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I hadn't seen the question that Will mentioned or the comments, but it seems rather extreme to "never use MO again" because of some sceptical comments on the first question and of a quote from her website, a link to which she provided herself. If, as others have said, this person had observed MO for a few days, she would have seen that the long time users do an important and thorough job, keeping the site useful for mathematicians by closing off-topic questions. To expect that at the same time, one's own questions will never come under any scrutiny is like saying I will never fly again, because the first time I flew, they checked me and my hand luggage, as though they assumed I was a terrorist. One doesn't have to think too many steps ahead to realise that this is done for one's own good (provided one really is a legitimate user).
I found the perspective of a new user as sketched by Thierry very valuable and interesting. But I agree with Will that for a mathematician it is not all that difficult to lurk for a few days or weeks and to do one's homework before participating, and to get a very warm welcome and to enjoy a smooth arrival.
@quid: Experience suggests that starting anonymously, one runs a much much higher risk of a bad start than by investing one's identity into the contributions.
We can't please or appeal to everyone. The situation Will mentioned was quite unfortunate, and should probably forgotten and put behind one. I will take away from it that there is great potential for misunderstanding, and that with new users who may bring benefits to the site, I should appear as much like a helpful and unassuming assistant as possible. I would attempt the care and consideration that Will showed, but I would also act in a way to allow the user to provide additional information at their desire.
This is with the intent of handling new questions that suggest mathematical maturity. Of course, if all new users read the FAQ and other helpful documents before acting, we would not have the number of discussions on meta that we have had.
Gerhard "Ask Me About System Design" Paseman, 2011.08.24
Alex, I do not think that one can infer from the, also IMO true observation, that among the new users one observes on the site anonymous ones have a much higher risk for a bad start that for one specific individual anon or not makes a large difference. In particular, if the new user pays some attention to the fact of being anon and therefore is moreorless forced to write some motivation (cf. my response to Tom Leinster above). In my opnion, but I have no data, a key point for a good first question is that the motivation is clear, and not in the mathematical sense, but really in the sense why did this user ask the question.
quid, I agree with your assessment that the motivation is often crucial for making a question suitable for MO or interesting for others. On the other hand, you explicitly suggest that one reason for a new user for being anonymous would be the possibility to just start over again unscathed if the first contributions go wrong. In my experience, this kind of thinking does not tend to improve the quality of the contributions.
@Thierry Actually, you made that point perfectly clear, and I found it very interesting, I probably just failed to address it properly.
Clearly, some people left who should have stayed, but in some (although not all) cases it seems an overreaction to me, and I don't see anything specific in the MO culture that would have to change for particularly faint-hearted people not to leave (except to allow everything and to never question the poster's background or motivation, which would of course mean the doom of MO).
@Nilima Of course, faint-heartedness is honest. Madelina perceived the environment as unfriendly and walked out, and that seems to be largely due to Will's comments. As I said, I haven't actually seen the comments in the thread that Will mentioned. But I have never seen Will attack anyone personally on this site, nor have I ever seen him say anything that would have made me leave the site forever, even if he had said that to me when I was a newcomer. Moreover, it seems clear that Will had the best intentions. That's why I used the word faint-hearted. This is of course a subjective term and is meant to implicitly contain a comparison with how I would have reacted. I simply haven't seen a comment of Will's that would have justified walking out (in my personal view), and I doubt that the ones under discussion were a big exception from the rule.
Now, if I were to leave a comment to the effect of "don't be fooled by the superficial appearance of the question, note that the OP has a good mathematical background, here are some quotes from her home page" and that would prompt the person to walk out, the incident would leave me puzzled as to what I was supposed to do differently. That's the point I was trying to get across.
Alex, I concur- in whatever I've seen, Will has always been both gracious and professional.
I agree with this statement.
I know that when one says something that inadvertently upsets someone else, then if one is denied the opportunity to set the record straight it can leave a bad taste. Those of us (perhaps myself, I'm not sure) who tend to leave curt comments can learn from this that if someone can take even one of Will's courteous remarks the wrong way, ours are probably more susceptible to misunderstanding.
Over on TeX-SX we have a list of "templates" for likely messages for new users. The point of these isn't that they be prescriptive ("you must choose one of these") but that they lay down a minimum level of politeness. Even if one doesn't use one of these messages, the fact that they are there makes one think a little about the message one is about to leave. Thinking about it, I probably am more polite on TeX-SX than here. That's something I should probably fix.
