|WikiProject Mathematics||(Rated Start-class)|
What does well-behaved mean in mathematics? It is used all over Wikipedia, but not defined.
Does it have something to do with how quickly a function changes, continuity, etc.?
— Rlschuh 20:48, 2004-03-31.
- It just means "behaves in a way which make it suitable for a certain field of study". It's the ordinary English meaning of "well-behaved". The nice behaviour is the one described in the original article. For example in Measurable function a well-behaved function is one that is measurable; in Newton's method a well-behaved region is one for which Newton's method works; in Monte Carlo method a well-behaved function is one for which the Monte Carlo method works. And so on. I suggest you remove the links you added, but leave this page for other readers who might imagine that there's a technical meaning. — Gdr 21:06, 2004-03-31.
- I hope you don't mind my turning your response into the article. I'm not convinced yet that the article should be unlinked.
- I have a vague impression from math, physics, and engineering classes I took that there is some more specific meaning. If the definition is just what you're describing, then many statements become circular if they use the description of something being well-behaved to explain why it works for that well-behaved thing. -rs2 21:37, 31 Mar 2004 (UTC)
- What you did is fine but you should have edited it. Properly understood, none of the uses of "well-behaved" makes a circular definition. For example, in Measurable function "well-behaved" isn't part of the definition, it's an additional piece of information. And in Newton's method the method itself doesn't use the term "well-behaved" so it's not circular for a "well-behaved" region to be one where Newton's method works. There's no single technical meaning of "well-behaved": each field of study has its own typical meaning or maybe several typical meanings. For example, in functional analysis, there are several kinds of "well-behaved" function: continuous, monotonic, differentiable, infinitely differentiable, integrable, etc. — Gdr, 21:47, 2004-03-31.
Whoever wrote this somehow expects it to be understood that mathmematics rather than some other subject is what this is about, even though it does not say so. That is a profoundly absurd assumption. Michael Hardy 23:25, 31 Mar 2004 (UTC)
I had rather put this article in VfD. Good behaviour cannot be explained as a mathematical concept more than in real life (it is something that depends on context, culture, time, space and probably colour). Pfortuny 11:35, 2 Apr 2004 (UTC)
- I'm almost convinced it should be unlinked from other pages, but it almost definitely has value as an entry. Unlike its common usage, "well-behaved" is often used as if both it had a clear meaning and the reader should know what that meaning is. I'm starting to suspect authors should use the term less, but the page definitely can serve the purpose of clarifying the term. --rs2 16:55, 2 Apr 2004 (UTC)
Someone inserted these lines before the initial informal list of examples:
- A well-behaved region is one for which Newton's method works;
- A well-behaved function is one for which the Monte Carlo method works.
Usually a mathematicians speaking of a "well-behaved function" means well-behaved in various other respects than that. Sometimes that is what would be meant. Similarly, the meaning of well-behaved region given above is just one of many; the meaning varies with the context, and the phrase is used when the context is expected to make clear which kind of good behavior is intended. It is highly misleading at best to say something that could give the impression that "well-behaved function" usually means one for which a Monte-Carlo method works. Michael Hardy 18:03, 19 Apr 2004 (UTC)
- Sorry about my "misbehavior" here. I see that I was not being a well-behaved editor. Thanks for (a) catching, (b) fixing and (c) telling me about my mistake. I will be more careful in the future. --Uncle Ed 20:29, 19 Apr 2004 (UTC)
I actually do not like the "Euclidean space being considered more "well-behaved" than the non-euclidean space" example. I dont think that it is better behaved. This may just my knee-jerk reaction though to someone saying something uncommon is not as nice or well behaved rather. What nice property is there that we have for a euclidean space that does not work elsewhere? I may very well be completely wrong on this, but at least i will learn something. Also, I like the use of the term nice, there is this idea of allowing people to lie, and using 'nice' works for that very well, it expresses what you want: "not the pathological counterexample you are concocting" -sean, math student
i feel that the page is too simplistic. what's well-behaved and what's not? it is not clear at all to me why a riemann-integrable function is better-behaved than a lebesgue-integrable function. fair enough, lebesgue integrability is a weaker notion, and therefore it's (in some contexts) more natural. in particular, via lebesgue theory it's possible to prove more results than via riemann theory. so what? i think there's a difference between truly well-behaved vs. ill-behaved cases (say, a stable system vs. an unstable system) and weaker vs. stronger properties (say, a once differentiable function vs. a twice differentiable function). --Deliou 08:13, 27 September 2007 (UTC)
Physicists also throw around the term "well-behaved" a lot. Usually they mean it roughly as discussed (following whatever rules are needed for easy analysis,) with an additional implication of "something I won't have to consult a mathematician about." :-) With respect to functions, it almost always seems to invoke continuity, and often also differentiability. Isomorphic 20:44, 19 Apr 2004 (UTC)
Reading the article again, it seemed to me that there are two slightly different definitions going on - one purely aesthetic, and the other practical. I put in the more practical definition that I'm used to hearing in physics, and separated it out from the aesthetic definition. Feel free to clarify further or change it if you feel what I wrote isn't correct. Isomorphic 21:14, 19 Apr 2004 (UTC)
Unreadability of this article
So, I came to this article trying to understand what Transportation problem meant with Borel-measurable function. The latter links here in saying it needs to be a 'well-behaved' function between measurable spaces. This article explains absolutely nothing about that requirement. What is it saying? Does the function need to be integrable? (how?) Does it need to be continuous? For all I know it needs to map from reals to reals and have a heart-shaped plot, and nothing else will do -- the page certainly gives no definition whatsoever.
