Wikipedia talk:WikiProject Mathematics

Mathworld neologism at Heegner number?

Dear all, it appears that the term Heegner number was most likely made up by mathworld. I've started a section on the talk page to discuss whether or not this is so. If it is, I believe the correct course of action is to delete and merge content into other articles. Opinions welcome. RobHar (talk) 00:36, 5 August 2009 (UTC)

IMO, determining that a name is nonnotable is not in itself a reason to refactor the article. What content belongs together in a single article should be determined by the content we have: choosing a name for the article should be a secondary concern. (If you had determined that most of the current content of Heegner number was nonnotable and should be deleted, that would be a different issue.) —Blotwell 18:43, 5 August 2009 (UTC)

I felt that the content of the article would be better merged into other articles anyway. As it stands this article contains a definition and two marvelous, surprising, disjoint results in number theory, one of which had its own article (that User:PMAnderson has now merged into Heegner number, despite the term Ramanujan's constant being more prevalent than Heegner number) and the other was described in an article on similar results. There is also a List of number fields with class number one. Anyway, I would've settled for a renaming of the article. RobHar (talk) 15:13, 7 August 2009 (UTC)

Manual of style disagreement

On Talk:Exponential_function#Overview and motivation an editor has replied to my objections citing the maths manual of style with 'I wipe my arse with the Mathematics manual of style!!'. I don't mind arguing about whether some ground rules should or should not apply or what they mean or whether they should be disregarded in particular instances, but this doesn't sound like a basis for constructive discussion. Dmcq (talk) 16:19, 5 August 2009 (UTC)

My sense (which may well be wrong) is that the remark was intended semi-humorously. The other editor does seem genuinely concerned about the state of the article. I suggest that you assume good faith, point out how easy it is to misinterpret such remarks in online discussions, and suggest that further discussion be carried out in a more restrained mode. —Dominus (talk) 17:12, 5 August 2009 (UTC)

The editor has stated, "The single best way of introducing the exponential function is by its Taylor series." In any case, this kind of singular view (POV) is not appropriate for writing on Wikipedia in general, and math articles too should try and comply with WP:NPOV as reasonably as possible. --C S (talk) 23:25, 5 August 2009 (UTC)

Words genuinely fail me. Having an opinion on the best way to write an article now constitutes a point of view, in contravention of some Wiki-law or other? Is this the way mathematics articles are going to be written around here? Will every entity from a triangle to tensor have to be introduced with a historical overview and given multiple equivalent definitions for completeness?

In my own opinion, which is now a point of view, people do not come to mathematics articles on Wikipedia for chit-chat. They come here to find out what mathematical objects are. In short they come here for definitions and properties. My views is you tell them, in order 1) What it is, 2) What it does, 3) What it's used for. If you try and mix up this order you will be helping no one; only confusing them.

With regard to the exponential function and Taylor series, it would appear my point of view is not an isolated one(Encyclopedia Britannica presumably qualifying as a prominent adherent), so perhaps we should all form a WP:Cabal of some description to better discuss this alternative world view of e^x and how it can be given due weight and properly included so as best to preserve the neutrality, impartiality and unbiased nature of the Exponential Function article. ObsessiveMathsFreak (talk) 15:20, 6 August 2009 (UTC)

There are two different cases of your opinion being dealt with apparently. The first concerns your opinion on how to organize the content in the article. This is certainly something that should be discussed and debated. A consensus will be reached and that will be how the article is written. There are several math articles that are structured as you'd like them to be, however each article is dealt with individually. If you'd like to make sure that consensus is against you on this article, you could attempt to attract more editors to the discussion. In the end, you do have to accept the consensus opinion (and, as a note, consensus is not simply a democratic majority).

Secondly, your opinion on how to define the exponential function is an issue. You seem to chide "completeness" in an encyclopedia article, but completeness is exactly the point. To strongly state that there is just one good way to do it means that you are pushing a point of view. Now obviously there are more common ways of defining things, and these should generally be given more weight in the article. As I recall certainly many basic analysis books define the exponential function as a power series. Demonstrating this is a good way to support your point of view. People don't want you pushing your point of view. Stating that there is only one good way to define exp is a point of view. I for one prefer defining it as the solution to a first-order differential equation with initial condition. This is simple and embodies the meaning of exponential growth, and the power series then comes immediately out of Picard iteration. But I'm an algebraist and this is my point of view on how definitions should be given: simple and meaningful. RobHar (talk) 16:21, 6 August 2009 (UTC)

So, Wikipedia is not a textbook, yet we must include enough definitions for a Treatise. This article must be complete to the point of confusing the reader, yet Trivia sections and Pokemon articles must be excised at all costs. Nigh every other mathematics article has a formal definition for its first section, yet I must obtain a consensus for the Exponential Function article, which I note is unsurprisingly one of the top 500 viewed mathematics articles, is of top priority and currently has a less than stellar B rating(who ever gives out those).

I'll repeat my assertion that people do not come to the mathematics articles for chit chat, historical perspectives, unexpected interconnections, or strict rigour; at least not initially. The cast majority come here for basic information, so that is what should come first on the page, albeit after a brief introduction. Unfortunately, since such opinions, now being "Points of view", put myself and anyone holding them in the same category as creationists, moon landing conspiracy theorists, and Holocaust deniers, I very much doubt they will be taken seriously by any of the gatekeepers around here.ObsessiveMathsFreak (talk) 17:44, 7 August 2009 (UTC)

{<<<<< Indenting back}. It is another point of view that people do not come here for chit chat or historical perspective, it may be true but the essential point is that wikipedia is a hyperlinked encyclopaedia so you can always jump to the mathematical definition if you want to go in at the deep end. If you want it easy you just read straight on. If you feel you would move the consensus then the Manual of style page is the one to work on. If you don't then you don't have a consensus and you are just engaged in a fight to push your own point of view. It is perfectly okay to go against the manual of style but you'll need to justify yourself. And you haven't made a good case that I can see. As to starting exponential function with either a series or a differential equation, if you know anything about those you probably already know quite a bit about the exponential function. It is putting the cart before the horse to require them right at the start. And the series definition is particularly unilluminating to someone without much maths, it is just a random sequence of terms. The exponential function is a very basic function that people with very little mathematics come across. That is why it is one of the most popular mathematics articles. Dmcq (talk) 19:11, 7 August 2009 (UTC)

It seems no matter what I do or suggest I'll end up breaking sacred neutrality. All while others (neutrally!) go about happily making and undoing edits left and right. Is the act of making an edit in itself a biased act? Do we not exclude alternative opinions and pass skewed judgements when we reformat paragraphs, correct spelling mistakes and excise material?

Apparently, suggesting people come here for basic information is a point of view. Naturally, suggesting otherwise is of course not. Hence, editing articles to place emphasis on basic definitions and properties is obviously culturally, socially or morally biased in some way; whereas providing inquisitive readers with a panoramic vista of connections, applications and history of a function they know little or nothing about, complete with hyper-links to more topics they didn't come to read about is accordingly the more inclusive option. The former merely provides information plainly and concisely. The latter allows the reader to experience the wonders and potential of web 2.0 enabled, worldwide community sourced knowledge through digital collaboration. Oh, and unicorns.

Now, since the Null hypothesis (by consensus!) is clearly that people come to Wikipedia for a magical wonder-trip of learning, my alternate opinion... sorry point of view, that people are simply looking for basic information is something that must be proven.. sorry, which a consensus must be reached on. Obviously the way to do that... sorry to make a good case, would be to conduct surveys, questionnaires or various other analysis. Luckily however, no-one around here bothers with anything like that, and so we can turn to existing consensus for the final answer; which is that I'm a terribly biased, uncooperative old grouch, too mired in my outdated opinions on mathematical exposition to see WP:SENSE.

Take for example the specific case of the exponential function. I've been stuck so long in my equations that I can't see the intrinsic superiority in presenting a mathematically lay person with the definition

$\,\exp(x)=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}.$

which despite Bernoulli's great difficulties can clearly be seen to converge for any x by anyone first presented with it. Any proofs which might be needed are of course trivial. Moreover it speaks to their intimate understanding of the principles of compound interest, which even starving street urchins know as well as they would their own mother's lullabies. Just look at the confidence with which people in our modern world borrow credit!

Instead I would subject the unhappy reader to an infinite sequence of essentially random symbols, namely

$e^{x}=1+x+{x^{2} \over 2!}+{x^{3} \over 3!}+{x^{4} \over 4!}+\cdots =\sum _{n=0}^{\infty }{x^{n} \over n!}$

which they could have no hope of ever understanding, seeing as how such esoteric mysteries as infinite series and the ratio test are reserved only for a select elite; namely 14 year old schoolboys(girls/persons/aspiring learners). Moreover, my subsequent appeal to the ease with which the function is shown to be its own derivative would only compound the folly, as defining the function as an eigen-function of differentiation makes this step confusingly redundant and is after all the more illuminating introduction the the whole topic.

So what is it that makes me persist in my disgraceful bias, my shameful lack of neutrality? What is it that drives me to try and improve articles with elitist notions of coherent mathematical exposition? Malice, perhaps? Or hate? Bile? No doubt my nightly sojourns with neo-Nazi's and crackpot Pseudoscientists have driven all the consensus out of me and left me a wretched point-of-view filled troglodyte.

So, it is good that one of the most popular mathematics articles on this site, nay on the web, is do diligently defended from the vile vandalisms which I might unleash upon it. A solid defence of rules, regulations, guidelines, customs and procedures, each more unbiased in intent than the last, stand squarely in the path of unhelpful interlopers like myself who know little and less of how an online repository of human knowladge should be properly run. Why, leave the page to my tender mercies, and people might log on, get the information they came from, and then leave! We can't have that! ObsessiveMathsFreak (talk) 22:58, 7 August 2009 (UTC)

You are not making allies. Ozob (talk) 00:38, 8 August 2009 (UTC)

I said illiterate peasants rather than street urchins and there was a couple of other small changes but yes I agreee with most of the stuff the sarcasm is about. I even read your reference We can't have that! and have come to exactly the opposite conclusion about what the problem was. The article that person looked up contained terms he did not understand before he came to a decent definition of what he was looking up. It may have been that it wasn't really possible to satisfy his requirement, I don't know as I don't know what the original query was, but for the exponential function I really do fail to see how a series which is essentially arbitrary as far as someone not in the know is concerned is better than compound interest. That it is easier for a mathematician to work with is not very relevant that I can see. A quick mathematical definition is given in the leader and the mathematical definition part gives things which are easier for a mathematician. The other articles which are more of the straight into the maths style are probably not ones that people with only very elementary maths will come across. 10:55, 8 August 2009 (UTC) —Preceding unsigned comment added by Dmcq (talk • contribs)

Everyone here supports the idea of coherent mathematical exposition. But inherent in the idea of "exposition" is that the reader will read the article, not merely skim it looking for a bulleted definitions.

Also, concerning the exponential function, there are numerous definitions, and we need to present them all. But unlike a textbook, there is no reason why our introductory articles need to pick one particular definition as the definition, above the others. There are at least three independent definitions of the exponential function, and each has its own role. — Carl (CBM · talk) 12:16, 8 August 2009 (UTC)

"reader will read the article, not merely skim it looking for a bulleted definitions"

Generally speaking, I don't think that is a valid assumption. Nor is it reasonable — as possibly mentioned in a previous discussion about infoboxes, there is no reason why articles should intentionally be made harder to use for readers just looking for something specific (that is why we divide articles into sections after all) and require everyone to read the whole article no matter what their level of understanding. Shreevatsa (talk) 15:21, 8 August 2009 (UTC)

Lambda calculus

(Cross-posting from Comp. Sci. wikiproject since activity is rather low there) Can someone with (at least) a graduate-level understanding of the topic take a look at the article, in particular the confusion with various typed lambda calculi; see the article's talk page for details. Pcap ping 17:33, 5 August 2009 (UTC)

Oops, the lead is written in an opaque and highly questionable fashion. The issue of consistency is possibly relevant to the foundational ambitions Church had; but it is not so relevant to introducing lambda calculus. This all looks wrong-end-of-the-microscope to me, as if the paradigm was Russell's theory of types, rather than functional programming. (That, historically, would make sense, but the weight of current attention would surely be in FP.) Since untyped lambda calculus is basically the situation with one type X that coincides with X → X, it is feasible to treat it in the foundational article as the case one should look at, so that the labelling with types is treated as at best a distraction. Anyway the historical introduction and air of paradox should be moved out of the lead, and the idea that "lambda calculus" is at minimum a notation for keeping track of higher-order functions should be given fair play. Charles Matthews (talk) 18:20, 7 August 2009 (UTC)

Deleting Pie method

I've put a prod on Pie method which is a putative method of fair division because I believe it is simply wrong. I actually found a place on the internet though where somebody quoted it though not as the 'pie method' and it probably didn't come from wikipedia! I sort of wonder if it is notably wrong and I should keep it and say it is rubbish? Perhaps I should put it under Proportional (fair division) as an attempt which is wrong and explain - but then the explanation could be counted as WP:OR. Dmcq (talk) 23:34, 7 August 2009 (UTC)

It's just Divide and choose, isn't it (with a completely wrong extension to >2 people)? Algebraist 14:49, 8 August 2009 (UTC)

It's a shame that divide and choose does not give any discussion to the problem of fair division between more than two parties. I gather that this has been the subject of considerable research over the years. —Dominus (talk) 15:01, 8 August 2009 (UTC)

Why should it discuss that? Divide and choose is about the divide-and-choose algorithm, which applies only to two-party division. The general problem of fair division is covered at fair division, as it should be. Algebraist 15:05, 8 August 2009 (UTC)

I see, thanks. —Dominus (talk) 05:52, 9 August 2009 (UTC)

I believe the last divider method in Proportional (fair division) is probably what the person who wrote the article meant butnthey left out an important part. Dmcq (talk) 16:50, 8 August 2009 (UTC)

MAJOR CHANGES: 4 hours ago

By a very simple verified equation we wiped out Prime numbers and Riemanns Hypothesis articles that are rendered obsolete and Wikipedia is the first place we went because you treated us with freedom and respect. The simple equation that is verifiable at face value was posted at the Math forums etc 4 hrs ago"IS 180-PRIME NUMBER(below180)= 180+PRIME NUMBER(any over 180) Till infinity ,So there is no need to be digging for these prime numbers now any more. See also the site Inverse19mathematics.com, or google inverse19 mathematics. THIS IS SIMPLE VERIFIABLE AS IT IS(ipso facto ). GO WIKPEDIA BE THE FIRST. Vinoo Cameron M.D , Theo Denotter.--Vinoo Cameron (talk) 05:38, 8 August 2009 (UTC)--Vinoo Cameron (talk) 05:38, 8 August 2009 (UTC)

This needs peer-review and publication by a reliable source before it is acceptable here as we cannot accept original research. Please propose this again when this has happened. Rodhullandemu 13:27, 8 August 2009 (UTC)

I do not think I quite understood that which was asserted, but if my interpretation is correct it seems either trivial or incorrect. Furthermore, I do not think that such a trivial equation can resolve the theory of prime numbers or the Riemann hypothesis completely (or to even a small extent). --PST 09:28, 9 August 2009 (UTC)

Shouldn't the title be MAJOR CHANGES: 2 days and 5 hours ago? Dmcq (talk) 08:19, 10 August 2009 (UTC)

Compact -> Compact Hausdorff

I would like to propose a change for the convention. Can we assume that a compact space is Hausdorff (and use quasi-compact for a space where an open cover has a finite subcover)? I think today this is fairly standard and helps to reduce clutters.

One problem with this change is what we do with other related notions like locally compact, or compact generated space (i.e., k-space): should we assume them also to be Hausdorff or not. I don't have a concrete idea for this problem. -- Taku (talk) 12:40, 10 August 2009 (UTC)

Is this fairly standard? I've only ever seen this mentioned as something Bourbaki does; I've never encountered anyone who actually used this convention themselves. Algebraist 13:52, 10 August 2009 (UTC)

I am looking at the textbook "Topology" by James R. Munkres (second edition, 2000); there, a compact space need not be Hausdorff. The same holds for "Introduction to topological manifolds" by John M. Lee (2000). Boris Tsirelson (talk) 14:35, 10 August 2009 (UTC)

So far as I know, the definition that includes Hausdorff is relatively standard in France and very widespread in Germany, but not at all standard internationally. I think it's best to use only the terms compact Hausdorff and quasicompact in topology contexts, except where the two notions are equivalent. This is analogous to how we are already dealing with $\subseteq /\subsetneq /\subset$. It minimises the potential for misunderstandings. And we can still say "Let X be a regular space. If X is compact...". Hans Adler 14:51, 10 August 2009 (UTC)

I think it may be a matter of the branch of mathematics in question. Certainly within algebraic geometry and number theory the Bourbaki convention is pretty strictly followed. Schemes are almost never Hausdorff, but often quasi-compact, so one often deals with quasi-compact spaces, which, frankly, are not nearly as nice as (Hausdorff) compact spaces. In a branch like topology, one can afford to allow "compact" to apply to non-Hausdorff things since one quickly restricts oneself to studying topological manifolds, which are generally taken to be Hausdorff (as they are in John M Lee's book). I am strongly in favour of the Bourbaki definition. RobHar (talk) 15:09, 10 August 2009 (UTC)

The "usual" definition of compact does not include Hausdorff. This is supported by the "standard" texts, Willard, General Toplogy (1970), Steen & Seebacch, Counterexamples in Topology (1970), Armstrong, Basic Topology (1997), Bredon, Topology and Geometry (1997), Munkres Topology (1999), etc., as well as references like Schecthter, Handbook of Analysis and Its Foundations (1997) and Hazewinkel, Encyclopaedia of Mathematics (2002). In my experience as a practicing topologist Bourbaki is definitely in the minority. Whatever our personal definitional preferences our, we should follow the standard sources. Paul August ☎ 16:01, 10 August 2009 (UTC)

The "usual" definition in topology does not include Hausdorff, but what I'm saying is that the "usual" definition in algebraic geometry does. Now, one can say that compactness is a topological notion, and that's fine, but despite this, I believe the bourbaki definition to be better. Questions: as a practicing topologist, how often do you study non-Hausdorff spaces? How many of the texts you list only refer to non-Hausdorff compact spaces to discuss their pathology? RobHar (talk) 16:14, 10 August 2009 (UTC)

Rob, I'm not sure what you are getting at. My only position here is that, based upon current usage, Wikipedia in its article compact space ought to continue to define a compact space as it currently does (every open cover has a finite subcover). Do you disagree with this? Paul August ☎ 16:32, 10 August 2009 (UTC)

Yes, I do. I'm not simply arguing for the sake of arguing. My position is that, also based on current usage, Wikipedia should change its definition. My argument is that though topologists often define compact without Hausdorff, they rarely actually use it (beyond basic pathological examples in textbooks), whereas algebraic geometers, who actually use non-Hausdorff compact spaces require the Hausdorff condition, and otherwise use the term quasi-compact. I may be wrong in the statement that topologists rarely use non-Hausdorff topological spaces, and this is why I asked you the above questions. You've shown that the standard definition in (basic) topology texts does not require Hausdorff, but that does not mean that the standard definition in mathematics is so. As an example that a major branch of mathematics does not use the same definition as topology, I point out algebraic geometry. I realize my opinion may (or may not) be a minority one, but I saw fit to express it. RobHar (talk) 16:50, 10 August 2009 (UTC)

My area of research was in categorical topology, so I'm less familiar with the current usage of the term in algebraic geometry, though I would guess that many texts will not in fact give a definition of something so basic. Note however Smith, An invitation to algebraic geometry, p. 9:

A compact space is a topological space for which every open cover has a finite subcover. Some authors call those spaces quasicompact reserving the term "compact" for Hausdorff spaces with this property.

Notice the word "some". Also that most modern research may be conducted in contexts where all spaces are Hausdorff is really neither here nor there.

Paul August ☎ 17:28, 10 August 2009 (UTC)

I happen to know that Karen Smith says "quasi-compact" just like all of us algebraic geometers do. And when she describes a variety as "compact" she, again like all of us algebraic geometers, means "complete" (i.e., the map to the one-point space is a proper morphism). This is simply a convention. The convention of using "quasi-compact" in algebraic geometry comes from two things, I think:

1. There is a strong French influence coming from Serre and especially Grothendieck.
2. There is more than one natural topology on an algebraic variety. If X is a complex variety, then you can either view it as a scheme (with generic points and the Zariski topology), or you can view it more classically as its set of complex points and their Euclidean topology. Whether or not a variety is compact changes depending upon your viewpoint: All varieties are quasi-compact in the Zariski topology by definition (and in fact, non-quasi-compact schemes are usually so pathological that they are not even considered); but to be compact in the Euclidean topology is to be compact in the traditional sense, and this is a strong assumption, just like compactness of a manifold is a strong assumption.

So what should Wikipedia do? I think we all agree that "compact" for "every open cover has a finite subcover" is the most common usage in English, and I, together with the other algebraic geometers here, can attest that "quasi-compact" is an active, but minority, usage. My own opinion is that we should stick with the most common usage of "compact" and mention "quasi-compact" as a notable minority usage. Ozob (talk) 17:45, 10 August 2009 (UTC)

Agree. And this is in fact what our article currently does. However more ought to be added about the "minority" usage, in particular reflecting what Ozob has written above. Paul August ☎ 18:06, 10 August 2009 (UTC)

I also agree. I work in both differential and algebraic geometry, but am also aware and interested in work in computer science, logic and category theory that uses topological spaces which need not be Hausdorff. Combining the two concepts in one definition is conceptually bad, in that Hausdorff-ness is a local property, whereas compactness is not. However, it is not Wikipedia's mission to change the world (even the mathematical world), only to reflect what we find in reliable sources. Geometry guy 20:13, 10 August 2009 (UTC)

(undent) Hausdorff is not a local property. If it were, non-Hausdorff manifolds would not exist. Plclark (talk) 21:54, 11 August 2009 (UTC)

I am somewhat inclined to the view that:

• Some spaces are Hausdorff spaces, but no space is Hausdorff, because Hausdorff was a person. In other words, "Hausdorff space" is a compound word, not a phrase in which "Hausdorff" is an adjective.
• Some compact spaces are not Hausdorff spaces. There's this nice little undergraduate exercise that says:

Suppose T is a topology on X and (X, T) is a compact Hausdorff space. Suppose S is some other topology on X. If S is finer than T, then (X, S) is not compact; if S is coarser than T then (X, S) is not a Hausdorff space. This makes me wonder if maybe there's some way of looking at it (e.g. something in category theory, maybe) from which "compact" and "Hausdorff" are some sort of duals of each other.

I'd rather keep the terminology as is.

Michael Hardy (talk) 21:25, 10 August 2009 (UTC)

Wow, I didn't expect this much response :) I agree that the convention depends on fields. But I would point out that you rarely see "compact Hausdorff" because what happens most of the time is that the underlying space is often assumed to be compact to begin with: manifolds, topological vector space/group, etc. So, compact groups or compact manifolds are Hausdorff. (Yes, I noticed I use Hausdorff as an adjective, despite Michael Hardy's objection, but I don't think it is only a minority who commits this misusage. It's similar to "Cauchy"; you say "Since the sequence x_n is Cauchy, it is bounded and ....) The argument for compactness subsuming Hausdorff-ness is therefore simple and natural in that it only tries to reflect the reality. By the way, there is a nice explanation for the duality that Michael Hardy wondered about: in a Hausdorff space an ultrafilter converges to at most one point: in a compact (not necessarily Hausdorff) space an ultrafilter converges to at least one point. Thus, a compact Hausdorff space is where an ultrafilter converges to exactly one point. Conceptually speaking, in other words, it makes no sense that there is no term for compactness plus Hausdorff-ness, as if we have terms "injective" and "surjective" but not "bijective". Since a non-Hausdorff compact space is pathological (without doubt?), "quasi-" is also a very appropriate prefix to use.

Of course, if the standard doesn't seem to have adopted this Bourbaki convention, we can't adopt it in Wikipedia either. But really? Yes, some textbooks on topology don't follow Bourbaki, but isn't that simply because they are old? I thought Bourbaki texts are "definitive accounts" on many things, including topology. In fact, many texts refer to Bourbaki for results on topology. -- Taku (talk) 22:23, 10 August 2009 (UTC)

The sources I cited above are not old, here are few more modern texts: Shick, Topology: Point-Set and Geometric (2007); Reid, Geometry and Topology (2005); Crossley, Essential Topology (2009), I'm sure there are many more. Can you provide any topology texts which follow Bourbaki? Paul August ☎ 01:09, 11 August 2009 (UTC)

I don't know of any topology textbooks that follow bourbaki, but then again I've never read a topology book. I also make no claim that topologists follow Bourbaki, though it's possible some do. From Hans Adler's comments, it seems like a good place to look would be French or German books. I know that EGA, SGA, Hartshorne and Mumford's red book all follow Bourbaki. RobHar (talk) 01:33, 11 August 2009 (UTC)

For some reason, I can't edit Wikipedia with my user account. (What did I wrong??) J.P. May's A concise course in Algebraic Topology follows Bourbaki. It is the only topology book I have ever read (or more precisely trying to read :). Maybe that's why I got a wrong impression. -- Taku —Preceding unsigned comment added by 67.186.28.195 (talk) 02:21, 11 August 2009 (UTC)

A Google books search returns a few more examples, such as: Topology by Horst Schubert, General topology by Ryszard Engelking, Handbook of the history of general topology by Charles E. Aull and Robert Lowen, Lectures on algebraic topology by Albrecht Dold. In Klaus Jänich's book "Topology", he doesn't assume the Hausdorff condition, but he says "Many authors call such spaces 'quasicompact'", whatever "many" means. RobHar (talk) 04:20, 11 August 2009 (UTC)

To add to the categorical point of view of "duality" between compactness: for schemes, the analogue of Hausdorff is "separated" and the analogue of compact is "proper". The so-called valuative criteria of separatedness and properness offer the same kind of relation that Taku describes above for ultrafilters. There's a nice discussion of the analogy between properness and compactness in 1.9 of Mumford's Red Book (where he's talking about complete varieties) and in Bourbaki's topology book. RobHar (talk) 22:48, 10 August 2009 (UTC)

In model theory, we have an invariant of complete first order theories that is called the Lascar group. Its inventor defined that a theory is called G-compact if its Lascar group is a compact Hausdorff group. Since the group is always quasicompact, this amounts to saying that it's Hausdorff. This may make sense in French, but based on observations on several occasions I would say it confuses most model theorists outside France, because they expect compact=quasicompact.

I still maintain that the best thing we can do is to use quasicompact or compact Hausdorff whenever there is a difference, and compact when there is none. Since we are writing for an international audience of people from different subfields of mathematics, this is the only way to make sure that our readers needn't guess what we mean. Even if we could agree on one of the two main conventions for the entire project, there would always be some articles that wouldn't follow the convention, e.g. because they are recent additions by a new author who doesn't know about the convention. And it still leaves the flexibility of defining compact as one of the two variants at the beginning of an article, if it's necessary to prevent awkward language. Hans Adler 23:26, 10 August 2009 (UTC)

Let's look at some other Wikipedias:

• FR: A topological space E is called quasicompact if it satisfies the Borel-Lebesgue axiom: of every open covering one can extract a finite subcovering. The space is called compact if it is also separated [i.e. Hausdorff].
• DE: Some authors such as for example Boto von Querenburg [an influential German Bourbaki-style topology text] use the term quasicompact for the property defined here and reserve the term "compact" for compact Hausdorff spaces; due to French influence this is customary especially in algebraic geometry.
• IT:Some authors require that a compact space be Hausdorff; in this case, a space that satisfies the present condition but is not Hausdorff is called quasicompact.
• ES:[Defines compact without Hausdorff. The French terminology is not even mentioned.]

Hans Adler 06:52, 11 August 2009 (UTC)

Very interesting. So, this is really a French convention. Let me add:

• JA: A topological space X is said to be compact if every open cover of any subset of X has a finite subcover. Bourbaki refers to a compact Hausdorff space by a compact space and use "quasi-compact" for a compact but possibly non-Hausdorff space.

(By the way, isn't this definition incorrect?)

-- Taku (talk) 12:21, 11 August 2009 (UTC)

(It is. According to this definition, any compact Hausdorff space is discrete, and therefore finite.) — Emil J. 13:08, 11 August 2009 (UTC)

This is the definition of a Noetherian space (aka Heriditarily compact). Since the spectrum of a Noetherian ring is a Noetherian space, what Emil J has pointed is out is why most schemes aren't Hausdorff. RobHar (talk) 13:29, 11 August 2009 (UTC)

I'm so sorry. I got the translation wrong :) For the record, this is the correct one

• JA: A subset of a topological space is said to be compact if its open cover has a finite subcover.

So, this is slightly more general but is actually equivalent to the usual one (and definitely correct). -- Taku (talk) 21:01, 11 August 2009 (UTC)

Not to pile onto Paul August's list, but Davis (2005, p. 87) also does not include Hausdorff. Unfortunately I can't seem to find my algebraic geometry text at the moment. CRGreathouse (t | c) 07:01, 11 August 2009 (UTC)

I very much favour Hans's position above - "use only the terms compact Hausdorff and quasicompact in topology contexts, except where the two notions are equivalent. This is analogous to how we are already dealing with $\subseteq /\subsetneq /\subset$." Where authors are inconsistent, the best way to avoid confusion is to rely solely on unambiguous terms, even if that usage isn't consistent with any particular author. Dcoetzee 07:37, 11 August 2009 (UTC)

Could anyone remind me how we're dealing with $\subseteq /\subsetneq /\subset$? My understanding is that we're dealing with it by not dealing with it. -- Taku (talk) 12:30, 11 August 2009 (UTC)

It's described at Wikipedia:WikiProject Mathematics/Conventions#Notational conventions:

Subset is denoted by $\subseteq$, proper subset by $\subsetneq$. The symbol $\subset$ may be used if the meaning is clear from context, or if it is not important whether it is interpreted as subset or as proper subset (for example, $A\subset B$ might be given as the hypothesis of a theorem whose conclusion is obviously true in the case that $A=B$). All other uses of the $\subset$ symbol should be explicitly explained in the text.

By the way, the most creative approach to these symbols that I have seen so far was when a single formula $a_{i}\subset a_{j}$ that occurred in a definition absolutely had to be interpreted as $\subseteq$ for certain values of i,j and had to be interpreted as $\subsetneq$ for certain others. Hans Adler 12:43, 11 August 2009 (UTC)

Ah, so here it is. Am I the only one to think that these guidelines would be much easier to find if they were integrated in MOS:MATH? — Emil J. 13:02, 11 August 2009 (UTC)

No. Good idea. Hans Adler 14:54, 11 August 2009 (UTC)

In Encyclopaedia of Mathematics, "Compact space" [1] has this comment:

In the West "compact" is used for both compact and T_2-compact, and the former is sometimes called quasi-compact. In topology the majority of Western authors equate compact and compact Hausdorff (T_2-compact), because the latter spaces are much better behaved; on the other hand in, e.g., algebraic geometry the term compact does not as a rule include T_2.

(The emphases are mime.)

I'm somehow unsure about the accuracy of this. I thought "quasicompact" typically appears in algebraic geometry. -- Taku (talk) 12:29, 11 August 2009 (UTC)

Having read this thread, I feel that some strong statements were made by User:RobHar, but that is simply my opinion. Firstly, "topology" is not exclusively reserved for manifolds; there are people who research point-set topology, set-theoretic topology and other related branches. Furthermore, non-Hausdorff spaces are not simply "pathological examples" in textbooks. There is an interesting theory behind non-Hausdorff spaces; one example is the result that the bug-eyed line does not have the homotopy type of any Hausdorff space (although it is a topological manifold). This, although fairly well-known, serves to illustrate how even "nice" Hausdorff spaces have a complex structure. I do not disagree that major areas of mathematics such as algebraic and differential geometry (and topology) assume all topological spaces to be Hausdorff, but in those branches, topology is a "tool" and not an object of study. By incorporating a Wikipedia convention that "compact" is to mean "compact Hausdorff" suggests not only the non-existence of many "pure topologists", but also the seemingly popular view that non-Hausdorff is pathological (and this is certainly incorrect). It may be that I am biased, by I vehemently oppose to the suggested convention. --PST 14:31, 11 August 2009 (UTC)

Here's an example of a non-Hausdorff topological space which arises naturally but which I think nobody has mentioned: L^p with the strong topology. As we all know, a measurable function R → R is in L^p if and only if its p-norm is finite; and in particular, all functions that are almost everywhere zero have zero p-norm for every p (even in the goofy cases when 0 < p < 1). Now, a lot of books take the quotient by the subspace of a.e. zero functions, but not all do; e.g., David Williams's Probability with Martingales doesn't, and even makes the claim that taking this quotient is never done when working with continuous-time objects in probability theory (I don't know enough probability to verify the truth of this claim, though).

More generally, of course, any seminorm which is not a norm will define a non-Hausdorff topology on a vector space. Ozob (talk) 15:18, 11 August 2009 (UTC)

My statements were made with the explicitly stated caveat that they may be wrong and were accompanied with questions. I realize that topology is not only algebraic and differential topology (subjects in which topology is the object of study), however I am pretty sure that point-set topology, set-theoretic topology, etc form a very small minority of the subject. In other words, in terms of usage of the term, compact most often means compact Hausdorff since the space is assumed Hausdorff to begin with. I also don't see how the "bug-eyed line" is not pathological. I believe that it is true that non-Hausdorff is pathological in most areas of study.

Anyway, I find Hans Adler's suggestion to use "quasicompact" and "compact Hausdorff" (mimicking the convention on subsets) to be quite sensible. Especially since in most cases it would be sufficient to simply say "compact" as the object in question will likely already be assumed Hausdorff. This is probably my first choice of convention.

However, it is true that different subjects have different conventions, and a possible compromise would be to say that in articles in topology (and someone mentioned some computer science) the term "compact" need not refer to a Hausdorff space, whereas otherwise (or something) it refers to compact Hausdorff. As most subjects other than topology (and algebraic geometry/number theory that use the term quasicompact) only deal with Hausdorff spaces only "compact" will be required. This would thus not represent a significant change to the wording in almost all articles. Thoughts? RobHar (talk) 15:14, 11 August 2009 (UTC)

So, are you asserting that point-set topology and set-theoretic topology are less important than differential geometry and topology? --PST 09:44, 12 August 2009 (UTC)

So much discussion. We have basic conventions to avoid getting sucked into such time-consuming stuff. "Quasi-compact" as used in scheme theory is a standard definition and means what you'd guess, but it is not a definition most mathematicians have to worry about. I think Bourbaki had a rather limited point in making that definition back in the day, and we lose little by ignoring the point in our conventions. Charles Matthews (talk) 16:07, 11 August 2009 (UTC)

Let me weigh in, as someone who has spent some time re/writing the articles on general topology. I have various comments:

1) Both conventions have a great deal of support. The use of quasi-compact is more widespread in French than it is in English (indeed, the above references seem to show that compact virtually always implies Hausdorff in French), but it is certainly widespread in English as well. As a rough rule of thumb, "quasicompact" is preferred by the algebraists (including algebraic geometers, algebraic number theorists, model theorists, etc.) whereas "compact" is preferred by the analysts and geometric topologists. There is enough use of each convention that it seems absolutely mandatory to mention both alternatives as being in common use.

