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    • CommentAuthorIan Durham
    • CommentTimeFeb 8th 2010 edited
    I have been using MO for a little over one week now and, combined with an experience I recently had at conference, I am disappointed in the extremely high threshold pure mathematicians and theoretical computer scientists seem to have for reasonable debate (note: I have a PhD in pure math, but have worked in physics for most of my career). The American Physical Society actually has a policy that all talk and poster submissions to their annual meetings are automatically accepted. Yes, we get a lot of cranks. But we also get some really good discussions going from papers that aren't perfect but where the core idea shows some promise.

    Regarding this particular site, the ability to close any question seems to go against everything the history of mathematics should have taught us about communicating with one another (Galois' story is one example that springs to mind). It stifles debate. What is particularly egregious about the system on this site, is that a certain subset of people could find a question interesting and worth debating whilst others might disagree and have the power to close that question, i.e. a handful of people who *want* to discuss a particular question could be prevented by doing so simply because others think it is not worthy of MO. If you don't think a question is worth debating, then don't participate. Nothing is compelling you to read every question.

    In addition, I am shocked at the level of myopia that some people have displayed on this site. The debate about "finiteness" on one of my questions was a good example. Firstly, it should be recognized, especially for a site like this, that sometimes the questions themselves are difficult to formulate. My question about Hausdorff topologies was along those lines. I was trying to make a broad question more specific in an effort to make it more palatable to people on this site, but was still unsuccessful. But rather than working to help me make it a better question, it was simply closed. I teach my students that there is as much value in the question as there is in the answer. That is exactly what science and mathematics are supposed to be about since if the questions themselves weren't important then we would never ask anything radical. We'd get stale. This is what I mean by my use of the word 'myopia.'

    I should also add that this site is often contradictory. People seem to hold questions to some rather nebulous level of rigor while simultaneously not holding their own judgment of said questions to the same rigor. In other words, you expect well-defined questions but you're definition of "well-defined" doesn't, itself, seem to be well-defined!

    With that said, I doubt this site is ever going to change and I further doubt I will convince those people who already disagree with me to change their opinion. Nevertheless, I would like to point out the need for a site whose rules foster more open discussion (while still not allowing homework problems) instead of one that seems to encourage animosity (no matter how hard you try, by making it so easy for a question to be closed, you are bound to anger someone and cause needless personal animosity).
    I wear glasses and I'm not into self-hate. It's a figure of speech. Please tell me you were joking with that first line since it's a pretty well-known figure of speech.

    Your comment about my question is a pretty broad generalization. A very bright mathematician (who is not on MO) understood the question when I posed it last week. Yeah, it was a little vague, but I didn't know quite how to word it. Nevertheless, I worded it the same to him and he got it. It's because he's a broader thinking mathematician. I've noticed this type of thinking (the broader type) is particularly prevalent in mathematicians who teach at liberal arts colleges (don't know where you're from, but they're somewhat unique to the US) and certain British universities (don't know why).

    And the tone of your above reply is the same as the tone of the replies (initial and otherwise) to my question: condescending and nasty. I have looked at other closed topics along with a few open ones and will note that the level of patience varies widely. I'm not suggesting it is personal, but what I'm suggesting is that it is *arbitrary* which is exactly what the site is supposed to be trying to prevent.
    And Harry, I just looked you up. I don't care how much of a genius you may be, but you're 20. You're an undergraduate. I mean no offense, but I'd buy your argument a lot more if you had a little more life experience under your belt. The world is full of vagaries. It's far from black and white. And I've spent the last ten years toiling over how to communicate highly complex math and physics to undergraduates. Judging from the success of some of my former students (and their general support for my methods), I'm a pretty good teacher. But I got here by recognizing that not all questions are easy to ask nor are they all easy to understand.
    No, Pete said: "I have closed the question. This is not intended as a judgment that it was intrinsically valueless, but rather that it simply wasn't clear as a mathematical question, and the sequence of comments was not making substantial progress towards clarification." That is a far cry from what you said above.
    And you still haven't addressed any of the real issues of this particular thread, of which my own personal issues are only an example.
    • CommentAuthorMariano
    • CommentTimeFeb 8th 2010

    I am all for debate, reasonable and sometimes a bit unreasonable, for it cam be quite fun. Yet MO is not a debate site: it is a place where you ask questions and answer questions.

    The decision to close questions is, in most cases, made by the actual users of the site (well, those that hace accumulated sufficient reputation). In my case, I tend to vote down and/or vote to close only questions that IMHO do not make any sense. Your question still does not make any sense to me. I suggested that you clarify it as the very first comment that question got, and I watched it get lots of comments and evolve into something that still makes no sense to me.

    There is always the possibility someone will come, read it and see clearly what you meant and, with luck, provide an answer. But I did not. My vote to close is not based on any unreasonable threshold for reasonable debate but on my opinion that the question does not make any sense, given its content, to me.

    I cannot vote based on what others think.


    Ian, I suspect you haven't quite understood how MO, by which I mean the underlying software and the entire structure of the site, has been optimized for one goal, and one goal only: To ask questions and receive answers. While questions and answers can surely engender debate, the site is not made to make debate practical. It's a question of focus, really. If you try to make a web site that is good for everything, you'll wind up with a web site that is good for nothing. Hence the resistance to questions that are more suited to a discussion forum. I wish MO had a sister site that could serve as that discussion forum, with easy linking back and forth between those two sites, but currently it doesn't. Maybe someone will create something like it one day.

    Meanwhile, we are still struggling with how to get the message across to newcomers to the site, explaining what it's for and what it's not for, and especially how to tell people their posts aren't welcome without implying that there is something wrong with said posts. If you have suggestions for how to convey that message, I am sure we're all ears. Or if you are convinced it is not a message we should convey in the first place, we'll listen, but please note that our disagreement is not because we are averse to debate in general, we're only averse to it on MO. I think you would, in effect, have to argue that such a tight focus as MO tries to have is a bad thing.

    While I'm in agreement with Harry here, he is unfortunately not our most tactful member. Let me save you some time and tell you that I'm also 20, but that doesn't mean our views are wrong, and frankly, I'm a little put off by your flaunting of your Ph.D. as a defense against our criticisms. Let me describe some of the problems I feel we all had with your question(s):

    1) Wrong definition of Hausdorff. Technical point, to be sure, but it didn't make things easier.

    2) Disconnect between the "Hausdorff" and "infinity" pieces of your first question. Your first question includes

    "The topology of the multiverse would thus be non-Hausdorff and, given these interpretations of QM, there ought to be an infinite number of branches. Given that, an infinite physical realization becomes possible."

    Everyone's point was that the topology has nothing to do with the cardinality, i.e., the number of branches - you would still have the "infinite physical realization" you were seeking regardless of the topology, which again, didn't make it easy to parse your question. Furthermore, your conflation of "a sequence of finite cardinality" (any finite collection of elements of a set) with "a sequence which converges to something finite" (which I believe makes sense only in a topological vector space) made it especially confusing - at least by my reading, this is what Harry and others were trying to clear up by asking what you meant by finite (Harry / others: feel free to correct me on this).

    Now, I think I made a good attempt at trimming down what you might have been asking in the comment thread on your first question,

    "are you *primarily* asking for a mathematical model of the scenario in your second example box (in which case the rest of the text of your question - Hausdorff topologies, mathematical philosophy, etc. - is somewhat superfluous)? Or are you *primarily* looking for an example of mathematical objects of infinite cardinality being essential to a physical theory, in which case I'm sure there are much simpler examples?"

    Perhaps, once you have answered this, we can together make progress on the matter.

    3) Disconnect between the "quantum" and "infinity" pieces of your second question. Theo's edits on your second question highlight this point. Either you mean to be asking about a claim in mathematical physics, or you mean to be asking about whether math is invented or discovered.

    4) Use of purely speculative cosmological concepts. Yes, I've only taken a basic quantum mechanics class - and somehow, I'm still pretty confident that asking about the "number of universes" is about as meaningful a question as the number of angels which can dance on the head of a pin. I'm not denying that "many-worlds" is a good model, a useful and interesting way of talking about the world - but since your question is so bent on having it correspond to physical reality, to the point where we can count off individual universes, I'm going to have to disagree with your use here. As I said in the comment thread on the first question,

    "The claim that such a thing as a "multiverse" exists, as a part of the real, material world, needs to have the experimental scrutiny any other scientific claim does."

    If I may just point out that, when encouraging you to "ask again", I personally said that I thought only the part about whether math was invented or discovered was a good question, whereas "I would hope that the "many worlds" idea doesn't muddy the waters again." Thus, I don't feel bad about the closure of your second question, though you may have a more substantial complaint with others who encouraged you to ask again without more qualification about how.


    That's my current understanding of this whole situation. Ian, I'm sorry if you didn't think we were being fair in the comments, but the fact remains that your questions contained significant errors and ambiguities. In my opinion, your explanation of these as being some sort of "disconnect" between physicists and mathematicians holds no water. I think the question of whether math is invented or discovered is an interesting one, which is welcome on MO (at least as a community wiki); the question of whether "tensoring such a channel with itself n times approaches unitary as n->infinity" is a valid question in mathematical physics, which is welcome on MO (though practically, I don't know how many people we have who could answer it); and how many universes there are to be a meaningless question, but one which you are obviously free to ask on your blog or elsewhere.

    And again, anyone else involved in all this, if I misrepresented your views, please correct me.

    @Zev: When you say you think

    the question of whether math is invented or discovered is an interesting one, which is welcome on MO

    I would agree (sort of) with the first part, but not the second, since the format of MO is not well suited to such a debate.

    @hanche: I was imagining it as a community wiki, with people posting links to papers or websites with the main arguments for each side, but I understand your view too.
    • CommentAuthorMariano
    • CommentTimeFeb 8th 2010

    By the way, Ian: there is no need to pepper completely unrelated questions with comments as to why your question was closed and not others... (I came across your comment on

    Restricting questions about MO policy to meta is the accepted practice.

    Let me first apologize for a misconception Zev seems to have gotten. I have not been flaunting my PhD, though, in hindsight, it appears as such. What I have been attempting to do is point out that the PhD I do possess is in math and not physics, i.e. I was trying to head off any criticism that assumed I was a physicist (which I also am) and thus didn't have the background or mentality of a mathematician. It has nothing to do with elitism, though I admit it came off that way.

    That being said, Zev, on this point, you are incorrect. Many-worlds is regularly invoked in discussions about practical, quantum information-related problems. Certainly there is no evidence yet of a multiverse, but a) that doesn't mean evidence won't be found (Deutsch proposed an experiment to test for it that, wild as it sounds, might prove possible depending on results from the LHC) and b) foundational and "speculative" questions in quantum mechanics such as this are exactly how quantum information theory got started in the first place, i.e. as crazy as it sounds, even if it is untrue it could still lead to some very interesting and potentially useful physical results and the single best example of this is the original EPR paper which led eventually to the entire field of quantum information and quantum computation.

    (Let me then note that the direction this thread seems to have taken only further confirms what I've been saying. It was never intended to be a place to discuss my MO question specifically. It was intended to bring up a discussion of what constitutes a valid question for MO and what the threshold for debate over this point is.)

    Regarding MO being a place for questions and answers rather than debate, I realize that. But I was not asking an open-ended, speculative question. I was asking (apparently using language more common to philosophers of science) for what I called a mathematical "object" (since I didn't know a better term and since that term has been used in debates surrounding the Quine-Putnam argument) that satisfied a condition. By object, I meant a group or a series or a function or something mathematical that satisfied the condition I had set forth. Yes, there was a mistake in my definition of Hausdorff. But people were picking apart the words "finite" and "infinite." Now, I know that "finite" could mean "a group with a finite number of elements" or a "convergent series" or something like that, but the basic idea is still the same - finite as opposed to infinite! I couldn't be much more specific since I was looking for anything that I could use to model what I was trying to model.

    Given that no one seemed to get what I was asking (because it was a broad, but not open question) I gave the motivation for why I was asking (the multiverse) thinking it might make it a bit more clear why I was interested in such a thing. Since this wasn't working, I followed the advice of someone who said my blog post was pretty clear and so I posted a new question based on this and made it a community wiki only to get slammed again.
    • CommentAuthorMariano
    • CommentTimeFeb 8th 2010

    Ian, the original question was:

    Does there exist some mathematical "object" (e.g. a set, group, category, etc.) that is known to be finite on a Hausdorff topology but infinite on a non-Hausdorff topology?

    I think it is quite natural that people---me included---wanted to know what you meant by 'finite', as it clearly it plays quite a central role in the question. Saying that we started «picking apart the words "finite" and "infinite"» is a rather bad description of the situation.

    The fact is that finiteness as in "a group with a finite number of elements" or in "convergent series" or in "something like that" are rather different notions of finiteness, are connected by little more than the choice of the word finite to denote them. Add to this the fact that it is quite non-obvious what Hausdorffness has to do with (in)finiteness in at least 2 out of the three examples you've given... (for "convergent series", one can imagine that you are asking if a series can be convergent in a topology and non convergent (or convergent to 'infinity', whatever that may mean) in another topology... but examples of these---usually phrased using sequences---are usually met when one first studies topology), and that it is also non-obvious what the connection is between the question and the motivation and background given, and you should see why the question was not liked.

    Let me just add that the points I'm trying to make here were articulated more clearly by apetresc in this post:, though I agree with several responders that I don't think elitism is the problem.

    If you want to attract applied mathematicians to the site (which you admittedly may not want to do), then telling them to go elsewhere to seek answers seems counter-productive.

    Also note that the line between pure and applied mathematics is quite blurry. Just because something looks like pure mathematics doesn't mean it won't find a physical application somewhere someday (or that it is being considered as an application right now and you're not aware of it).
    • CommentAuthorMariano
    • CommentTimeFeb 8th 2010 edited

    I do not see in what way is the putative applied-vs.-pure discrimination connected to this thread, which is AFAICT, on the more or less established fact that MO is not a debate site.

    • CommentAuthorIan Durham
    • CommentTimeFeb 8th 2010 edited


    The fact is that finiteness as in "a group with a finite number of elements" or in "convergent series" or in "something like that" are rather different notions of finiteness, are connected by little more than the choice of the word finite to denote them.

    I wholeheartedly disagree. You are correct on one level that they are different, but there is more connecting them than a simple word. Perhaps it is easier to look at this from the standpoint of the opposite word "infinite." Now imagine trying to explain to an elementary school student the difference between a group with an infinite number of elements and a series that doesn't converge. The notion of finite and infinite on a very broad level is still the same in both cases and a young child, innocent enough not to know the subtleties we add to these terms, would see the cases as being the same.

    Nevertheless, if mathematicians decide that these things are completely different then they shouldn't be using the same word to describe them (and they do - I taught Modern Algebra and Real Analysis at the same time last year and remember both textbooks discussing finiteness).


    I do not see in what way is the putative applied-vs.-pure discrimination connected to this thread, which is AFAICT, on the more or less established fact that MO is not a debate site.

    It is intimately related (which goes to show that you don't understand much about how non-mathematicians use mathematics). What you see as unwarranted "debate" is the inherent "messiness" that accompanies all applied mathematical problems. To quote Mike Fortun and Herb Bernstein, in science, eventually "[s]omewhere along the line - no matter how long that line is - every experiment, every mathematical equation, every pure numerical value will have to find its way into words."

    "Now imagine trying to explain to an elementary school student the difference between a group with an infinite number of elements and a series that doesn't converge. The notion of finite and infinite on a very broad level is still the same in both cases and a young child, innocent enough not to know the subtleties we add to these terms, would see the cases as being the same."

    Could you please clarify in which sense an infinite set (such as N) and a divergent series (such as $(-1)^n$) are "the same" to anyone, including a schoolchild?

    "Nevertheless, if mathematicians decide that these things are completely different then they shouldn't be using the same word to describe them"

    In an ideal world, maybe. In a real world, same terminology is reused for different things all the time, consider e.g. words such as "program" or "machine".
    • CommentAuthorIan Durham
    • CommentTimeFeb 8th 2010 edited

    Could you please clarify in which sense an infinite set (such as N) and a divergent series (such as $(-1)^n$) are "the same" to anyone, including a schoolchild?

    An infinite set has an infinite number of elements. A divergent series is an infinite series meaning it has an infinite number of terms. The word "infinite" means "that which has no boundary or end." In both cases we have an infinite number of "things" (elements in the former case, terms in the latter).

    In an ideal world, maybe. In a real world, same terminology is reused for different things all the time, consider e.g. words such as "program" or "machine".

    In an ideal world, then, my question would have made sense. Incidentally, just because colloquial English (and presumably most other languages) is full of mis-used terminology doesn't excuse mathematicians and scientists from being more careful (particularly if they are using that non-rigorous language to judge the rigor of a mathematical problem).


    Yeah, Ian, I don't get it. They're not the same at all. First of all, a convergent sequence need not be finite. A divergent sequence need not take on an infinite number of values. A group is a set with a composition operation fulfilling certain axioms. A sequence is a function N->S. You're comparing apples to oranges, then you're requiring that they in addition be finite.

    A divergent sequence need not have an infinite number of values but it will, by definition, have an infinite number of terms.


    All sequences have an infinite number of terms. A finite sequence is an equivalence class of infinite sequences sharing the first n terms. Convergence does not play a role in this definition.

    Let me clarify, then. A series converges when the sequence of partial sums has a finite limit. If the limit is infinite or doesn't exist then it's divergent. There is some debate about whether "doesn't exist" can be associated with "infinite" though, perhaps, this is more theological than mathematical.

    • CommentAuthorIan Durham
    • CommentTimeFeb 8th 2010 edited

    There's quite a big difference between a function having a limit and a set being finite or infinite, as I've described.

    Let me re-clarify yet again: a series converges when the sequence of partial sums has a finite limit. If the limit is infinite or doesn't exist then it's divergent.

    Thus, in both cases, "infinite" means "infinite" and "finite" means "finite" on a very broad level.


    Let me just say, I'm so glad we started meta. It's serving its purpose (of diverting pointless argument away from mathoverflow) perfectly! :-)

    The following was not easy for me to write, but upon reflection I think it is the lesser evil to say it openly. I will be very happy if I am proven wrong in what I am about to say.

    I am concerned that Dr. Dunham has some serious gaps in his understanding of basic concepts of topology, especially limiting processes. He really didn't seem to understand that his definition of Hausdorff space was obviously incorrect (the condition he gives holds for all topological spaces). Surely most readers (myself included) assumed at first that it was just a typo, until his response that the difference between his definition -- taken, apparently, from a single physics text -- and the standard one was evidence of a disconnect between the mathematical and physical communities. This is simply embarrassing all around.

    In order to get a better sense of what Dr. Durham was trying to ask, I went to his blog, which contains numerous complaints of his work not being accepted and published. He refers in particular to the paper

    The main result of this paper is Theorem IV.3 on page 10. In the proof, Dr. Durham argues that a group G of permutations on an infinite set is a limit, in some sense, of finite groups, and that consequently any representation of G is unitarizable.

    This argument makes no sense to me. I would very much like to think that I am too narrow-minded to understand it; can anyone here justify it?

    [Edit: I have removed the last paragraph since Dr. Durham wrote to me to reiterate that he was distressed and offended by it. As others have said, it is essentially the logical conclusion of the rest, so that it is not a significant loss of information to omit it.]

    Let me just say, I'm so glad we started meta. It's serving its purpose (of diverting pointless argument away from mathoverflow) perfectly! :-)

    Was that really necessary?

    Ian, I was discussing your questions precisely because I wanted to make the point that, regardless of how biased the users of this site may be towards pure mathematics, your question was still closed for valid reasons, instead of being indicative of any problems in the community of pure mathematicians. Perhaps I should give up on this point already, but: I listed what I see as the pieces of your questions. I think we would all very much appreciate it if you indicated which one you intended, or want answered most, or - if neither of those things - at least described why you don't think my analysis of your question is correct. In particular, please describe why any matter of convergence, regardless of whether it is in a Hausdorff topology or not, has a bearing on your question.

    Also, I hope the discussion which has transpired since my last post has convinced you of the very important distinction between the different possible meanings of "infinite". For the purposes of the Quine-Putnam argument, Qiaochu's answer on your second question should suffice.

    EDIT: Pete's certainly taken it up a level. I'm in no position to judge either way, so I'm out of this thread.

    I agree with Harry here. These notions of finiteness are completely different. First of all, even if a sequence converges to a finite value, the limiting process itself is inherently infinite. I did not understand the question the first time I read it, and after 45 comments on MO and a whole thread here I still don't understand what you are looking for.


    I think intellectual honesty requires that we let Dr. Durham know that he is making arguments that the communities of pure mathematics / applied mathematics / theoretical physics [I do not think it makes any difference] do not find valid. That is, he is not successfully presenting himself to the community as a mathematician or physicist.

    Now, I find that uncalled for. If you'd like evidence that I am, indeed, taken seriously, please click here:

    Yes, the argument you point out in my paper is flawed in its wording. I have since made changes but have not uploaded them yet.

    The other paper I have complained about on my blog ( suffered from bad luck since it was good enough to have been thoroughly picked through by several leading quantum physicists who attempted (and are still attempting) to assist me with publishing it.

    Regardless, Dr. Clark, I think that was an utterly unjustified personal attack.

    To Dr. Durham:

    It's not an attack, and it's certainly not personal, since I don't know you at all or have any stake in your career success.

    I edited the remark that you quoted so as to remove the last two words: I am not a physicist, so I don't know who looks like a physicist or not. (The only evidence that I have that you were not being taken as seriously as you wished by the physical community was from your own blog entries.)

    In mathematics circles, it is considered to be a favor to explicitly point out what one perceives to be mistakes in others' arguments. So I was treating you as I would a professional mathematician who (I sincerely believe) has made a serious mistake. I also firmly believe that what I am saying is what a lot of other people are thinking, so that by telling it to you outright I am respecting you enough to take that information and handle it in an appropriate way. I didn't see anywhere in your blog entry where you said that you knew there was still a mistake in your paper: rather, you were expressing great confusion as to why it had not been accepted.

    Don't you want to know whether professionals understand or find validity in your arguments?
    @Harry: Briefly, no, I know about profinite groups and they are not relevant to the matter at hand in any way that I can see.
    • CommentAuthorIan Durham
    • CommentTimeFeb 8th 2010 edited

    Don't you want to know whether professionals understand or find validity in your arguments?

    Yes but there are far more tactful ways of doing this. My blog is exactly that - my blog. Which means my contributions are sporadic and I don't necessarily update things. Thus I did not say anything on my blog about my corrections to that paper.

    But that is beside the point. Your comments, particularly the paragraph I highlighted in my original reply (i.e. the last paragraph), were offensive and uncalled for.

    @Harry: Right. The group in question is not locally profinite.

    @Dr. Durham: As I said, if our roles were reversed I would have wanted the mistake pointed out. It is relevant to your MO questions because you were asking about limiting processes in a way that the mathematicians on this site found difficult or impossible to understand. In order to give a response to a questioner, you have to know where they are and what they are thinking of. As a convenient example of this, Harry Gindi is guessing that you are alluding to the theory of infinite dimensional representations of p-adic groups. I am guessing that you're not. My understanding is informed by the arxiv preprint of yours I read. Am I wrong?

    Anyway, as I have said, it is not my intention to distress you, and I'm sorry if I offended you. I will not say anything further about the matter unless you ask me to.

    (Did anyone else think that this thread was going to attempt to set the threshold for reasonable debate on meta, decide that it had itself crossed that threshold even without knowing exactly where the threshold is, and subsequently stop before the threshold could itself be determined?)

    In all seriousness: Ian, I see no indication in Dr. Clark's comments that he is in any way mounting a personal attack on you. I understand that criticism is difficult to take, but if the goal of this discussion is to foster an atmosphere on MO in which we give each other the benefit of the doubt, wouldn't the most appropriate way to inaugurate such an effort be to start doing it yourself?


    @Dr. Clark: To re-iterate, my problem was not with the fact that you pointed out an error (which I am already very much aware of thanks to the meeting I referenced in the blog post). It was your offensive final paragraph and your general approach to the matter which has been utterly unprofessional. It is rather unfortunate that you can't seem to see that (Zev apparently noticed the change in tone, by the way, so I'm not the only one). My only hope is that someone with more sway than me will someday convince you that your tactics were unprofessional.


    @QY: My only comment to this is that Zev noticed a change in Dr. Clark's tone too, so I'm not the only one. It was the last paragraph that was particularly offensive.

    I announced that I was leaving to avoid getting sucked into this any further, which I have obviously failed at. In fact, things like this are why I generally stay off of meta. In no way did I mean to imply that Pete was unprofessional; I could not have said what he said any better than the way he put it (and Pete was in fact saying what I was thinking). I am merely acknowledging that I have neither the technical knowledge nor standing in the community to participate any more on this matter, now that Pete has explicitly stated (in a very respectful, calm way) this concern.
    Not to beat the horse further, but I looked at Dr. Durham's blog just now and wasn't able to find any statement that the paper in question was flawed. It might well be there; I just haven't found it. Could someone provide a link?

    The blog post that I did read -- which seems to be the latest blog post mentioning the paper -- is from December 29:

    It contains a link to the latest arxiv version of the paper; the text that you click on to get to this link is

    "you may judge for yourself if you think the paper qualifies me as a lunatic".

    I hasten to add that I don't think Dr. Durham is a lunatic or that he has no good scientific ideas, but I hope that this helps to explain why I thought that he was unaware of the mistake [again, in my opinion] in his paper.

    Where did I ever say that there was anything on the blog that said there was a mistake? Didn't I say that I know there is a mistake but haven't taken the time to update my blog?

    Zev: sorry for misinterpreting. Apparently I am alone in thinking his final paragraph - which everyone is ignoring in this thread - was unprofessional and uncalled for. He could have gotten the entire point across without that last paragraph.


    On the grounds that discussing/debating the wider issues may be even more fractious than the particulars....

    Having looked at Theorem IV.3 in the paper which Pete Clark as linked to... I think the paper could benefit from the attention of some of the people on the site who have worked with areas bordering quantum information / quantum communication theory, and more importantly have talked to people working on related things but with different backgrounds. Greg Kuperberg (the dark lord on his dark throne, as I like to imagine him) was my first thought, but perhaps he has recused himself from MO to let a few more people catch up... [That should have been enclosed in <jocular> tags, but I don't think XHTML recognizes those yet.]

    More seriously: people in operator theory and matrix theory have worked on quantum channels, and the statement of Theorem IV.3 does not need categories (a one-object category is just a monoid, so introducing them as one-object categories is perhaps using unnecessarily sophisticated machinery). It seems to be a statement of finite-dimensional Banach spaces or geometric matrix theory. I have to agree with Pete's mathematical dissatisfaction/confusion as regards the proof (which isn't to say that the result isn't true, and isn't to say that the proof as given couldn't be honed into something more precise; although my gut feeling is that if it is true as stated, then once can get rid of the epsilons and "well-approximated" by a compactness argument). Specifically, the invocation of Cayley's theorem leaves me puzzled; and the point at which a certain representation of an infinite group is claimed to be \epsilon-isomorphic to some representation of a finite one for any \epsilon, leaves me even more confused.

    I am offering these comments, despite the obvious demerits of doing so in a forum such as this. because Pete Clark raised a technical (as opoosed to conceptual, ethical, ontological, political, whatever) point, which I didn't want to leave hanging. Ian Durham has said in response that he has already made revisions to the paper, so I trust that it will over the usual life-cycle of a research paper get checked, modified, reviewed, etc; the remarks that PC and myself have made are (I think) not intended to be any summary judgment or tool for point-scoring.

    (By the way, I am actively irked by a perhaps unintended undernote to some of the discussion, as if it is in the nature of pure mathematics people to disdain applications and fetishize precision, as opposed to applied mathematicians who are interested in probing foundational questions with not-quite-nailed down definitions. I got to know several applied mathematicians during my time as a PhD, many of whom were working on complicated simulations of a fluid-dynamical nature; and they were by and large just as concerned as I was with precision. Their lack of concern for foundational issues in physics didn't make them less erious as applied mathematicians, in my view, any more than their lack of interest in recondite vocabulary from general topology or algebraic widgetry or anything else.)

    • CommentAuthorYemon Choi
    • CommentTimeFeb 8th 2010 edited

    Oh, and as regards the last paragraph of PLC's post

    I think intellectual honesty requires that we let Dr. Durham know that he is making arguments that the communities of pure mathematics / applied mathematics / theoretical physics [I do not think it makes any difference] do not find valid. That is, he is not successfully presenting himself to the community as a mathematician or physicist.

    I think it was blunt; but I don't think it was out of order, in the present context. It wouldn't be a polite thing to say on a first encounter/discussion; but the way it's phrased takes great pains to say that the things PLC finds lacking are particular to the present discussion, not to ID's esteem or qualifications or body of work. That is: the difference between saying to someone "I think you're not making your case well enough", and saying "I think you don't understand what you're doing". The 2nd is much harsher, less fair, and probably is out of order; the 1st, in my own Eeyorish/Benjaminesque view, is legit, and - dare I say it? - more professional.


