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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.

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    $\begingroup$ I must admit that this is a question that I had for a long time, too, but I don't think there's a good answer. It leaves me absolutely speechless that there are people asking about loop groups and Eilenberg-Mac Lane spaces but unable to compute the fundamental group of the diagonal of a space; or asking about adelic groups, trace formulae and bornologies, but unable to prove that a holomorphic function can be developed up to its nearest pole. I don't know how one may even know that the former concepts exist without knowing where to look up the latter if unable to prove them immediately. $\endgroup$ – t.b. Jul 18 '11 at 15:05
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    $\begingroup$ I have to admit that I am a very lucky person that my teachers forced me to hold back and take it one stepping stone at a time (I still jumped around, and often fell, but eventually the message stuck and I figured slow can be good too). Also, it is a good thing I only found out about MO on the very last week of my B.Sc. degree (and that MSE opened a month later), or else I would have been one of those people you ask about :-) $\endgroup$ – Asaf Karagila Jul 18 '11 at 22:50
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    $\begingroup$ Another possible type of explanation involves lack of educational opportunity. Someone who has been prevented by circumstance from studying mathematics formally develops a fascination with a certain mathematical problem. Lack of background produces lack of perspective, but the fascination remains. $\endgroup$ – André Nicolas Jul 18 '11 at 23:53
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    $\begingroup$ @Asaf, I have to admit, you should have started your comment with "I must admit" to keep it in line with my and Theo's previous comments. $\endgroup$ – Eric Naslund Jul 19 '11 at 15:56
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    $\begingroup$ @Eric: I have to admit that you are correct. However I had to admit despite the fact it was optional so using "I must admit" seemed a bit strong :-D $\endgroup$ – Asaf Karagila Jul 19 '11 at 16:19
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    $\begingroup$ Would this be an instance of the en.wikipedia.org/wiki/Dunning%E2%80%93Kruger_effect Dunner-Kruger effect? $\endgroup$ – Jonas Teuwen Aug 17 '11 at 18:52
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    $\begingroup$ @Andre Nicolas I am one of those who is in such circumstances yet I feel very insane to talk about something I do not understand. I 'was' a person who did that during my school days. But as time went on that insanity faded and apparently when I started my B.E. I was ready to focus on real and basic mathematics starting from the definition of a group etc.. :-) I too had a lonely 3 years almost with no help yet there are sources: summer schools, e-mail advice from Profs, Grad students in premier institutions in my country which helped me a lot. $\endgroup$ – Dinesh Sep 6 '11 at 6:14
  • $\begingroup$ continued.. Today I am very glad i am out of 'that' bizarre world as I was in it in my school days! $\endgroup$ – Dinesh Sep 6 '11 at 6:14
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    $\begingroup$ I ask stuff about topoi and non-standard models of arithmetic, but every time I read "compact" I have to check wikipedia for the definition. I used to have problems to distinguish it from "bounded" too. I can fish for arguments, but I couldn't prove any relevant theorems in complex analysis. I'm a physicist for god's sake. :D $\endgroup$ – Nikolaj-K Nov 3 '12 at 1:49
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    $\begingroup$ Und Sie, lieber Felix, was sind Ihr Hintergrund und Ihre Beschäftigung ? Infolge der Qualität Ihrer Antworten bin ich schon lange darauf neugierig... $\endgroup$ – Georges Elencwajg Dec 14 '12 at 12:38
  • $\begingroup$ And more to the point, your question and conjectures are incredibly relevant, interesting and challenging. $\endgroup$ – Georges Elencwajg Dec 14 '12 at 12:48
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    $\begingroup$ I have had this exact though, in many technical areas (not just mathematics). In particular, on the MATLAB forums, near the end of terms you will often see users asking for "the codes" for complex projects -- not simple things like the implementation of an LU factorization algorithm, but fairly involved projects such as automated character recognition of license plates, or something along those lines. I've always wondered what goes through someone's mind to think: a) anyone cares that it's a homework project with a deadline, b) the solution is so trivial as to be sitting out there, prepackaged $\endgroup$ – Emily Dec 14 '12 at 18:27
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    $\begingroup$ BTW, I am much more concerned with the fact that these people already occupy an absolutely disproportionate of attention in real-life and on the net than with the supposed need for more "help" that this question posits. $\endgroup$ – Phira Dec 17 '12 at 18:13
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    $\begingroup$ A non-empty (there is at least one person belonging to it: I am not attending any course but I try to amend my ignorance a little bit by myself, while waiting for the opportune time to enroll at university) subset may be the set of self-teaching people from non-English speaking backgrounds who are not accustomed to the use of English in mathematics. $\endgroup$ – Self-teaching worker Nov 8 '15 at 14:11
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    $\begingroup$ You know those cranks who think they can disprove special relativity by thought experiments about airplanes on treadmills and rotating mirrors? They exist in math too. Even ignoring the completely nuts ones, there are still quite a few posters who think that, say, the Birch-Swinnerton-Dyer conjecture is totally within their capacity to prove if someone would patiently explain the definitions and symbols to them. They're the same sort of people who insist that everything is explainable to a 5-year-old, and mathematicians use complicated jargon and definitions just from sheer bloody-mindedness. $\endgroup$ – anomaly Jun 6 '16 at 21:00