It tries very hard to say something meaningful, and then fails. The partial ordering at the bottom (with which various people here disagree, clearly!) is not helpful either. It's like saying we want to minimize some number, but we're not going to tell you what domain that number is in (N, R, Z, some subset, whatever), or how that domain is defined. We can tell you, though, that 3 is less than 5, and 6 less than 10.
The worst offense is the page saying "Of course, in these matters of taste one person's "well-behaved" vs. "pathological" dichotomy is usually some other person's division into "trivial" vs. "interesting"."
It's like this page is the entry for Pretty and editors are trying to explain the notion by giving examples of objects which are prettier (uglier) than other objects, and then ultimately saying that there is no absolute way to know, good luck. I am pretty sure that mathematical proof and requirements for functions are not about taste.
I am not a mathematician proper, so I am not going to even try to fix this article, but someone better had. I am going to take out the aforementioned phrase right now, because it is so clearly utter rubbish that even I can tell. Gijs Kruitbosch (talk) 16:40, 17 July 2009 (UTC)
- As I thought this article makes clear, 'well-behaved' is not a precisely defined term in mathematics. Despite your objections, mathematicians use a lot of non-precise jargon, and the use of jargon is often a matter of taste. As for [[[measurable function]], that article gives the precise definition in the third sentence. Algebraist 20:18, 17 July 2009 (UTC)
- I see. So, I guess the main reason I am not satisfied by this article is that, although I speak a reasonable amount of English, it is not my native language, and in mathematics "ordinary" words do not necessarily mean what they would mean in normal conversation. In particular, if the article in question wikilinks something, I take it there may be some meaning I am not necessarily familiar with. "Well behaved" is vague at the best of times, but I would have expected that it would be well-defined when used in mathematics. That definition may be different in different use cases; fine. This article, if it should exist, should then state either this, or the different meanings. Now it attempts to do a little bit of both and ends up saying, in my eyes, nothing at all. The partial ordering at the bottom does not actually answer the question what "well-behaved" means, of course.
- As for this being jargon, and use of it being a matter of taste - point taken. However, jargon still (by definition) has a definition. As you say, its use may be subject to taste, fashion, etc.; however, its meaning should not be (and that is what the article says at the moment!). The definition of eg. the reals does not change overnight, and while I would be perfectly entitled to hold a strong personal belief that , I would be wrong. While there may be paradigm shifts, and these could be documented where and if necessary, it seems peculiar to just have an article that says that whatever the term means is up to the writer, but then still tries to say something.
- Perhaps that is what confused me about the article in its current state more than anything else: that it intends to accurately describe a topic, but then says on the one hand that there is no definition, and on the other that there are examples in which there is some agreed-upon meaning to "better-behaved", which (along with the introduction to the list) begs the question where the 'line' for "well-behaved"ness lies. That may well be a nonsensical question from a mathematical point of view (I am still not quite sure!) but if there is no relation, the information should not be in this article. If there is, the relation should be explained better.
- With regards to the measurable function article, all first three paragraphs seem to be three separate definitions for what "measurable function" means. It is not obvious to me what "well-behaved" as stated in the first paragraph means as related to the two other, distinct, definitions in the other paragraphs. Is its use in the first paragraph actually identical to the adjective "measurable"? If not, how does the article define well-behavedness as necessary but not sufficient for measurable-ness? If so, why is using the term enlightening in this case?
- I should apologise for ranting, I suppose - that was not my intention. I am glad you took the time to reply to my concerns, and hope we can figure out what, if anything, could be changed in this and/or the measurable function article to make it easier for weird people like me to understand the concepts more quickly, while maintaining accuracy. (Again, I would be bold, but I do not pretend to be a great mathematician, so I'd rather not make things worse than they are now) :-) Gijs Kruitbosch (talk) 22:10, 17 July 2009 (UTC)
- "Well-behaved" is mathematical jargon (as the term is used in our article), it is not precise mathematical terminology. As such, its meaning is highly variable depending on context and speaker. A "well-behaved" object is simply one which has some property which, in the context we are talking about, we have decided is desirable. In some cases, it may be fairly clear-cut which properties are the desirable ones, in other cases less so, but the concept is not precise and there's always some room for argument. Thus in measurable function the first paragraph notes that the property of being measurable is the correct notion of desirability for functions between measurable spaces (this is one of the clear-cut cases; there's not really any other sensible notion of well-behavedness for such functions), the second paragraph gives the precise definition of what a measurable function is, and the third paragraph gives an important special case. Whether it's a good idea for that article to start with a couple of vague intuitive sentences and then give the precise definition I don't know. Algebraist 15:16, 18 July 2009 (UTC)