It seems to me that the general subject of communication on MO, in particular regarding those involving new users, can be an important discussion. However, discussing it along the lines of one particular incident (in particular this one) seems, for various reasons unfortunate to me (including, but not limited to, the fact that this thread is the meta thread of a particular question yet not the one discussed now; the actual incident is not known and is not knowable anymore to everybody, and what is still visible is so incomplete that it is misleading and, in a negative way, misrepresents Will Jagy's contribution).
I thus created a thread
http://tea.mathoverflow.net/discussion/1124/communication-on-mo-in-particular-with-new-users/ for those wishing
for those wishing to continue the general discussion, independent of the specific incident.
Gil, it seems to me he [Andrew] explained in quite some detail why he does not/cannot answer the question [73246], for example he says "it is possible to read it in too many different ways and each has a subtly different answer." Do you want him to write an answer for each interpretation? (Leaving aside the fact that he, in this discussion and frequently before, expressed his believe that MO is not the place for lengthy and general expositions.)
What I do not understand is why the questioner or those who think the question is 'good' do not edit the question (or ask another question); as several people suggested quite some time ago. Or, at least explain in detail what they mean, or where they disagree with the reasoning of those who think otherwise.
To repeat and rephrase what I said in an earlier comment, my personal opinion is: The original question (as written) was unclear/vague. Ryan and Donu correctly guessed the intent (as confirmed by latter comments of the questioner). So, the original question (in its spirit) was answered in the comments (the first ones!).
Those interested in answers to follow-up question could simply ask them.
@Gil: The question wasn't clear enough for many people. I think when you get questions that are sufficiently outside the scope of MO's mandate you invariably get confusion. The question has several flaws that have been pointed out already, and the OP appears to have abandoned it so it's not clear to me why the question was re-opened, especially without editing.
Gil, let us simplify this discussion: In the second comment of SPG to SPG's answer two different interpretations are given. Which one is the one that was (originally) intended by the questioner?
Gil, yes this comment seemed clear to me (and further clarified an initially vague or if you prefer fairly clear question). As such it might have been useful to edit this information into the question rather than to only have it as a comment in a fairly long comment thread. Perhaps there was some need for editing after all. Yet, in my opinion this question got already answered within the first three comments, before that comment was even made. So, I did not understand the continuing discussion. Except if some people still read it differently than I read it, which in turn would imply the situation is still not clear. Or, the request is for elaboration on the comments, but I honestly never understood the discussion to be mainly about elaboration on these comments, except until your last comment. [Added: where by 'the discussion' I mean your comments and the ones by Andrew you refer to as well as my contribtions; of course earlier parts where in some sense about elaboration or 'level'.]
SPG, thank you for the information. But this starts to be a bit confusing for me. The following questions seem quite different to me:
a. What is the/a motivation of M&S to use the definition they use [direct sum]?
b. Could one use this specific alternative definition [direct product] instead and still do what M&S do?
c. Could one use some other alternative definition to do what M&S do, or does one have to use the one they use?
d. Is the specific alternative definition used/usefull anywhere else in this context?
e. Are other alternative definition used/usefull anywhere else in this context?
All of which I could imagine, in principle and abstractly [I do not have the expertise to truly judge this], to be things on could answer 'around' this question. I believe that the main intent was to ask a., in particular based on the comment of the OP quoted by Gil Kalai, I repeat the keypart "[...]I would like to understand why (for the purposes of the theory developed in the book) we define infinite dimensional R(inf) the way it is defined in the book[...]"; and before that suspected something like this based on the first reaction to comments. However, it seems to me you think b. is asked. While some of Andrew's comments perhaps suggest that also an answer to c. would be interesting. So, I am confused.
I still believe the question is a bit too vague. But I decided to answer what I assume is the OP's main concerns, given that they're reading Milnor and Stasheff.
I agree with Gil and SPG's comments above, to the effect that the question was pretty unambiguous (why consider the direct sum rather than direct product of counably many copies of R), and that a context was provided (a non-expert reading Milnor and Stasheff), and that furthermore this context placed the question squarely in the (stated) purview of MO, namely a question come across while reading a graduate-level book.
I don't see why this question generated so much fuss.
I also wonder what the (actual, rather than stated) point of MO is at this stage: if someone can't come and ask a (possibly confused, but still essentially unambiguous) question about Milnor and Stasheff, what is the minimum technical level of question that people (say those participating in this thread) regard as appropriate?
Why not consider the direct sum?
Ill-fitting parable: when thinking about C*-algebras, you can consider the c_0-sum of a (countable) family of C*-algebras. Why is this defined the way it is?