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I don't think anyone has mentioned the role of the internet here. Even ten to fifteen years ago, it would have been very difficult for a non-expert to even see the terminology of advanced mathematics (the same is true for other fields). The only written descriptions were in books in university libraries, journals, and privately-distributed lecture notes, and few people had access to these.

Popularizations were almost the only way for a non-academic to get a glimpse of a field. For example, I know several non-mathematicians who read Gleick's Chaos to get a glimpse of that area. These people would never look at advanced expositions of the field, so they would not be able to ask advanced questions about dynamical systems.

Today, many advanced books are available online, preprints of research papers are everywhere, and lecture notes are easy to find on google. This speeds up access to new research. It opens up the huge amounts of knowledge to a wider audience, some of whom would otherwise not have been able to access it (for example, students in developing countries where budgets are tiny). And it gives people a better perspective on what mathematicians actually do than books like Chaos.

But the old system had the advantage of guiding students into the advanced material. Students couldn't ask about things they hadn't seen, and they usually wouldn't see things before they reached the right stage in their studies. Now, a student can jump directly into a graduate-level preprint without going though the courses that led up to the material. Of course the student will not know the basic background results of the field if they do that, but at the same time they may be more motivated to learn more mathematics if they can see where it goes.

It used to be that to see the view from any of the Adirondack mountains in New York you had to climb it yourself, often on unmarked trails. In 1935 they finished a paved road to the top of Whiteface mountain. I'm sure there are people who climb other mountains in the area after driving up that one and seeing how nice the view is from the top. And now we can show children who are too young to climb the sorts of views they will be able to access once they grow up. But there are still some people who grumble about how the road makes it too easy for tourists to get to the top, and how people who drive up miss the climb, which is just as interesting as the view.

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    $\begingroup$ What I like (and dislike) about this answer is that it makes these questions seem as both a good thing and a bad thing... :-) $\endgroup$ – Asaf Karagila Jul 20 '11 at 16:19
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    $\begingroup$ To say nothing of the effect of Wikipedia! I learned most of the differential geometry I know from there (and various introductions to general relativity), but my understanding is patchy. To the extent that I didn't realise it: for instance, it wasn't until this year that I understood that there is a difference between the tangent bundle and the vector space of its global sections. (Obviously, the notion of a non-global section never really crystallised in my head. Oddly enough, it was studying sheaves in algebraic geometry which finally led to this personal conceptual breakthrough.) $\endgroup$ – Zhen Lin Jul 20 '11 at 17:06
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    $\begingroup$ @Zhen: You should be aware, however, that wikipedia articles about mathematics sometimes contain severe mistakes. You might just learn wrong things. $\endgroup$ – Hendrik Vogt Jul 21 '11 at 7:50
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    $\begingroup$ In general the articles about mathematics have less mistakes than about other disciplines, I think. And it's a normal practice eminent mathematician providing links to Wikipedia in their blogs. $\endgroup$ – Américo Tavares Jul 21 '11 at 11:55
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    $\begingroup$ Very good answer. $\endgroup$ – Américo Tavares Jul 21 '11 at 11:57
  • $\begingroup$ @Zhen I was interacting with this fairly famous algebraic geometer who made fundamental contributions to it. When he asked me something about analytic continuation I fumbled and he immediately asked where did you learn this from, I said Wikipedia. He advised me not to use it any more for learning mathematics and even mocked me for doing so in some funny discussions :-D His opinion is that anyone can write and edit in wikipedia and it makes it vulnerable to lack of precision and/or completeness and/or mistakes. $\endgroup$ – Dinesh Sep 6 '11 at 6:23
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    $\begingroup$ @CarlMummert It is funny how you mention Gleick's Chaos, which is the book I read and that made me take up mathematics even more vigorously than before. And what you say is exactly true, good textbooks are very expensive to procure and the ones that people in developing countries can procure are 20 to 30 years old atleast. For up-to-date material, arxiv and similar sources are the best source, and hence the effect. $\endgroup$ – user14082 Feb 11 '13 at 19:36
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I suspect that we are all guilty of this. It is just that when we overextend ourselves in to regions of mathematics that are not yet supported by our skills base we don't know it anymore than the people you have noticed. There is a mathematician somewhere who likely would say the same about your questions and about mine. Who are these people? They are us, in the past and in the future. They are me at age five telling my peers breathlessly about infinity even though I couldn't multiply three digit numbers. They are me on the subway straining to read Baby Rudin in 9th grade, and feeling very proud to have such a "serious" book in my lap.

