete Wow... Pete you have pushed me into a "limbo"... I dont want to give a long reply... But it seems i have no other choice...

I dont want to discuss any personal thing here, but I would like you to know, that I work from home, and am not a professional math. student... But I am a computer science grad and we have had several math papers as subjects of study... I am about to take math, but I am doubtful, because academics dont let you think out of the box... which is why I have delayed taking the course... I want to explore math but I don't believe in mathematical rigor when I am exploring (Chaitin's fan you may say)...

Regarding your comments:

(i) I am new to MO... as such I read the faqs, and accepted wherever I think the answer's correct... Is there something else I need to do (I mean click tick marks, etc.) ?

(ii) I have done lots of mistakes, and I acknowledge that and have acknowledged that before also... thats is why I have 7 versions of a paper to arxiv... I will even withdraw my paper if someone points out where I am wrong... But when the academic community grips you like this and tells me to master something and pointing out that I need to give attention, I feel a bit intimidated... Because thats exactly what I want to do...

All I want to know is what is true... and when discussions turn into a flame game, I really hate that... because that delays knowledge exchange...

"academics don't let you think outside the box" "I want to explore math but I don't believe in mathematical rigor when I am exploring"

I would normally try not to be a stereotypical academic when responding, but am rather tired after a couple of days thinking inside the box and trying to make a certain Ansatz rigorous. So: harrumph. And what Harry said.

(A more constructive comment may follow, but given the nature of rpg16's postings on MO, I can't bring up the necessary enthusiasm or sunny disposition at present.)

@rpg16: You seem to have construed my comment:

"@Roupam Ghosh: It is fine to be a learner. However, some of your behavior is alienating the research community: (i) not taking time to understand when people point out errors in your reasoning. [(i)' You have asked questions containing significantly flawed reasoning, which has been pointed out and corrected by several MOers, including me. You have never upvoted an answer, accepted an answer, or even clearly acknowledged your mistakes.] (ii) You have posted 7 versions of a paper to the arxiv, most recently tonight, on a subject you have not mastered. This is not what the arxiv is for."

as some kind of personal attack. That was not my intention. I don't know you at all, so I don't have any opinion of _you_ whatsoever, only of your mathematical writings. In that regard, what I described is not solely or even primarily my own personal response to what you have posted on MO and the arxiv, but rather is my educated and experienced guess as to how many professional mathematicians will respond to your writing. (This comment has several approvals attached to it so far, indicating that I was probably correct in my assessment that at least some other mathematicians feel this way.)

The reason I wrote that comment was that I see you headed down a path which will result in your being largely ignored by the professional mathematicians, in particular here on MO. (Note that already the leading analytic number theorists on this site are not the ones responding to your questions.) Your messages above display a sort of ambivalence towards that outcome -- maybe you can fly better if you are free and unfettered by the mainstream of mathematical orthodoxy -- but honestly that sentiment doesn't make a lot of sense in your case, because if you didn't wish to interact with professional mathematicians then you wouldn't be posting to the arxiv and MO (that's the whole point of both of those sites).

About the acceptance of answers: I see that one of the effects of my comment was that you promptly accepted answers to some of your questions, including my answer that as k approaches infinity, p_{k}/p_{k+1} approaches 1. Thank you for that -- accepting answers and upvoting is a good way to encourage people to answer more of your questions. However, in the deeper mathematical sense you haven't really accepted these answers, i.e. you haven't internalized that they are correct and modified your work accordingly. On page 5 of the latest version of your arxiv preprint, at a key step in a proof you write

limsup_{k -> \infty} (p_k/p_{k+1})^{1/k} < 1.

But my answer implies that this limit is 1^{1/\infty} = 1^0 = 1.

Do you see now why people feel that you are not listening to what they're trying to tell you? We're not trying to impede your creativity: we're trying to push you back on track.

Again, I hope you will take this message in the spirit in which it was intended, namely as information and possibly helpful advice. I am sorry if my frank talk dismayed you, and I expect that this will be the last I'll have to say on the matter.

Best regards,

Pete L. Clark

On two-way linking: I suspect I am not alone in coming here and being totally mystified as to what this discussion is all about. In order to save others the trouble, I will note here that this is a continuation of a discussion started at this question. Now, I only wish there were a way I could bump this comment up to the top of the thread …

@rpg.

Please go through Rudin's "Principles of Mathematical Analysis", and solve the exercises.

For elementary number theory, Hardy & Wright's book also might help you.

Of course, it will be good to learn some algebra, topology, etc..

After you do that, you will gain the quality called "rectitude of thought". Also you might benefit from joining a formal course somewhere.

You will have to ask your questions at the "ask a topologist" etc forums, though, if you decide to work through these books. MO is not the place, unfortunately.

With regards to the orthodoxy and thinking outside the box, let me just quote Carl Sagan.

They laughed at Galileo. They laughed at Newton. But they also laughed at Bozo the Clown.

rpg16, I was once stuck in a similar mindset as your own. I felt that doing problems out of books wasn't necessary - I just wanted to "understand" the minimum number of concepts necessary to "work" on unsolved problems myself, and I also wasn't rigorous in my investigations, I just wanted to get at the ideas and leave the "proving" part to others. I thought I had many great ideas to offer the mathematical community, and when people responded with (kudos to them, respectful) criticism that I wasn't ready to be doing such things, I was deeply hurt. I know from personal experience that it is a very difficult and depressing way of learning mathematics, since you are constantly viewing yourself as being outside of the community and only really understanding concepts at a very slow pace, if at all (due to the lack of solving problems from books). But with the help of some great, supportive people in the PROMYS program (Dave Hansen, especially), I got set on the right path, and realized that the "understanding" I thought I had and "work" on unsolved problems I thought I had been doing was hardly any such thing. It brought me down several notches in my own mind, to be sure, but that is nothing compared to the knowledge that one is finally doing things right. I worked through tons of problems, even ones I had previously dismissed as too easy or uninteresting, and (pardon the cliche) I eventually got to where I am today (and I don't mean in quantity of knowledge - being a sophomore, I obviously know very little in the grand scheme of things): I am a serious student of mathematics.

Harry's absolutely right: "You're right now at the point where you need to learn some mathematics before you continue to do mathematics." Just sit down, pick one of the books suggested (perhaps Number Theory by Andrews, or Introduction to Analytic Number Theory by Apostol), and start reading the book at the beginning, and doing the problems. If you need to learn some of the easy stuff in the beginning, then this will certainly help; and if not, then it'll be a breeze (and the practice won't hurt anyway). It may feel like attempting big things on your own, and trying to learn the small things on the way, is the best way of learning math; but this is simply not the case - you wouldn't decide to build a house and work on your hammering technique along the way.

Just to sum up what everyone is saying: We aren't making a personal attack on you - we are trying to guide you in the right direction. If you acknowledge that a great deal of work (doing plenty of problems, reading and rereading a proof until you understand it, etc.) is necessary to understand the big concepts, and are willing to accept the advice of those who are further along in their studies than you are, you're off to a great start in your mathematical journey.

(I do agree with Harry's point about not pronouncing on a subject one is a beginner in, but I mainly wanted to share my personal experience of being in this mindset - hopefully it will help rpg16)

@rpg: very nice result of this discussion. I'll share a drink in the evening on this :-)