Put another way: I am sure there are theorems about the Grassman-ish object defined as the set of n-dimensional subspaces of $\prod_{n=1}^\infty {\mathbb R}$, equipped with suitable structure. I suspect they are not the same theorems as the ones about the Grassmanian of n-dimensional subspaces of ${\mathbb R}^\infty$. Without having a copy of Milnor-Stasheff to hand, and not being familiar with the book, it is not clear to me which of these theorems would be the ones of interest.
I think I can see both sides.
On the one hand, when you're having trouble understanding a new piece of mathematics, it can be hard to formulate a really focused question. You're fumbling around in the dark, and probably you don't know exactly what it is that's blocking your understanding. So the best you can do is "why is this defined the way it is?", and you hope that someone knows what you mean well enough that they can tap into the source of your confusion and enlighten you. I'm sure there have been questions like this on MO before, and everything's gone just fine.
On the other hand, there were some particularly unfortunate circumstances in this case. Several people genuinely found it hard to know what kind of answer the OP wanted. (The first person to say so on this thread was Qiaochu, who I have never seen being petty or anything other than level-headed.) Ordinarily that would be OK: if commenters request clarification, the OP generally clarifies. That's all part of the process. But in this case the OP didn't, and in fact reacted in a quite emotional way. If he/she had promptly edited the question, or even just written "sorry, I'm a beginner at this stuff and don't know how to make my question any more precise", that would probably have defused things.
I haven't communicated with Andrew about this, so the following is pure guesswork, but I wonder whether for him it was uncomfortably close to a question of the type "write me an expository article about such-and-such". Evidently he could think of lots of things to say on this topic, but he didn't know which ones would be useful to the OP. And when he asked, he didn't get a reply that helped him to narrow it down. So I think I can understand his frustration.
Later on, unpleasant things were written by two other anonymous users, on this forum and in a swiftly-deleted answer on the main site. Even if you think that some people were simply pretending not to understand the OP, there's no one to blame for those pieces of nastiness other than those who wrote them.
Emerton, this relevant context was provided after the closure, in fact I guess as a consequence of it; and it wasn't a quick closure, and the OP commented before it. Might I ask that Gil, you, and whoever else in addition wishes to tell the closers how wrong they were to at least acknowledge this fact.
SPG, a. or b. or is this the same?
SPG, thank you for the response.
Dear Quid,
I think "evil" may be a bit of an extreme adjective to introduce into the discussion; I haven't seen it used, or intimated, before now (unless I missed something).
I looked through the question again at the various timestamps, and saw that you are right and I was wrong vis a vis the non-expert/Milnor and Stasheff material. Nevertheless, the question of direct sum vs. direct product is expressed from the very beginning, and the additional material providing context was posted in under 24 hours. And although it wasn't originally made explicit, the fact remains that this question did arise in a legitimate way from reading graduate level texts. (And I find it hard to think that anyone would regard this question as dealing with undergraduate level material.)
In any event, it may be that such questions, asking for very basic clarifications of graduate-level subjects, no longer belong on MO, but then, as SGP suggested, perhaps the FAQ should be updated to reflect this.
Best wishes,
Matthew
I'm sorry to interrupt this interesting discussion, but let me throw in a somewhat related "MO rule of thumb":
MO is a good place for mathematicians to ask basic questions in a field outside of their area of expertise.
The reason for this rule is mainly sociological, but it is nevertheless a viable rule. Many professional mathematicians would feel uncomfortable asking questions on MSE or similar sites since those sites are primarily intended for less a experienced audience. Indeed, the most suitable responses to such questions is most likely above the usual standards of these alternate sites. (Note that MSE often has very excellent answers to questions, so don't take this last sentence to mean that MSE is not a good place to ask such questions!)
I don't like throwing big names around just for show, but let me give this example. Last year, Terry Tao asked some relatively basic questions on ultrafilters. Granted that ultrafilters aren't a standard part of the graduate curriculum, but any expert on the topic will concur that these questions are basic knowledge for the area. Terry asked because these questions were relevant to his current work but outside his current knowledge base. Would anyone refer Terry to MSE or elsewhere in such circumstances? Of course not! Is Terry the only mathematician worthy of this exception? Of course not!!!
Dear Emerton,
it seems 'evil' has a stronger meaning than I thought. I retract it and appologize. Thank you for confirming the time-line.
Regarding the general question whether typical graduate-level material should be on-topic on MO or not, I agree that a clarification and/or discussion could be useful; indeed, I made a somewhat similar observation a week ago in this thread. Personally, I would not have anything against MO being (or perhaps again being) more open towards this. However, it seems to me this particular question is not the best example for making a case for it.
Thanks again and best wishes!
Quid gave five distinct interpretations of the question. From the question itself I cannot tell which the OP is asking and each, in my mind, requires a different answer. That is why I do not like this question. I think that the level is absolutely fine for MO and that any of those five questions would be perfectly acceptable here. But unless or until the OP clarifies which question they mean, then I cannot see how it is answerable.
My best guess is that the first is the right answer, and then Ryan's answer comes the closest to answering it; though I still think that he isn't clear enough. The key to answering the first version would be to explain exactly why this particular model is a good one to use. Allen says "Because then it is a CW-complex", but so what? Why is actually being a CW-complex better than having the homotopy type of a CW-complex (this would be a very good question, I think)? (Then he goes off with some irrelevance about U(∞) versus U(ℋ)). Ryan at least says:
A key nice result about the weak topology on ℝ<sup>∞</sup> is that any continuous function from a compact space to ℝ<sup>∞</sup> has an image in ℝ<sup>k</sup> for some k.
which to me, at least, is the heart of the matter. It says that when dealing with ℝ<sup>∞</sup> then you are effectively dealing with "very big (but finite) ℝ<sup>k</sup>", at least if your source space is compact (say, a closed manifold or finite CW-complex). So although we want to deal with the classifying space BU(n), we can pretend in any given circumstance that it is a finite dimensional manifold/CW-complex.
But the rest of Ryan's answer, and of what just about everyone else says, is pretty much model independent and so talks about properties of BU(n) without saying why one particular model is preferable to any other. If the OP is truly asking "why this model and not another" then the answer has to address some property of this model that is not held by another, and explain why that property is important.
In response to other remarks, no matter how often I read the question I do not see any mention of the direct product. There are a heck of a lot of spaces between the direct sum and the direct product which I would expect Jo Mathematician to be vaguely familiar with, far more familiar with than the direct product. With no information as to the field of the OP, I don't see how we can assume that he or she means to compare the direct sum with the direct product. (For what it's worth, the direct product is countable infinite dimensional: the direct sum is dense in it.)
This is the sort of question where even if I don't answer it myself, I feel I am competent enough to judge what is a good answer to it. There is not enough information in the question for me to be able to do that! Allen's answer I just do not like, Ryan's is okay, SPG's doesn't address the issue of different models, Paul Garrett's could be taken in one of two ways: either it is about the homotopy type (in which case it doesn't address the issue of different models) or it is about the specific model (in which case it doesn't address what is special about this particular model), Yemon's is - sadly - also missing the point: most of the time we only care about the homotopy type of BU(k) so the particular model doesn't matter, it would surprise me if M&S's book couldn't work with a different model.
To summarise: there is nothing in this question to indicate that it is of a level below that of MO. That part of the debate I find quite bizarre. However, there is also nothing in this question to indicate exactly what sort of answer would satisfy the OP. I was commenting on it because this is the sort of question where I might have been able to contribute, but without knowing more then I wouldn't know exactly what to contribute. That the OP was satisfied with Allen's and Donu's answers (though exactly how, I have no idea) means that the OP needn't bother responding to my comments. But I still maintain that it is not a good question, and we have not have any good answers yet (though I applaud those who tried for doing so).
Andrew, since I also mentioned 'level'. The question is not the problem, but what about this comment:
"[...] It seems to me that the right hand side of the equality is (possibly) "bigger" than the left hand side. Let $1^k \in \mathbb{R}^k$ denote the finite, constant sequence, $(1,\ldots,1)$. The limit of $1^k$ as $k$ tends towards infinity does not lie in $\mathbb{R}^{\infty}$, even though we can identify $1^k$ with $(1,\ldots,1,0,0,\ldots) \in \mathbb{R}^{\infty}$ for all $k < \infty.$ I can see that all of the elements of $\mathbb{R}^{\infty}$ can be constructed by the union, but we seem to be able to construct other elements too."
In my opinion, this confusion is the root of the question.
ADDED: I should say 'was' instead of 'is' as it got resolved. But, in my opinion, this 'answered' the original question.
Yes, that comment did reveal considerable confusion on behalf of the questioner. Given that the questioner was confused about that, I'm surprised that the apparently satisfactory answers were satisfactory. But I may be being uncharitable, this might have just been one of those "hadn't thought it through" incidents.