Wikipedia talk:WikiProject Mathematics/Archive/2015/Jun

Formal linear combination

This notion is used, for example, in "Chain (algebraic topology)", with a link to "Free abelian group"; there, "formal sums" are defined (but "formal linear combinations" are not). In some books I see "formal linear combinations" used with no definition (and often, with no explanation). Once I used it on an undergraduate lecture and was asked by students: "what's it?"

Should we redirect "Formal linear combination" to "Free abelian group"? Should we enlarge the latter, including non-integer coefficients? Boris Tsirelson (talk) 17:27, 31 May 2015 (UTC)

Free module seems like a better _target, although that article could use improvement. Sławomir Biały (talk) 17:56, 31 May 2015 (UTC)

For now it does not mention "formal linear combinations". Boris Tsirelson (talk) 19:02, 31 May 2015 (UTC)

The book Introduction to Topological Manifolds by John Lee introduces "formal linear combinations" in the context of free Abelian groups. --Mark viking (talk) 18:39, 31 May 2015 (UTC)

I guess, it uses only integer coefficients; but (at least) real coefficients are really needed in the context of Stokes theorem.

I also bother that "free module unique up to isomorphism" is demanding for some readers that could be satisfied with "finitely supported functions on the set of (...)". Boris Tsirelson (talk) 19:02, 31 May 2015 (UTC)

I have rewritten the corresponding section of Free module for defining explicitly formal linear combination (and also for being less technical). I have also created the redirect Formal linear combination. D.Lazard (talk) 09:17, 1 June 2015 (UTC)

Very nice; now I can recommend it to my undergraduates. Boris Tsirelson (talk) 15:32, 1 June 2015 (UTC)

Now if only someone could do something about that lead paragraph ;). --JBL (talk) 19:44, 1 June 2015 (UTC)

Mathematical logic and Goedel sentences

Please take a look at mathematical logic. --Trovatore (talk) 18:30, 26 May 2015 (UTC)

Also axiom, please. --Trovatore (talk) 04:19, 27 May 2015 (UTC)

What is the problem with those articles? — Arthur Rubin (talk) 23:53, 28 May 2015 (UTC)

If I read the talk page correctly, it involves a philosophical debate between a Platonic position in which one can say that a first-order sentence about the integers is true of "the" integers (even if it might be false for some models of the Peano axioms), and a radically relativist position in which sentences may be stated to be true of models but not true in any absolute sense and in which no model of Peano's axioms (not even the model given by the finite ordinals) is privileged as being "the" integers. This affects the article in terms of whether it is more accurate to say that the Gödel sentence is "true, but unprovable" or "neither it nor its negation are provable". —David Eppstein (talk) 01:13, 29 May 2015 (UTC)

Not quite: it's about whether a first-order sentence should be said to be true without qualification, or true of the integers. But anyway it's resolved. --Sammy1339 (talk) 04:13, 29 May 2015 (UTC)

Sorry that I missed this debate :-) --GodMadeTheIntegers (talk) 18:00, 2 June 2015 (UTC)

Oh yes; according to your username, you should be platonistic. :-) Boris Tsirelson (talk) 18:10, 2 June 2015 (UTC)

Fundamental lemma of calculus of variations

Much weakened version of that lemma is formulated in our article. The reason (articulated on the talk page) is that the simple proof given in the article does not give more. As for me, irrespective of the proof, a stronger formulation should be given. But for now I have only lecture notes as sources. I wonder, do you know better sources? Mine are:

• M.Bendersky, course, pp.10-12 in [1];
• G.K. Ananthasuresh, course, Lecture J, the four lemmas in [2].

Boris Tsirelson (talk) 11:53, 29 May 2015 (UTC)

Wikipedia isn't in the business of giving proofs we should be pointing to textbooks for that, though of course trying to explain it is an important part of Wikipedia's business. I must admit I can't see why it has been split or forked out of the article on the calculus of variations, it isn't as though that article is too big and this is a very important part of it. Dmcq (talk) 12:36, 29 May 2015 (UTC)

Of course if you can expand that article with another formulation then of course it deserves its own article, it just seemed rather short and a duplication of bits of the main article to me at the moment. Dmcq (talk) 12:53, 29 May 2015 (UTC)

Well, I am neutral about possible merge. I bother that the optimal function should be proved to be smooth, rather than assumed. That is, in the fundamental lemma, the given function "orthogonal" to all smooth functions must vanish even if not assumed smooth. I could expand; the problem is, whether "my" sources are reliable enough. Boris Tsirelson (talk) 13:44, 29 May 2015 (UTC)

I'm not entirely clear on your objection. Do you want to strengthen the lemma by saying that $h\in C^{\infty }$ instead of just $C^{k}$? That would be the grownup way to write this; it would involve bump functions instead of the current proof (which is kind of cute nonetheless). --Sammy1339 (talk) 15:06, 29 May 2015 (UTC)

Oh I see. Yes it should be strengthened. --Sammy1339 (talk) 15:06, 29 May 2015 (UTC)

Smoothness of h matters, too. But I mostly bother about smoothness of f. Boris Tsirelson (talk) 15:18, 29 May 2015 (UTC)

(Edit conflict) Really, that lemma should be treated as a special case of the fact that every weak solution of a linear ODE is also a strong solution. Regretfully, I did not find this fact in our article "weak solution" (nor in "distribution"). I also did not find this connection in texts on calculus of variations. Is it my Original Research?! Boris Tsirelson (talk) 15:36, 29 May 2015 (UTC)

I just looked in Gelfand & Fomin and interestingly they follow the exact same steps as your sources above, but this formulation is neither stronger nor weaker than what appears in the article. You require a stronger hypothesis ($\forall h\in C^{2}$ instead of just $\forall h\in C^{k}$.) Then there is Lemma 4 in your sources above (which is also Lemma 4 in G&F on p. 11) which is just integration by parts basically showing that if $f$ is continuous and weakly differentiable then it is differentiable. --Sammy1339 (talk) 15:28, 29 May 2015 (UTC)

Ah, yes, it is nice to know that Gelfand & Fomin is a good source! Given that you have this book while I do not (for now), maybe you'll improve the article accordingly? Boris Tsirelson (talk) 15:39, 29 May 2015 (UTC)

And yes, "if $f$ is ... weakly differentiable then it is differentiable" is just what I want to see. Boris Tsirelson (talk) 15:42, 29 May 2015 (UTC)

I'm looking for a source for that which states it full generality. --Sammy1339 (talk) 15:58, 29 May 2015 (UTC)

It seems, you are busy... Meanwhile I took books (including Gelfand & Fomin, thanks for the advice), got bold, and rewrote the article. Still, do better, as much as you can (and want). Boris Tsirelson (talk) 20:13, 2 June 2015 (UTC)

See theorems 1.2.4 and 1.2.5 in volume 1 of Hörmander. If f is a continuous (resp. locally integrable) function st for all compactly supported smooth φ

$\int f\phi =0$

then f=0 (resp f=0 a.e.) Sławomir Biały (talk) 16:47, 29 May 2015 (UTC)

poincare conjecture incident

There is an ongoing and somewhat agitated discussion of the Poincare conjecture here. Tkuvho (talk) 13:37, 3 June 2015 (UTC)

Vietoris–Rips complex

The page on the Vietoris–Rips complex lacks a formal definition written in a readable format. Please fix thanks. — