Vanilla 1.1.9 is a product of Lussumo. More Information: Documentation, Community Support.
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.
@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.
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 http://mathoverflow.net/questions/14680/)
Restricting questions about MO policy to meta is the accepted practice.
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.
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.
Mariano,
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."
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.
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! :-)
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 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: http://fqxi.org/community/essay/winners/2009.1
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 (http://arxiv.org/abs/0801.0403) 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.
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.
(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.
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.)
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.
@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)?
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.
Let me also fall in behind Ben Webster. This sort of innuendo is completely slimy.
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:
http://books.google.com/books?id=z5ik85_bIMsC&printsec=frontcover&dq=quantum+gravity+rickles&lr=&cd=1#v=onepage&q=Durham&f=false
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.