And, as long as it isn't the result of students being passed along in programs without understaing the course work, then fearfully searching for solution online once in over their heads (a sad state of affairs) -- I think it's a good thing that mathematics has shiny objects that attract admirerers.

I can't wait to ask and answer questions about higher topics, I love taking a peek at them. It keeps me going. When I work with someone who is doing the same thing, a grade school student who wants me to teach them calculus, I'm very grateful to have a curious mind.

Some folks grow up in small ponds, they think because everyone around them can't even do algebra they must be very brilliant, and their friends and parents say so. My Husband thinks anyone who can master Calculus must be some kind of mastermind. until I was connected to a community of mathematicians I might have thought he was right.

There should be nothing exclusive or eleist about mathematics. And I don't think we should be surprised that many folk are isolated from what comes to be common knowledge to anyone who is a grad-student or a regular in a college mathematics department.

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    $\begingroup$ I somewhat disagree with what you said. I met quite a few professors of mathematicians that when extend themselves into unfamiliar regions, they pick up an introductory book (or ask introductory level questions, they might be directed to a certain purpose - but they are introductory level nonetheless). I can say that when I try to understand something which is completely unfamiliar to me I keep in mind what is the thing I'm after, but I do try to build the required basic knowledge basis first. I am surely not the only one doing so. I disagree that we are "all guilty of this", this is untrue. $\endgroup$ – Asaf Karagila Jul 19 '11 at 7:13
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    $\begingroup$ Right. Asking a naive or ignorant question about a subject that one is ready to study is very different from asking a genuinely meaningless question about a subject that one is not ready to study. (That said, I also don't think that meaningless questions are the worst thing in the world, as long as the questioner is open to the fact that maybe he is in over his head.) $\endgroup$ – Jesse Madnick Jul 20 '11 at 6:33
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    $\begingroup$ This answer brings me back to buying baby Rudin in 9th or 10th grade at a giant used book sale at the fairgrounds. I didn't get very far into it at the time, but it was only 10 cents, which saved me a lot of money when I got to college. $\endgroup$ – Noah Snyder Jul 26 '11 at 17:28
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Lieven le Bruyn has some remarks related to this issue in this blog post:

When given the option, students prefer you to tell them monstrous-moonshine stories even though they can barely prove simplicity of $A_5$, they want you to give them a short-cut to the Langlands programme but have never had the patience nor the interest to investigate the splitting of primes in quadratic number fields, they want to be taught schemes and their structure sheaves when they still struggle with the notion of a dominant map between varieties...

In short, students often like to run before they can crawl.

...

Perhaps, it is time to promote slow math...

So perhaps this phenomenon is more universal than you think. Its prominence on math.SE may simply be a matter of selection: given that such students exist, they are more likely to ask questions on sites like math.SE than other students.

Anyway, here is my own speculation: graduate programs in certain countries seem to thrust their students into research without seeing if they have adequate preparation for such a thing, and the support system does not seem to be as strong for such students. Of course this is just speculation based on some stray observations.

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    $\begingroup$ Then why have exams if slow is better most of the times? $\endgroup$ – Damien Jul 18 '11 at 23:37
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    $\begingroup$ Exams force students to be honest about their studying efforts. In an ideal world they would not be needed. $\endgroup$ – Noteventhetutorknows Jul 19 '11 at 1:15
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    $\begingroup$ @Damien: did I say anything about exams? I don't really believe in exams myself. The ideal way to judge if someone has learned something is oral interview, I think, but of course that doesn't scale. $\endgroup$ – Qiaochu Yuan Jul 19 '11 at 1:54
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    $\begingroup$ @Qiaochu Yuan: No you didn't...I was just asking a rhetorical question based on what you wrote. $\endgroup$ – Damien Jul 19 '11 at 2:00
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    $\begingroup$ @Qiaochu: At least in these parts graduate level courses are open to undergrads also - there is no sharp division. The students are expected to get the hint from the listed background requirements. Here you don't need to formally sign up to a course, because (unlike in the US IIRC?) the tuition fee (non-existent here) does not depend on the number of classes you attempt to take. The professor will gently suggest to such a student that may be they should wait a year or five, but this type of a person will likely view that as a challenge... $\endgroup$ – Jyrki Lahtonen Jul 19 '11 at 7:49
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    $\begingroup$ I think it's especially bad in the UK where graduates go into a PhD without proving they have the necesary knowledge in all areas via an exam, as in the US. I'd say the US graduate programs are the best in the world, and it shows with the top 20 universities in the world being placed by 17 US ones. $\endgroup$ – user10389 Aug 25 '11 at 12:31
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    $\begingroup$ @Qiaochu Yuan I second you in the opinion about 'oral interviews' $\endgroup$ – Dinesh Sep 6 '11 at 6:44
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Who are these people?

Well there was a user named "trust god" on Math Overflow, who had an entire meta thread created solely to discuss whether or not his behavior was acceptable.

Why these questions?

I think part of it is ego, part of it is lack of patience. When you see all of these prominent mathematicians talking about advanced topics, and discussing material you don't understand, it is natural to want to learn about those topics too. This happens for many areas, a young hockey player will try to copy the moves of one of their idols, and it is no different for mathematics. The problem is you often need patience as well, and you need to spend a lot of time before you will be successful, and it is just patience these people do not have. They want to understand it now, and saying that it requires years of study first just isn't what they want to hear. This goes into the all too common general mentality of wanting things immediately, without work, without effort, and without spending time.

"Patience is waiting. Not passively waiting, that's laziness. But to keep going when the going is hard and slow - that is patience."

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  • $\begingroup$ I see that Theo tried the "giving advice to study other things before returning to the advanced subject" approach with a nice quote but limited success :-) $\endgroup$ – joriki Jul 18 '11 at 14:29
  • $\begingroup$ @Eric: Two bits - firstly, how did you know that he was both trust god and iyengar? Secondly, I agree fully that people are impatient. Math is hard. One of the problems I encounter most when I am teaching math is a profound lack of patience. But how does one make someone more patient? $\endgroup$ – davidlowryduda Jul 18 '11 at 21:19
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    $\begingroup$ @Mixedmath: To answer the first one: He kept asking about the Birch and Swinnerton dyer conjecture, and has the same description in both. For the second, I wouldn't just say patience is important in mathematics. It is important in just about every aspect of life. I don't know how you can teach it, it is similar to maturity. "Have patience. All things are difficult before they become easy." I also quite like: "Patience is waiting. Not passively waiting. That is laziness. But to keep going when the going is hard and slow - that is patience.” $\endgroup$ – Eric Naslund Jul 18 '11 at 21:39
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    $\begingroup$ This coupled with how easy it is to trick yourself into believing you understand a concept and I can see how these questions arise. $\endgroup$ – ttt Jul 18 '11 at 21:50
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    $\begingroup$ @Eric: see en.wikipedia.org/wiki/Stanford_marshmallow_experiment (and the references therein). $\endgroup$ – Qiaochu Yuan Jul 18 '11 at 22:20
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    $\begingroup$ @Qiaochu: Thanks for the link, that is an amazing experiment!! $\endgroup$ – Eric Naslund Jul 18 '11 at 22:50
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    $\begingroup$ @Qia Consider an analogous experiment here. Delay the posting of your answer by 15 minutes and get a random rep boost. Could the "fastest guns" resist? Would we get higher quality answers or toasted marshmallows? $\endgroup$ – Bill Dubuque Jul 19 '11 at 3:10
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    $\begingroup$ old saw: "Patience is a virtue. / Possess it if you can. / Seldom found in women. / Never in man." :-) $\endgroup$ – Mike Jones Aug 24 '11 at 23:17
  • $\begingroup$ Well, to say something @WillieWong was the only person, who never tried to hurt others. A moderator should be like that. I always like his attitude, even though some times I crossed my limits, he said in the way that don't hurt others. There is a way of saying everything ! I really thank you Wille $\endgroup$ – IDOK Dec 14 '12 at 12:25
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    $\begingroup$ @Iyengar: while I thank you for your confidence, as many will point out that such sentiments are rather undeserved on my part. Separately, however, I implore you to tone down your language; at least give Eric a chance to clarify what he means. $\endgroup$ – Willie Wong Dec 14 '12 at 12:34
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    $\begingroup$ @Iyengar: note that Eric's comment is not recent. He posted this more than one year ago. This is not a case of someone dredging up your old behaviour and flogging you again for it. This is simply the case of "things posted on the internet lasts a long while." I'll let Eric talk about the content of what he wrote himself. $\endgroup$ – Willie Wong Dec 14 '12 at 12:46
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    $\begingroup$ @WillieWong Is there a new rule that mods have to post a "This is my personal opinion and does not necessarily represent..."-disclaimer? I always assumed that is understood. $\endgroup$ – Michael Greinecker Dec 14 '12 at 13:12
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    $\begingroup$ I will shut my mouth, once if Eric edits the answer removing my name, and continue writing it in a normal way, without pointing out some specific X or Y . $\endgroup$ – IDOK Dec 14 '12 at 17:01
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    $\begingroup$ @EricNaslund : Well, removing iyengar is not enough, trustgod and iyengar are same , both refer to me, its like exploiting your MO moniker and then arguing that this is Math.SE not MO. Well, can't you frame out an answer without using any references to people ? This will have a serious impact on my name. If it is your thought, keep it to yourself. $\endgroup$ – IDOK Dec 14 '12 at 17:46
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    $\begingroup$ I find that this last of Iyengar's comments contains all the constructive content of the others but is far less rude and offensive. I have taken the liberty of removing the others. Gentlemen, play nice. $\endgroup$ – davidlowryduda Dec 14 '12 at 19:09
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I think I have met a couple of students like this in real life. I'm bad at reading people's motives, but I think basically they want to impress. Either the lecturer or their peers or themselves. All without realizing that their questioning reveals these huge gaps in their background, and has rather the opposite from intended effect. Also they cherish a mistaken idea of what math is all about. They think that citing (or copy/pasting) deep theorems adds to the scholarly value of their essays and such.

As an example I remember one such eager beaver student coming to my office to discuss a practice essay I was assigned to supervise. Normally I need to chat with a student a while to figure out a topic that fits their background and that would interest them enough so that motivating them would not become too much of a burden. Not this guy (IIRC in his sophomore/junior year). He expressed in unmistakable terms that he wants to write an essay about primality testing (this was shortly after AKS had gained some media attention). Ok, fine that saves me a bit of work. So I grab a suitable book and pick a topic that fits his background. The assignment is about Rabin-Miller (or something else at about that level). He looks (in retrospect) a bit disappointed, but starts working.... A few weeks later he comes to my office asking for a good reference giving a more accessible description of the Artin symbol...

Some of them may learn. They are interested, but they are just channeling their interest in the wrong way.

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There are popularized books on advanced topics, much more so in physics than mathematics, written by famous, by the standards of science, faculty at prestigious institutions. If you read any of these books and then ask yourself what problems you can actually solve using what you've learned from this book the answer is almost always none. If a person does not have the background to realize this then that person might actually believe they have some understanding of the topic. To paraphrase a former U.S. Secretary of Defense, "They don't know what they don't know."

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I know full well that I am one of the perpetrators. And while I am delighted to have "found" math at this stage of my life, I bemoan the circumstances that I self-study on my own, with no viable math program geographically at hand.

I do miss the lack of a rigorous study regime. There is definitely an advantage studying in an academic context where virtue is rewarded and the table is set for you.

But I know full well my own failings and continually look for an entry point for material that is interesting and doable. My typical experience is that I am good for the first 20-30 pages of a text, and then hit the wall. So your general theme pervades my thoughts on a daily basis.

In my own defense, I don't think I am at the most deficient end of the spectrum you lay out in that I feel I can articulate proficiently, and I do not delude myself. I know when I am confused and lack understanding.