    Let me just say, I'm so glad we started meta. It's serving its purpose (of diverting pointless argument away from mathoverflow) perfectly! :-)

    Was that really necessary?

    I was reiterating a point that I've made elsewhere on meta, in regards the level of civility we insist on at meta and at mathoverflow. My pet theory is that on the internet, people are going to want to do a certain amount of bickering and fighting. Given that, I'd prefer it doesn't happen on mathoverflow itself, but rather here, in the echo chamber of meta.

    I think this whole thread has been an enormous waste of people's time: your question was closed for good reasons, I don't particularly see any underlying problem, and if you have good questions in future I don't think previous questions, or indeed this thread, will be held against you. A certain fraction of the mathoverflow community investing their time in this fight is an unfortunate side effect of trying to do something good and useful on the internet (ie mathoverflow). While it's a waste of people's time, it's also probably in some sense unavoidable: I'm just glad that we can segregate it out from the main site.


    OK, it's probably too late for this, but let me try to cool things down, since I really prefer to not have flamewars here.

    Ian- I recognize your frustration at having your question closed, and of course, it's reasonable for you to disagree with the site's philosophy about closing questions. But I disagree that the intent or the effect is to stifle debate; closure is an operation on questions, not on users. That particular question was closed because a number of the sites users felt (I can only assume, I wasn't one of them) that as it written it could not be usefully answered. But that's not an irrevocable state; there's a very recent example of a question being reopened after a discussion on meta. Not to mention that you're free to post any time you want. It's not as though we have banned you from the site.

    Everyone- Can we all just please stop with the accusations? Everything sounds more offensive than it was meant to on the internet, so can we please just press the reset button? I'm not sure there's much more to discuss (as far as I can tell, there's just a reasonable disagreement between Ian and everyone else on this thread about what sort of website MO should be), but if there is can we get back to it?

    Harry- I thought you were sticking to math. That sounded like a really good plan to me, rather than trying to start flamewars on meta.


    "Trying to start a flamewar" was probably uncharitable (I'm not editing above in order to be sensibility). On the other hand, I really hope you don't think writing a post like your first one is a way to start productive dialogue. I don't agree with a lot of what Ian said, but that doesn't make sarcasm the right response.

    • CommentAuthorIan Durham
    • CommentTimeFeb 9th 2010 edited

    @Mike: I'm having a hard time deciphering whether your comment was intended to be somewhat tongue-in-cheek or not. Just so I can clarify, are you talking about my picture from several years ago that actually has the correct definition of a Hausdorff topology in the background on my whiteboard?


    OK, so could you be more specific then since you are accusing me of dishonesty (which is one of my major pet peeves)?

    • CommentAuthorBen Webster
    • CommentTimeFeb 9th 2010 edited

    Ack! So much for stopping the accusations. Meta is supposed to be a discussion about how to run the site; I don't care if Ian shot a man in Reno just to watch him die, just whether I find his arguments about how to run the site convincing or not.

    Thanks for trying Ben, +1.

    Let me also fall in behind Ben Webster. This sort of innuendo is completely slimy.

    • CommentAuthorIan Durham
    • CommentTimeFeb 9th 2010 edited

    This will be officially my last post but I want one accusation out in the open. I have been accused of not actually having a PhD in Mathematics. That accusation is patently false. My degree reads Mathematics. My advisors were John O'Connor and Edmund Robertson (now retired) who are group theorists. Yes, the general crux of my thesis had a historical bent to it, but as proof that it is a serious work of mathematical physics, click on this link:

    Edit: I realized I never said where it was from: U. of St. Andrews up in Scotland.


    I had told myself I wouldn't comment further, but: despite what one might read into my last couple of comments on this thread, I think Mike's earlier comment ("I know there is something...") was out of line. Whatever your views Mike, that wasn't the way to go about broaching them. (If you feel strongly then I'd be happy to discuss this by email; I think Google will usually find from my name some page with contact details.)

    Note that this is not having a go at Mike, saying that Mike is a "out-of-line-kind of guy", &c &c. Just saying that there are various ways of voicing misgivings, and I don't think he chose a good way.