tea.mathoverflow.net - Discussion Feed (Choice vs. countable choice)

François G. Dorais comments on "Choice vs. countable choice" (5308)

2010-05-02T00:21:16-07:00

Gil, note that with the possible exception of "off topic" (usually for elementary math questions) and "too localized" (usually for homework questions), none of the reasons to close a question are based on academic or mathematical merit. The other reasons ("exact duplicate," "subjective and argumentative," "not a real question," "no longer relevant," "blatantly offensive," and "spam") for closing a question do not require much expertise on the subject matter. See the whatnot section of the faq for details on reasons to close a question.

Here is the paragraph of the faq most related to vagueness.

MathOverflow is not an encyclopedia. MO is a site for questions that have answers. MathOverflow visitors should know how to learn new things and do mathematics on their own, but we all get stuck sometimes, and this is where MO saves the day. When you're stuck, you can come to MathOverflow and say "I'm trying to do X. How can I do that? Does this work? Does anybody have a reference?" The idea being that for an expert, it should take very little effort to understand your confusion and set you on the right path. Or maybe a non-expert has come across the same sticking point and can explain how she resolved it. MathOverflow is not the appropriate place to ask somebody to write an expository article for you. If you want somebody to write an article about some subject, you should make a stub on Wikipedia or make a query block on nLab.

As you can see, the vagueness issue is not a judgement of academic or mathematical merit. The issue is that vague questions interfere with the functionality of MO.

MO can be very time consuming, partly because it is so addictive and partly because it requires time to write a thoughtful answer. Vague questions make this even worse, no matter how interesting or meritorious. Closing vague, discussiony, argumentative, and subjective questions is essential to maintain the functionality of MO. It ensures that experts like you can drop by when they have spare time, answer a question or two without having to worry about extraneous context, and then leave to go about their regular business without lingering aftertaste.

gilkalai comments on "Choice vs. countable choice" (5307)

2010-05-01T22:40:49-07:00

Anton, for me another principal merit of a question is the academic quality of its content. Evaluating questions cannot be made just on procedural "context-free" basis. I like your heuristic about vagueness, and I think in each case we should let the people with real knowledge (not just strong opinions) on the subject to make the call.

The problem I had with your example is this. All the discussion of "merits" is statistical in nature. So a few counterexamples do not make much of a difference. And certainly a hypothetical <\i> counterexample is not very useful. (I wonder if mathematicians' training is a disadvantage in dealing with real-life statistical statements.)

Anton Geraschenko comments on "Choice vs. countable choice" (5303)

2010-05-01T16:56:12-07:00

@Gil: yes, I still stand by that example, but I don't claim that it applies well to this situation. "Tell me something interesting" indeed has the ability to lead to very good answers, but only because it is extremely vague. I choose this example to be so vague that it has no actual content, so I would be very surprised if somebody tried to argue that it isn't a terrible question. Of course, a small amount of vagueness is sometimes necessary.

I propose the following heuristic: if somebody with vast knowledge of the subject (who would clearly know the answer if there were such a thing) is likely to waste time trying to decide on an answer to give (among many possibilities), then the question is probably vague enough that it should be closed.

gilkalai comments on "Choice vs. countable choice" (5297)

2010-05-01T12:44:46-07:00
I have a mixed feeling regarding Andrew's famous post: Andrew wrote
"The strengths of MO lie in the interaction between asker and answerer."
And to this I completely agree. However, he then continued to say
" If an answer does not engage with the asker, then it's a Bad Answer, no matter how fantastic the rest of us think it is."
Which I regard as an extreme opinion to which I cannot agree.
In any case, the context was completely different. (It that case the question was not detailed, while here it was perhaps too detailed.)

In "real life" mathematics the ability to lead to very good answers is indeed a principal merit of a question. I believe this is also the case for MO. There are, of course, other merits for good questions.

Anton, I could not follow your example. Do you still stand by this example?

Anton Geraschenko comments on "Choice vs. countable choice" (5277)

2010-04-30T08:26:45-07:00

Of course, a principal merit of a question is the ability to lead to very good asnwers.

This is going a bit off the topic, but I want to stress that I don't think this is much of a merit. For example, consider the question "Tell me something interesting." Ignoring the fact hat it's not a question, it can certainly lead to fantastically interesting answers. However, it's a terrible question. Those fantastic answers could not properly engage the question or the questioner: they would be better off as articles. See Andrew Stacey's excellent post on this.†

†This is meant to say that I think that particular post is excellent, not that Andrew has a unique excellent post.

François G. Dorais comments on "Choice vs. countable choice" (5272)

2010-04-30T01:36:31-07:00

I think this experiment was a great success. In a very civil manner, we managed to save this question from wrongful closure. We should keep doing this.