So my constant inclination is to want to start over. Thus my problem is coming up with practical solution.

I am perfectly willing to start over. In that regard, where should that point be? And I am somewhat unenthusiastic studying from 500 - 1000 page theorem/proof textbooks where there is little motivation, although they do have some very beautiful material.

Two recent experiences of things I loved and would like to pursue were Samuel's "Algebraic Theory of Numbers," where I did OK till almost the end of Chapter 2, and then was completely lost. Or Marcus's "Number Fields," which looks great, but every inch of the way he says "prove it." Wish I could, so maybe that shows I have no business being there.

Yet I found "Atiyah & Macdonald" undoable when trying to upgrade my algebra skills.

So for me, realistically knowing where I stand, I long for someone to give me an explicit directive: learn this specific material, from this specific source, do "x" problems, a don't show up till you've mastered it.

That would be a great relief, motivation, and inspiration.

Thanks for opening this discussion.

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    $\begingroup$ Dear Andrew, It is not easy to learn to construct your own proofs in commutative algebra or algebraic number theory (or any part of mathematics) if you haven't trained to do this, so don't feel too discouraged. Have you tried working through more elementary algebra texts, such as Dummit and Foote, or Fraleigh, or similar? If you are reading (even parts of) Samuel and contemplating Atiyah--Macdonald, then the material in these easier books is probably familiar, but perhaps the exercises will be helpful (D&F especially has many good ones), and could be a way to work at improving your ... $\endgroup$ – Matt E Feb 12 '13 at 6:15
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    $\begingroup$ ... skill at arguing. While A&M is a beautiful book, it is also notoriously terse. Have you looked at Eisenbud? It is much longer, but is gentler on a line-by-line basis. Also, have you looked at Ireland and Rosen? This is a beautiful introduction to algebraic number theory, which is perhaps gentler than some others. (I forget whether or not it has exercises.) I think that if you asked a question like this on the main site (but slightly more specific --- perhaps just focusing on one topic a time, maybe algebraic NT or commutative algebra) as a reference request, with some background ... $\endgroup$ – Matt E Feb 12 '13 at 6:18
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    $\begingroup$ ... and explaining what has and hasn't worked for you so far, you might get other good suggestions (probably more than you'll get posting here on meta). Best wishes, $\endgroup$ – Matt E Feb 12 '13 at 6:19
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    $\begingroup$ Dear Andrew, I do not think you're one of the perpetrators, I think that one quality that people (if they do indeed exist) described in here lack is the ability for serious self-doubt. As your introspection in this post shows, you do not lack it, and you're pretty aware of your limitations. I think that your example can be inspiring to all of us, and personally I can only hope that I will have the same persistence and will to learn new things (difficult as mathematics) at your age. (I hope this isn't offensive in any way, since I surely didn't mean it that way) $\endgroup$ – user5501 Feb 12 '13 at 7:45
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    $\begingroup$ @MattE Thanks very much for generously taking the time to consider my situation and give me your advice. I really appreciate it. Best regards, Andrew $\endgroup$ – user12802 Feb 12 '13 at 12:12
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    $\begingroup$ @Q__ Dear Q__, Thanks for your warm encouragement. Best regards, Andrew $\endgroup$ – user12802 Feb 12 '13 at 12:13
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    $\begingroup$ @Andrew I have one word to say: respect. I really admire these qualities, even though I am an undergraduate I fell nowadays not many students have the patience to work through most background material before tackling more advanced subjects. Post more questions/comments if you have, and try I will to answer them! I hope my post on the class group was helpful to you. Regards, $\endgroup$ – user38268 Feb 13 '13 at 14:57
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    $\begingroup$ @BenjaLim Thanks very much for your kind words. I have often seen your excellent answers (both in the math and in your manner of expression), and am sure you will do great things in all regards. All the best. $\endgroup$ – user12802 Feb 13 '13 at 18:21
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I don't think it's a social phenomenon specific to maths but something all human beings engage in within a social construct: They modify it via their stupidity, ignorance, misunderstandings, creative brilliance to create a new ones, with the older generation tearing their hair out at the social vandalism.