2) In terms of authoritative texts on General Topology, in my opinion (as someone who has spent some time perusing them) the following are the most authoritative, in historical order:

1955 Kelley's General Topology 1958 Bourbaki's Topologie Generale 1970 Willard's General Topology 1975/1977 Engelking's General Topology

I find it strange that nowadays people seem to name Munkres' book as the definitive reference on the subject. It is a very nicely written book (it was used for my first course on topology, and I had an entirely positive experience with it), but it does not have the scope of a reference. From the author's preface: "This book is intended as a text for a one- or two-semester introduction to topology, at the senior or first-year graduate level."

In terms of the authority of the above books, I would rank them in descending order as: Bourbaki, Engelking, Willard, Kelley. Note that the first two of these use the term "quasi-compact".

3) Except for the fact that it is probably not in majority use among English-speaking mathematicians, I have never heard a reasonable argument against Bourbaki's convention. There are many arguments for it, most of all the fact that it clues the student in to the fact that many of the nice properties of compactness in metric spaces hold only when the Hausdorff axiom is assumed. Moreover, the alternate terminology gets awkward when one is seriously interested in non-Hausdorff spaces. For instance the term "compactification" is used in every text I have ever seen to mean "Hausdorff compactification", but the fact that this is not built into the terminology can cause confusion.

4) I would myself prefer that wikipedia adopt the quasi-compact convention. This would be a progressive move: choosing terminology that we feel is best even if it is not in the majority use. I appreciate though that this is a big step for wikipedia to take. I think that Hans Adler's advice is ultimately best: mention both conventions in the foundational articles, and then in the applications try to phrase things so as to make sense independent of which convention has been chosen. Plclark (talk) 21:54, 11 August 2009 (UTC)

What a plethora of opinion. Plclark, I gave a perfectly good argument against the Bourbaki convention above, but you didn't read it: Hausdorffness is a local notion, whereas compactness isn't. Conflating the two is logically confusing and unacceptable in most fields. Bourbaki and others were simply seduced by the happy combination that compact and Hausdorff spaces provide (uniqueness of the compact Hausdorff topology etc. etc.) You can pick and choose your references according to your prejudices, but you can't argue against the principle that there is no such thing as a locally Hausdorff space, whereas locally compact spaces are prevalent in mathematics. Geometry guy 23:23, 11 August 2009 (UTC)

Actually he did read your argument, and he commented on it above. He said Hausdorffness is not a local notion. RobHar (talk) 23:42, 11 August 2009 (UTC)

My apologies: we evidently have different conceptions of what "local" means. It is precisely the nonexistence of locally non-Hausdorff spaces which differentiates Hausdorff-ness from compactness. Geometry guy 23:49, 11 August 2009 (UTC)

Geometry guy, I'm afraid I have no idea what you're talking about. Are you familiar with local property and locally Hausdorff space? I can't tell from what you write. Certainly there are spaces which are not locally Hausdorff (or even locally non-Hausdorff), for instance the trivial topology on a set with more than one element. Plclark (talk) 07:08, 12 August 2009 (UTC)

My apologies again. I'm only used to dealing with topological spaces in specific arenas (albeit several unrelated ones that include Hausdorff and non-Hausdorff spaces). The notion of local you describe is entirely the right one, and I use it regularly myself. So I should not have caused confusion by using the word "local" in this context. What I meant was that a witness to non-Hausdorffness only involves two points and arbitrarily small neighbourhoods of these points. In contrast compactness is a feature of the whole space. One can delete a point from a non-Hausdorff space to make it Hausdorff, while one can also delete a point from a compact space to make it noncompact. This seems very different to me, but perhaps you can clarify. Geometry guy 20:58, 14 August 2009 (UTC)

To respond to Plclark's 3), I'm no expert but I believe that the strongest argument against the inclusion of Hausdorff-ness would be that the category of Hausdorff spaces is not well behaved (I never quite understood what topologists mean by "not-well-behaved"). Hence, it is important to work with -- in algebraic topology in particular -- the category of spaces with some weaker separation axioms such as weak Hausdorff space. (See also [2]) This is not surprising since Bourbaki introduced their convention before the category theory became mainstream. -- Taku (talk) 11:59, 12 August 2009 (UTC)

It is usual at some point to "appeal to common sense". As far as I can see, typing "compact Hausdorff" is not really so bad if you know you have to do it; and the advantage seems to be that with the two terms "compact" and "compact Hausdorff" in use, the worst that happens is that someone may read a proposition like "if X is a compact space then blah" in too restrictive a sense. Which cannot be said if Hausdorffness is tacitly assumed. Charles Matthews (talk) 14:11, 12 August 2009 (UTC)

The consensus seems clear by now, but let me indulge a bit more, because I don't understand Charles Matthews' comment at all. Yes, it is true that there is no serious problem that could be solved by adopting a new convention: there is nothing wrong with "compact Hausdorff", especially because like 99% of times spaces are Hausdorff and so this is usually simply non-issue. (Likely, I was bored before a new semester, which started the whole thing :) But, but, why reject the idea of having a discussion on conventions at all? It is important to adopt a correct convention; not just because that helps the reader but because that's the whole point of this project. Isn't it? We strive for the accurate description of (contemporary) mathematics, and the choices of conventions are therefore extremely important because they're reflection of philosophy. It is possible that, as PST pointed out, adopting the Bourbaki convention gives a wrong impression that certain materials in topology are unimportant (because they are?) I don't see why we, as the authors of this encyclopedia, can't have a long discussion then choose conventions that best reflect views that we think correct? Because we can't agree ever or why try? (Excuse me for ranting.) -- Taku (talk) 01:34, 13 August 2009 (UTC)

Well, the discussion seems to show that your initial comment I think today this is fairly standard is simply not correct. Of course conventions can be discussed - there is a talk page for the conventions page. I thought you were misunderstanding a little "quasi-compact" as it used in scheme theory, and its significance. The Bourbaki style is (was, I think) to be very aggressive in discussions on terminology and conventions. That is not very suitable for us, and we have to compromise a little between reflecting the terminology used by mathematicians, and being self-consistent. Sometimes this means accepting "least bad" solutions to convention issues. Charles Matthews (talk) 16:26, 13 August 2009 (UTC)

Conventions

Following a suggestion of Emil J., I've created a new section of the math MOS: Wikipedia:Manual of Style (mathematics)#Conventions. This is mostly a link to the current page on conventions, Wikipedia:WikiProject Mathematics/Conventions. I feel like it would be a big improvement if the conventions page were merged into the MOS: The conventions page is short, is highly relevant to the MOS, and would be easier to find and maintain. Does anyone else have an opinion on this? Ozob (talk) 15:57, 11 August 2009 (UTC)

In the past sometimes people have objected to including things in MOS subpages that are not strictly speaking style issues. But I think that was often motivated by unrelated political concerns, and I hope there are no such politics involved here. I support the move. It's not entirely clear to me whether the logic conventions should be merged as well, since they are also used by philosophers, who might not otherwise be interested in MOSMATH. But that can be decided later (and in the logic project). Hans Adler 16:10, 11 August 2009 (UTC)

Probably inevitable. We do need to recognise the implications, and that the MoS generally has taken on a much more prescriptive role in recent years. Which is not always for the best. Charles Matthews (talk) 17:00, 11 August 2009 (UTC)

It is not clear to me how many of our articles follow the conventions page, and when they do it is possible they do so only because the conventions described are already somewhat common elsewhere. But I don't have any strong objection to the merge as long as some cautionary language about not applying them blindly is present. I added that to the MOS just now. — Carl (CBM · talk) 12:15, 12 August 2009 (UTC)

I'm sorry if this is the wrong place to write this (please delete if so), but there needs to be more consistency with respect to how formula are presented. For example, consider the difference between how relations are written in the definition of an asymmetric relation and an anti-symmetric relation (i.e. aRb vs. R (a, b)). Conventional consistency seems to always be preferable here. —Preceding unsigned comment added by 72.90.67.27 (talk) 18:27, 17 August 2009 (UTC)

This is a fine place to write it, but I don't agree with you. Yes, there is some advantage to keeping an eye on how things are presented in different articles, to avoid confusing readers who click on a link where "compact" implies "Hausdorff" and arrive at an article where it doesn't, especially if the difference is not mentioned. But trying to prescribe prefix versus infix notation for relations is a waste of effort in a project as sprawling as this one, and will just annoy contributors. Any reader who has a chance of understanding the material in the first place, will be able to handle notational diversity at this level. --Trovatore (talk) 18:50, 17 August 2009 (UTC)

GA Review of Proof without words

Edge3 has started a GA review of Proof without words. Their main concern so far is that the article does not give sufficient coverage of its topic. Review status is "On hold: this article is awaiting improvements before it is passed or failed". If anyone has the time and inclination to expand the article, please do so. Gandalf61 (talk) 10:57, 11 August 2009 (UTC)

Composition of functions

This is perhaps a trivial topic but I feel that some discussion is necessary. In calculus, functions are often composed from right to left and this is therefore the convention with which most people are familiar. However, group theorists prefer to compose from left to right, and in general, many influential algebraists have selected this convention. Consequences of this convention include the consideration of only right modules (rather than left modules) and specific cases of this (for instance, right ideals rather than left ideals). However, in Wikipedia, for the most part, only left ideals, left modules and related concepts associated to "left" rather than "right" are considered. In my opinion, this is an inconsistency, and at can at times lead to incorrect assertions (in the context of rings, only, since a ring need not be isomorphic to its opposite ring). Should something be done about this? --PST 14:58, 11 August 2009 (UTC)

I believe that the "opposite" composition descends from Philip Hall; it is true that it is used by group theorists, mainly in that tradition. I think if you broaden to "influential algebraists", it is very much a minority point of view, which is why it isn't much represented here. Obviously we do have bimodule as an example against your thesis that the point is neglected. In computer science there is a convention of writing f;g for gof as the "opposite" notation for composition. I don't see much virtue in carrying around extra verbiage about composition across mathematics generally; Wikipedia does what almost all mathematicians do in this area. Category-style R^op notation can say enough where necessary, I think, where close attention is required. Please flag up any actually incorrect statements so we can deal with those. Charles Matthews (talk) 15:57, 11 August 2009 (UTC)

Isn't this discussed under Composition of functions#Alternative notation? In computer science, the (fg)(x) = g(f(x)) is preferred to the point that they made an ISO standard for it (the Z notation), which introduced the "fatsemi" symbol as the dual of the circle. Pcap ping 17:18, 11 August 2009 (UTC)

P.S. As you can see below the LaTeX \fatsemi does not work on Wikipedia because it doesn't include the right package. If you read the Composition of functions article on Linux, or on anything else with decent Unicode fonts (Mac?), the Unicode fatsemi appears correctly; but not on Windows XP. Pcap ping 17:21, 11 August 2009 (UTC)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \fatsemi}

P.P.S: I see that I had added some of these details to the article on Composition of relations last year, but completely forgot about it... Pcap ping 17:31, 11 August 2009 (UTC)

Also looking at something I added to that article, you could use $\circ _{l}$ and $\circ _{r}$, as done in Kilp et al., if you really need both notions, and need to distinguish between them in an article. Pcap ping 17:39, 11 August 2009 (UTC)

Equality (mathematics)

Can someone review that article and remove, or at least frame properly, the ramblings that permeate it? I've done a little work on it, but I have the rewriting fish to fry, for which there are way fewer knowledgeable Wikipedians around (as far as I can tell given how bad the article was). Pcap ping 17:07, 11 August 2009 (UTC)

P closures

I've added a section to the article on Closure (mathematics) article describing a related notion. The name used by Baader and Nipkow is somewhat non-descriptive. Has anyone encountered it under some other name? Also, is that article the best place to discuss it? Pcap ping 18:52, 11 August 2009 (UTC)

Also, can these be somehow defined as closure operators? Pcap ping 19:17, 11 August 2009 (UTC)

According to your definition, the P-closure of a binary relation R on S is a subset A of S. An obvious "P-closure operator" maps R to A×A.

But I think the definition makes no sense because it only depends on P union all first or second elements in pairs from R. Shouldn't it be "the least set with property P that is closed under R"? Then the P-closure operator would map R to the smallest symmetric and transitive relation that extends both R and P². Hans Adler 20:29, 11 August 2009 (UTC)

No, the P-closure (say Q) of R is a subset of S×S, just like R. The P here is a predicate; e.g. for the symmetric closure the predicate P_sym is "forall x,y in S: (x,y) in Q implies (y,x) in Q". I guess you could see the P-closure as an operator on S×S parametrized by some predicate P. Pcap ping 20:42, 11 August 2009 (UTC)

Oh, I see. P is actually a property of subsets of S×S? I didn't read this literally. Seems quite a technical notion to me then. Hans Adler 20:57, 11 August 2009 (UTC)

Well, Nipkow's specialty is higher-order logic (one of his celebrated results is extending Knuth-Bendix to higher order logic, and he's the guy behind Isabelle/HOL). But the "Term Rewriting and All That" book is written for undergrads, so I guess he didn't want to complicate "P closures" with a formal definition, which appears to require higher-order logic because the predicate P appears as an argument. Pcap ping 21:07, 11 August 2009 (UTC)

I am sure these can be defined as closure operators provided the property in question is nice, for example monotone. But the property P(Q) = "Q is empty" is not going to appear as any sort of closure operation. This all seems to be a special case of the fixed-point theorems (e.g. Kleene fixpoint theorem). — Carl (CBM · talk) 01:50, 12 August 2009 (UTC)

I think "nice" (as you put it) is "closed under arbitrary intersections" (as Baader and Nipkow put it). Pcap ping 09:29, 12 August 2009 (UTC)

You're right that it is also enough for P to be satisfied by the full relation on S × S, and for P to be closed under intersections. But the intersection of any number of empty relations is empty, so some variation of the first condition would also be required. I said nice because there are several fixed-point theorems with slightly different hypotheses. For example, one can assume the map is expansive instead of monotone, and neither of these properties is implied by the other. Also, to use the fixed point theorems one has to convert P into a map, which is easy to do in the examples given (symmetric relations, transitive relations, etc) but nonobvious in general. — Carl (CBM · talk) 11:43, 12 August 2009 (UTC)

By the way

Did anyone see how terrible our article on predicate (mathematical logic) is? Pcap ping 20:49, 11 August 2009 (UTC)

Oh dear. Hans Adler 20:57, 11 August 2009 (UTC)

The real issue with the title of that page is that predicates are trivial from the point of view of mathematical logic, because all the interesting questions have been collapsed by the use of set theory. There is a lot to say about predicates from the point of view of philosophy, but that isn't mathematical logic. We can discuss what to do with that article on its talk page in any case. — Carl (CBM · talk) 01:39, 12 August 2009 (UTC)

Aren't there non-trivial things one can say about predicates in contexts where set theory is not assumed as a foundation? E.g., in categorical logic, each topos has an internal language in which one ought to be able to formulate predicates. Also, isn't there some notion (again in categorical logic) called a "theory"? (Or maybe I'm thinking of a "sketch"?) Which, if I have the right intuition, would have a notion of a predicate. I remember skimming Borceaux's Handbook and seeing these sorts of things. (Much more so than I saw geometric things! Unfortunately, as far as I know, the only decent book about the geometric aspects of topoi is still SGA4.) Ozob (talk) 13:35, 12 August 2009 (UTC)

Equational logic?

Am I just blind, or we don't have an article on this? Equational theory redirects to Universal algebra, which sort of touches on the idea of a model theory, but I don't see the fundamental result that equational logic is sound and complete mentioned there. I was trying to find something to link to from rewriting in order to explain what the motivation is, but no luck... Compare with [3]. Pcap ping 02:45, 13 August 2009 (UTC)

You've probably found a gap in the coverage. http://eom.springer.de/E/e120140.htm may be more useful than the MathWorld page (EoM is a reliable source in just about all areas, I think). Charles Matthews (talk) 16:18, 13 August 2009 (UTC)

Help needed

Could anyone give some feedback on the discussion under Talk:Bijection#Terminology? It might not be a very deep discussion, but I think it's important nontheless. Some extra views would be more than welcome. Thanks! 145.88.209.33 (talk) 08:27, 13 August 2009 (UTC)

Reorganize the Computability articles

We have these "general" articles:

• Computability, which incorrectly implies that computability in CS and Math are somehow disjoint. This should not be a dab or a stub!
• Computability theory (computer science), a poor quality article that focuses narrowly on automata and Turing machines (despite the name). Barely mentions lambda calculus in the history section and makes no mention of recursion theory or random access machine!!
• Model of computation, a sort of WP:DICTDEF

We also have much better articles on the important topics, recursion theory, lambda calculus, Turing machine, and random access machine; we also have a decent overview article on register machines in general.

The way I see it computability should be is a high-level intro to the often encountered equivalent models of computation: recursion theory, lambda calculus, Turing machine, and random access machine. This is along the along the outline of S. Barry Cooper's Computability Theory (see pp. 7-8), which despite being written by mathematician was quite satisfying for me as a computer scientist (despite the many misprints, and his insistence on calling RAMs URMs, but that's another matter).

(I will cross-post to the CS wikiproject to attract participants from there too, but that project is nearly dead.)

Thoughts? Pcap ping 11:22, 13 August 2009 (UTC)

I don't know why there was originally a split between computability theory (computer science) and recursion theory. However, I know why I have never tried to merge them, which is that it seemed too difficult to write a single article on "computability" that gave the proper focus to both areas.

At some point I worked on getting the recursion theory article up to shape. This article does not describe a model of computation; "recursion theory" is a field of mathematical logic concerning computability and definability. The model of computation is described at μ-recursive function. The article recursion theory gives a pretty reasonable overview of that field of logic.

It would be possible to merge computability theory (computer science) into the article recursion theory, but I have never been convinced that many computer scientists would be happy with the result. I could work on that next week if there is a desire for it among the computer scientists.

We should discuss that at Talk:Recursion theory. — Carl (CBM · talk) 11:43, 13 August 2009 (UTC)

Okay, discussion continues there. Pcap ping 12:45, 13 August 2009 (UTC)

New behavioral guideline: Wikipedia:Editing scientific articles

See discussion here Count Iblis (talk) 15:01, 13 August 2009 (UTC)

Why do you say it's a guideline when it's an essay? Charles Matthews (talk) 16:15, 13 August 2009 (UTC)

Because of a misconceptions about the way we create new guidelines. [4] Hans Adler 16:38, 13 August 2009 (UTC)

Yeah, we gotta be careful with every comma in a science article on this wiki or the world might explode! Seesh... Pcap ping 17:22, 13 August 2009 (UTC)

It is necessary, because the usual wiki rules are not enough to prevent nonsense from being edited in articles. The reason why it isn't usually a problem has a lot to do with the fact that most editors intuitively stick to my propsed guidelines and in cases where editors do not and thatleads to conflicts, Admins intervene in a reasonable way. But, strictly speaking, you can find yourself on the wrong side of wiki rules when some editor tries to edit in nonsense in articles that seems to be supported by sources if you can't simply quote from a source a direct refutaton of these edits.

This is a recent case. You can see that the existing wiki rules and guidelines leave too much room for quack editors to argue and put the editors who want to keep nonsense out of articles too much on the defensive. All I want to do is have a few guidelines that that points out some pitfalls when arguing like: "My book says X, the wiki article says Y, so I'm right and you're wrong".

Besides conflicts between editors, there have been cases of seriously flawed wiki articles. In most cases the articles would have been ok if editors had stuck to the simple rules in my guidline. Count Iblis (talk) 18:34, 13 August 2009 (UTC)

We don't need a new guideline just because a clueless editor tried to defend the valuable resource that Wikipedia has in the person of Randy in Boise against exasperated experts. If we augment our guidelines each time an editor makes a serious error of judgement, they will become even more contradictory than they already are and they will only be useful as something the parties of a dispute can point to, but not helpful to anyone who wants to be guided. Hans Adler 18:58, 13 August 2009 (UTC)

(ec) "There have been cases of seriously flawed wiki articles." O RLY? What else is new? Did you look at the recent history of lambda calculus? Reliable sources occasionally get some stuff wrong too, or they may express something in a misleading fashion even when it's not downright wrong. An example from about a year ago is here. But you can't really enforce clue with a guideline; you will always have editors making use of appeal to authority, because that's a basic pillar of the wiki. Thankfully, we're not writing math articles that way here, despite what the rules say. (Yes, Wikipedia has an inconsistent axiomatic system.) So, next time when someone is a clueless WP:DICK, remember there's a policy against that, making your guideline superfluous. :) Pcap ping 19:00, 13 August 2009 (UTC)

Sorry for being off-topic... I just want to thank Count Iblis for his contribution to the article Helmholtz free energy and its talk page; these are very helpful to me. Probably, editors in physics have more problems with quack editors than we mathematicians.Boris Tsirelson (talk) 06:43, 14 August 2009 (UTC)

Agree with Pcap. There's no point in having further policies like this. A manual of rules needs to be as short as possible whilst including everything that's really necessary. This policy would be like telling grandmother how to suck eggs as far as anyone with a clue is concerned and would be ignored by the idiots with a mission. Besides it's unworkable - how would an administrator decide it was broken except by asking the parties to discuss the matter which is what happens for edit wars anyway? Making the rules any longer diminishes the ones already there. This could be put into a tutorial text but that's about it I think. Dmcq (talk) 09:33, 14 August 2009 (UTC)

Boris, thanks! To the critics here, an Admin who intervenes in a dispute would have the option of pointing one or both editors to these guidlines. The word "edit war", could be avoided in some cases where now Admins would have to use that word. That could prevent tensions being raisd. Some newcomer who thinks that he his right and that his opponent is removing his "sourced edits" will have difficulties accepting the judgement that he is "edit warring", given that the wiki rules seem to support his condiuct and not the conduct of his opponent.

An expert at Wikipdia would also have something concrete to point to when someone complains that he is removing sourced edits, and that he is violating wiki rules. Many experts have left wikipedia after a few days out of frustration. Count Iblis (talk) 16:08, 14 August 2009 (UTC)

Not every expert is able to "cooperate" with Randy, although some are quite good at it. But some simply want to write a complete new article from scratch and upload it. I think it would be better overall if those experts who are really stressed by the Wikipedia editing environment simply went to Citizendium and worked on writing and approving articles there. As far as I know approved Citizendium articles can be used as reliable sources for Wikipedia. And since we are now using basically the same licence, large passages can simply be copied. The Citizendium article can work as a known good state of the Wikipedia article, and consensus will autmatically crystallise around that in many cases. Conversely, improvements that happen here can be folded back into Citizendium and formally approved there after a while.

The complaints about removing sourced facts are a general problem of Wikipedia. They have nothing to do specifically with science articles and need to be addressed sooner or later. AFAIK it's not written policy or guideline but there was an old Arbcom decision saying persistently removing sourced statements is disruptive. I believe they were careless with their formulations at the time, so that now it can be quoted as justification to defend nonsense or irrelevancies that are properly backed with formally reliable sources.

Even if your text were policy it would have no effect whatsoever on edit warring. Your text might conceivably help to find out who is right in an edit war. But that's irrelevant because it's a purely behavioural concept. There are exceptions for vandalism, BLP and probably OTRS and copyvios, but your text won't add a new justification for edit warring. It might conceivably help admins to stop or prevent an edit war by deciding which side is right. But I think that would be very controversial. Hans Adler 16:50, 14 August 2009 (UTC)

GA review of "Mathematics and art" and "Jeep problem"

Protonk has started a GA Review of Mathematics and art (review page) and Jeep problem (review page). In both cases Protonk feels that the articles are some distance away from GA quality. Mathematics and art has "many challenges", which Protonk has listed in detail; Jeep problem "requires a substantial rewrite" and so Protonk has given it a more summarised review. Both reviews have a status of "On hold: this article is awaiting improvements before it is passed or failed". If anyone has the time and inclination to improve these articles, please do so. Gandalf61 (talk) 10:25, 14 August 2009 (UTC)

Protonk's review Talk:Mathematics and art/GA1 is thoughtful, detailed, and well written, addressing the content of the article rather than just stylistic issues. It is the type of review I would be happy to see more often. — Carl (CBM · talk) 11:51, 14 August 2009 (UTC)

I agree, and that is partly because the article is close enough to the GA standard that detailed suggestions for improvements might lead to a GA list. I encourage editors here to contribute to achieve such a result. Geometry guy 21:03, 14 August 2009 (UTC)

Theta function

I just wrote Theta function (disambiguation) as a possible expansion to the hatnote at Theta function. I think that the list of functions should be split into the true theta functions (Jacobi's, Ramanujan's, and the q-theta functions, at least) from the other functions that merely use (or are called) theta. Something like

A theta function is a special function in complex analysis. the Jacobi theta function $\vartheta (z;\tau )$ Jacobi theta functions (notational variations) $\vartheta _{ij}(z;\tau ),\vartheta _{j}(z),\theta _{i}(z;q)$ the q-theta function $\theta (z;q)$ Other theta functions include the Chebyshev function $\vartheta (x)$ the Riemann theta function $\mathbb {H} _{n}$

Other possibilities: leave all 10 functions in one large list; split by field (analysis/number theory/set theory).

Here's a list of possibly-related pages for comparison:

• Theta function
• Theta (disambiguation)
• Psi function
• Psi

Any thoughts? I wanted to at least let some other people look it over before I put an {{about}} tag on Theta function.

CRGreathouse (t | c) 18:11, 14 August 2009 (UTC)

Fine good, I never knew there was so many of them as that disambiguation page has grown to now! Dmcq (talk) 07:47, 15 August 2009 (UTC)

Morally the Riemann theta function probably encompasses the Jacobi functions. It's for several variables, and if you make it for one variable and tweak it a bit you presumably find all the one-variable special cases. In other words there is the family of theta functions associated with abelian varieties, and they are all related in a structural way. Charles Matthews (talk) 09:27, 15 August 2009 (UTC)

Links to Bonse's inequality

I've just created a new article titled Bonse's inequality. It's a stub. So:

• Expand and otherwise improve it if you can;
• Help figure out which other articles should link to it. I've created about three or four links (I can't count that high at the moment). (I'm surprised we have no list of prime number topics. If we did, it would belong there.)

