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.

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).

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.

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.

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?

@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.

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.)

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.

"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.

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

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.