Pete L. Clark comments on "Choice vs. countable choice" (5267)

2010-04-30T00:09:29-07:00
I don't see anything objectionable about this question, and Gil Kalai's comment that it has led to good answers hits home with me. (A bad question *could* lead to good answers, but a question that *has* led to multiple good answers seems to have a reasonable expectation of leading to more good answers, or so says Bayesian probability.)

As I have said before, I wish people would go a little easier on their votes to close.

Emerton comments on "Choice vs. countable choice" (5266)

2010-04-30T00:01:43-07:00

Dear Harry,

There are many results in number theory that I accept as proven because they (ultimately) argue from basic properties of the standard model of the natural numbers. The arguments are often second order (e.g. involve inductions which quantify over subsets of the standard model of the natural numbers). I don't know whether these can all be rewritten in some first order way, and to what extent they can be reduced to Peano arithmetic. (For example, I don't know whether FLT is first order derivable from the Peano axioms, but I certainly accept it is as a true result about the standard model of the natural numbers --- which are the numbers I personally care about.) The point is that, in the end, these results are true because they argue from true properties of numbers; not because they argue from certain axioms. (After all, we know by Godel that the true properties of numbers can't be captured by a recursively enumerable axiom scheme.)

As for (b), just as I don't think that number theory is the same as Peano arithmetic (number theory is about the standard model of the natural numbers, which are not captured by the Peano axioms), I don't think that set theory is necessarily what is captured by ZF, or other related axiom schemes. (I certainly have less intuition here than for number theory, but to the extent that I understand Godel's position on this, I probably agree with it. So I would probably be labelled a Platonist.)

gilkalai comments on "Choice vs. countable choice" (5265)

2010-04-29T23:55:57-07:00
I think that the question "why in the minds of some people ACC is "unproblematic" but AC's validity is not?" is a good question and indeed it led to very good answers. The question should be left open.

Of course, a principal merit of a question is the ability to lead to very good asnwers.

It is often the case that people based their suggestion to close a question on their expectations from the answers like " it will lead to subjective and argumentative answers" and "it does not contribute more to the discussion than what is already available on previous questions". Given that, it makes much sense to look at the quality of the actual answers when you evaluate a question.

Harry Gindi comments on "Choice vs. countable choice" (5261)

2010-04-29T23:11:41-07:00

Dear Emerton,

It seems to me that taking a constructivist position is nothing more than saying that one doesn't like a certain choice of axioms, and therefore that all results derived from those axioms are meaningless, regardless of the work it took to get those results. It's an inherently argumentative position.

To respond to a.) I pose the following question: Would you accept a poposition as proven if it doesn't follow from a combination of axioms and hypotheses (which are also axioms in the theory defined by adjoining the hypotheses)?

And for b.), I was using ZF to mean classical ZF on one side and constructive ZF on the other as "common ground", so to speak. AC is independent of both, and adjoining AC to CZF gives you a theory equivalent to ZFC by Diaconescu's (spelled incorrectly?) theorem.

Emerton comments on "Choice vs. countable choice" (5260)

2010-04-29T22:44:04-07:00

Dear Harry,

To say that "The former is simply a statement of fact, while the latter is a statement of opinion about things about which one has no business giving opinions (namely the validity of axioms that are otherwise independent of set theory)" is to state an opinion, namely that (a) mathematics is about something described by axioms; (b) set theory is about what is described by ZF. Not every mathematician believes either (a) or (b) (personally, I believe neither). There are other views of mathematics besides the formalist view, and questions of this kind invite people to explain (albeit in a sometimes indirect way) the various views of mathematics that they hold. I think that such explanations can be valuable and interesting, and I would guess I'm not alone in that.

Anton Geraschenko comments on "Choice vs. countable choice" (5254)

2010-04-29T21:10:28-07:00

Though there have been a number of questions about AC, I don't think this question is a duplicate. The original form of the question was a bit argumentative and subjective (for some reason, people tend to get angry when discussing "whether AC is true"), but not too much, and it has been edited to be less so. The only remaining issue is whether it's too much of a discussion question. A rule of thumb that I like for deciding whether a question is so vague or discussiony that it should be closed is, "Is it clear what the question is, and is it clear what constitutes an answer?" I think it is clear what the question is, even though it's not terribly precise, and I feel like I can tell what a good answer would be. So I would vote to keep this question open.

(+1 François. Thanks for starting this thread. I hope this system catches on.)

Noah Snyder comments on "Choice vs. countable choice" (5253)

2010-04-29T21:07:47-07:00
I don't like these sorts of argumentative questions, but enough people do seem to like them that I don't see what the harm is in having one or two of them active at any given time. My opinion would be different if they were starting to take over (which I recall being a worry several months ago).

Harry Gindi comments on "Choice vs. countable choice" (5252)

2010-04-29T21:04:09-07:00

The general policy here is:

Questions should be closed or left open based exclusively on their own merits (or given recent events, also on the merits of the questioner), not the merits of their answers.

stankewicz comments on "Choice vs. countable choice" (5251)

2010-04-29T20:59:55-07:00
I would be quite disappointed if this question were closed due to the excellent answers it has elicited.

Moreover, although it is somewhat argumentative it is only in revealing an issue in mathematics we all must grapple with: Do we invent structures or do we discover them?

For the latter, Andrej Bauer's answer shows how in studying intermediate forms of choice, we can discover something that is amenable to "real world" phenomena.

For the former, Pace Nielsen's answer reminds us that no matter how unreasonably effective mathematics is at modeling "real world" events, mathematics does not describe reality in any reasonable way.

Let's recall why argumentative questions should usually be closed: they invite any actual mathematics to be covered by a pile of arguments. This question, while somewhat argumentative, does the opposite. It reveals something about the practice of doing mathematics.

Harry Gindi comments on "Choice vs. countable choice" (5245)

2010-04-29T17:33:10-07:00

I feel like saying things like "I'm a constructivist" or "I'm a classicist" are annoying and their inclusion in a question is pointless. I feel like taking philosophical stances about the validity of certain areas of mathematics that are proven with as much rigour but using different axoims is just being argumentative.

Mathematics is big enough for constructivists and classicists to live. There's a difference between saying "I do constructive mathematics" and "I am a constructivist". The former is simply a statement of fact, while the latter is a statement of opinion about things about which one has no business giving opinions (namely the validity of axioms that are otherwise independent of set theory).

Emerton comments on "Choice vs. countable choice" (5244)

2010-04-29T17:23:33-07:00

I don't see what is wrong with the second question (the one this thread is about). There may not be a single correct answer (which is at best an argument for community wiki, but not in itself an argument for closing), but nevertheless, in one answer to the first question (the earlier AC question), a prominent MO contributor (Pete Clark) quoted a strong assertion of Kaplansky about countable choice vs. choice, and indicated that he agree with that assertion. As Tom Church noted in a comment on the second question, at least four other MO contributors seem to agree, since they upvoted Pete's answer.

Given Kaplansky's comment, and some sign that several people agree with it, I don't think it's unreasonable for someone to ask "what's the deal with countable choice vs. choice", which is essentially what the second question asks. (Incidentally, I don't see how it duplicates the first question, which was not in the least about the issue of countable vs. full choice.)

If this question were closed, I would immediately vote to reopen it.

Harry Gindi comments on "Choice vs. countable choice" (5240)

2010-04-29T17:02:40-07:00

I voted for closure because I agreed with Theo (I was the second vote).

Jacques Carette comments on "Choice vs. countable choice" (5239)

2010-04-29T16:56:04-07:00
Sorry, I needed to vent, just once. And I did it on meta rather than on MO! I was much more civilized in what I wrote than what I was tempted to write.

The system really needs to have more checks-and-balances, so that there ought to be a vote-against-close possibility. Or simply that you need max(votes-for + 5, 5) votes-to-close for a question to be closed rather than a flat 5. And maybe vote-to-close should be -20 on your reputation too, just so that it's see as a 'big deal'.

François G. Dorais comments on "Choice vs. countable choice" (5238)

2010-04-29T16:43:28-07:00

Jacques, one of the ideas behind the "Is this question acceptable?" category to give the opportunity for people to voice their opinions for and against closing. Since the voting mechanism doesn't allow votes against closing, this is a great place to make your opposition heard! However, it is better to do it in a civilized manner if you want to get your point across.

Jacques Carette comments on "Choice vs. countable choice" (5237)

2010-04-29T16:36:27-07:00
And it has now garnered some excellent answers. I really wish you guys would lighten up and allow somewhat subjective questions to not be closed a little longer, to see if they get 'good' answers or just generate noisy argument.

I am all for closing questions which seem to spin out of control. I am flabbergasted by the self-righteousness of some high-reputation people about what is a 'good' MO question.

François G. Dorais comments on "Choice vs. countable choice" (5232)

2010-04-29T15:44:45-07:00

This is regarding 22990, which currently has two votes to close. Theo posted the following comment explaining his vote:

I think this question should be fairly heavily edited. As it is, it is argumentative and subjective, and does not contribute more to the discussion of AC than what is already available on the other questions on the subject, including the linked question. I vote to close.

The "other question" is 22927.

(I'm not the other voter, but I want to give this system a try.)