Michael Hardy (talk) 00:16, 15 August 2009 (UTC)

Is it clear that this topic is notable enough for its own page? Searching MathReviews, I found that it intervenes in four reviews. If am not mistaken, can't one immediately deduce better estimates by applying Chebyshev's result that there exist explicit constants C_1 < 1, C_2 > 1 with $C_{1}<\pi (x)/(x/\ln x)<C_{2}$ for all (explicitly given) sufficiently large x? Chebysehev's result was proven at least 50 years before Bonse's, and has an elementary proof. (For that matter, I had some trouble finding this result on wikipedia: where is it?) I think it would be more efficient to have a page devoted to inequalities involving $\pi (x)$. Plclark (talk) 03:45, 15 August 2009 (UTC)

I'd have thought it followed even quicker from Bertrand's postulate which really is elementary. Once the factors get more than 8 you know that four times the second last prime and twice the last prime must both be bigger than the next prime. I wonder why it was thought interesting. Dmcq (talk) 07:31, 15 August 2009 (UTC)

Sorry I see that Chebysehev did actually use his results about bounds for the prime number theorem to prove Bertrand's postulate so it should really now be called the Bertrand-Chebyshev theorem or Chebyshev's theorem. Dmcq (talk) 07:40, 15 August 2009 (UTC)

It does look like it should be merged in somewhere discussing the initial segment of their primes and their distribution. Charles Matthews (talk) 09:30, 15 August 2009 (UTC)

Differential of a function

The definition of differential of a function that appears in that new article has appeared in calculus textbooks for more than 30 years now, and that's an unfortunate gap between mathematicians and authors of calculus textbooks. You'd hope that authors of calculus textbooks would be mathematicians, but it seems they're a different culture (I don't mean Spivak and Apostol, and I think there are a few others....). And they write books by zeroxing each other's books. It might not be politic to propose burning them at the stake as heretics, so I won't mention anything like that. But I've made some comments here.

Would other mathematicians here agree with me that this abomination is an abomination? Michael Hardy (talk) 02:36, 17 August 2009 (UTC)

Yes, I completely agree. In fact, what is presented in the article is rather worse than what I've found in most calculus textbooks I've looked at lately. In my experience, most 21st century calculus textbooks are written so as to never say something that is mathematically incorect, because mathematicians who teach calculus complain more vocally about actual mathematical errors than other deficiencies. Plclark (talk) 03:58, 17 August 2009 (UTC)

Thank you. Now if possible, can you add some comment to the linked-to talk page? I'm not at all sure the creator of that article is reading this present page. Michael Hardy (talk) 04:46, 17 August 2009 (UTC)

Yes, I'm reading. I'm following all the disscusion [5] [6] [7] [8]. You may read my last input in the discusion on the article's talk page. Usuwiki (talk) 02:27, 18 August 2009 (UTC)

I think there is a bit of a culture clash here. As far as I can make out, and I could very easily be wrong, this has come from an analysis/numerical viewpoint and may have started in Russia investigating linear differential operators including both Δx and dx and suchlike, and they'd want them in the same terms and comparable. I'd guess more people here see differentials as being more part of studying manifolds and start with a topological outlook and aren't so interested in finite differences. You got them both using linear maps and the same symbols so it grates. Dmcq (talk) 06:22, 17 August 2009 (UTC)

Sorry I see I should have gone to that page, okay will copy my comment there. Dmcq (talk) 06:26, 17 August 2009 (UTC)

I for one have tried to redirect the new article to a section of the existing article, plus I have made some other comments in the new article's page. As for calculus textbook, I can't say much: I am Italian, and textbooks when I was a student had, if anything, the opposite problem, being a bit too formal for, say, first-year students. But I see that presently there is a tendency towards "American" calculus, using new books translated from English and even renaming courses from "Analisi matematica" to "Calcolo". [[::User:Goochelaar|Goochelaar]] ([[::User talk:Goochelaar|talk]]) 07:34, 17 August 2009 (UTC)

A triviality: are tuple and word (mathematics) the same concept?

Clearly the notion of an n-tuple is distinct from that of a word, but I but a quick search in google books failed to find a set theory definition for tuple; only n-tuple is defined. This is related to a debate on List (computing), but the article on tuple could use some clarification as well. Pcap ping 12:49, 17 August 2009 (UTC)

Word (mathematics) is not being explained in String (computer science) and shouldn't redirect there.  Cs32en  13:01, 17 August 2009 (UTC)

Whether it should redirect there or not, it is explained there. Pcap ping 13:02, 17 August 2009 (UTC)

I've just changed the redirect to point to that section. Pcap ping 13:05, 17 August 2009 (UTC)

Normally one does not use the term "word" unqualified, it is always a word over some given finite alphabet. But otherwise there is no real difference, both word and tuple mean a finite sequence. — Emil J. 13:07, 17 August 2009 (UTC)

I agree. It is mostly a matter of consuetude: probably one would not say that a vector space such as ${\mathbb {R} }^{n}$ consists of words. Similarly, the operations on tuples one would spontaneously think of are mostly termwise ones, while two words tend to be concatenated, or shuffled, and the like. So, in a sense, if you use one of the two terms rather than the other, you predispose the audience to a certain set of properties and operations. [[::User:Goochelaar|Goochelaar]] ([[::User talk:Goochelaar|talk]]) 13:13, 17 August 2009 (UTC)

No, because "words" in ${\mathbb {R} }^{n}$ are all n-tuples, i.e. the words have all the same length. Pcap ping 13:42, 17 August 2009 (UTC)

Indeed. One would not usually describe ${\mathbb {R} }^{n}$ as consisting of all words on $\mathbb {R}$ of length n either. [[::User:Goochelaar|Goochelaar]] ([[::User talk:Goochelaar|talk]]) 14:41, 17 August 2009 (UTC)

Actually, I think that the definition of tuple from that article is a Wikipedia original, and that it was caused by renaming the article some four years ago from n-tuple; according to MathWorld "tuple" means just n-tuple for some fixed n obvious from context; it does not mean word. See further discussion at Talk:Tuple#Problem_with_def_of_tuple. Pcap ping 13:42, 17 August 2009 (UTC)

Apparently the word "tuple" (without n) is mostly used by computer scientist, especially in Python programming language, where it is the actual name of a data structure. [[::User:Goochelaar|Goochelaar]] ([[::User talk:Goochelaar|talk]]) 14:44, 17 August 2009 (UTC)

Indeed. In Python a tuple is an immutable list. Gandalf61 (talk) 14:54, 17 August 2009 (UTC)

The term is also in wide currency among relational database theorists, who use it to refer to a row of a table. (Each table is a relation, in the mathematical sense, and rows in the table are elements of the relation, and so are tuples.) —Dominus (talk) 16:10, 17 August 2009 (UTC)

There is a distinction: perhaps it should be clarified by means of the concepts of internal operation and external operation. The "point" of words is that concatenation is an internal binary operation - we are living in the free monoid. Obviously you can concatenate tuples of any finite length, but this then appears as an external operation on two Cartesian powers ending up in a third. In other words (in other tuples?) as soon as you write * for concatenation with its type data you become conscious of an overloading of the notation. Charles Matthews (talk) 14:54, 17 August 2009 (UTC)

Matrix calculus: Definition of the matrix derivative

Content from the archive. The issue is still unresolved.  Cs32en  13:04, 17 August 2009 (UTC)

We could use some help to resolve a controversy about the correct formulae for the matrix differential and the matrix derivative at the article Matrix calculus. See the talk page, especially the section Disputed information: Matrix derivative.  Cs32en  22:52, 11 July 2009 (UTC)

I concur we need assistance, primarily as to the notation(s) actually used in serious mathematical works. — Arthur Rubin (talk) 15:49, 13 July 2009 (UTC)

See Talk:Matrix calculus#Scope of questions for my view as to the matters in dispute, and my take on them. My desired outcome is not necessarily represented in all cases. — Arthur Rubin (talk) 21:19, 13 July 2009 (UTC)

This really should be resolved by verifying that the formulae stand as stated in the references (and noting the conventions in operation, per reference). I edited the section on the nature of the so-called "matrix derivative" - and there doesn't seem to be controversy about that. So that leaves only the formulae collected from the literature. Charles Matthews (talk) 14:47, 17 August 2009 (UTC)

Leibniz function

With respect to article Leibniz function, can someone please verify its meaning in regards to its derivative ( f ( x ) f ' ( x ) = 1 ). Not familiar with the term in this context and the word "Leibniz" is not found anywhere inside the books listed as references.

My addition/contribution to the article is with respect to Lie groups/algebra, with cleanup under the good-faith assumption that such an identity exists and is named after Leibniz. Henry Delforn (talk) 16:56, 17 August 2009 (UTC)

I haven't seen that usage before, and I find it at least a little bit implausible that any such convention is widespread. Michael Hardy (talk) 22:41, 17 August 2009 (UTC)