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    $\begingroup$ Cicero: "O tempora! O mores!" - I've up-voted this answer. $\endgroup$ – Mike Jones Aug 24 '11 at 23:28
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    $\begingroup$ @mike I honestly don't know why this got down so bad, but thanks for the up vote $\endgroup$ – user10389 Aug 25 '11 at 12:24
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I actually was (and maybe still am) one of those people whom you are talking about. What happened to me is that I started studying my bachelor's degree in mathematics at one university, where they didn't teach very well and therefore, most students were happy to get a 60% (passing grade). I really didn't feel like I understood very much, so I began to study my bachelor's degree from zero at a different university (Open University), where I feel like I've been learning much more deeply, and receiving better feedback from my lecturers which makes it easier for me to truly understand a subject. Many people simply made the wrong choice of university or chose to learn in a way that doesn't suite them and therefore, understood nothing. Also I've noticed that some of the users on the site are still in high school and therefore, usually don't yet have a formal education in mathematics.

In regards to the bad English, as someone who has no experience beyond this site in reading or writing mathematical proofs and questions in English (I'm an undergraduate, and at my university, all of the books and lectures are in Hebrew and I don't know any mathematicians who don't speak Hebrew), sometimes my lack of mathematical terminology and proof writing experience in English (which I've received multiple down votes for) can be seen as ignorance or lack of understanding in the subject I'm writing about. I'm pretty sure that many people learning some form of mathematics, who have little experience writing mathematics, can relate to what I'm saying here.

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    $\begingroup$ Thanks for this answer! I can assure you that if you ever were, you're no longer one of the people I was talking about :-) I checked out your questions, and they mostly seem exemplary to me, with only minor English glitches here and there and very clear expositions of what you tried and what you need. Also you seem to know what you're taking about, and to be aware of what you know and what you don't know; whereas the people I meant talk about very advanced stuff without understanding even very basic things. (By the way, I only found one downvote on one of your answers and compensated it. :-) $\endgroup$ – joriki Jun 6 '16 at 20:46
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    $\begingroup$ דרך אגב, אני מדבר עברית, אז אם את צריכה עזרה עם משהו באנגלית, אל תהססי לפנות אליי :-) $\endgroup$ – joriki Jun 6 '16 at 20:46
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    $\begingroup$ @joriki: $\Huge\odot\odot$ $\endgroup$ – Asaf Karagila Jun 7 '16 at 5:42
  • $\begingroup$ קטעים. Ich spreche manche Deutsch, aber mein Deutsch ist sehr schlecht $\endgroup$ – Noy Soffer Jun 7 '16 at 8:20
  • $\begingroup$ דרך אגב, תודה, אשמח לפנות אליך, אבל אין לי מושג איך עושים את זה באתר הזה. $\endgroup$ – Noy Soffer Jun 7 '16 at 20:02
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    $\begingroup$ בגדול, משאירים תגובה לפוסט כזה או אחר. האתר הזה הוא לא ממש רשת חברתית -- וזו אחת הסיבות שאני אוהב אותו. $\endgroup$ – Asaf Karagila Jun 7 '16 at 21:06
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I'm probably one of them. I know, probably I don't have any idea of what I'm talking about on SE. But I can say you what bring me there every day.

I studied a bit of algebra at school, easy stuff and I never went to the university. But since I was young I always destroyed my brain with impossible mathematical problems, and played with abstract structures. I really love the abstraction, finding structures between abstract objects. So when I face the huge amount of advanced math. on internet I'm totally enchanted. When I was a school professor make me "almost" to fear advanced topics only because I must start from the basis before..probably they where always right.

But I needed few time to understand the math they gave me , and faster then other classmates (7 years old I was thinking about monomial and how $ \lim\limits_{x\to 0^+}1/x=+\infty $ even it they said that $b\over 0$ has no meaning, or at 13 years about abstract algebra and trigonometry)...

exercise on obvious things and listening 200 times the same lesson made me bored. So I just jumped with shoes and cap in the "advanced" argument thinking my own problems. Math is an hobby for me so I dot have to study very hard all the concepts. I just start with a idea finding a solution alone without help, search something about and understand all I can..then I try play on a blocknotes, if my results are wrong I return and read again and again till I understand.

With this way is sure I have a lot of holes in my education (weak point) but I have reasons to go back to the learn what I left back every time I think I need more powerful tools for my "more interesting" problems (good point).

I don't say this is good for a student... but school form 6y.o. students to 18y.o. don't motivate students to study the easier arguments..I think students need to face alone hard problems, abstract ideas, then when they understand their knowledge is not enough they are more motivated to continue study the basis.

sorry for my English, I'm using a translator.

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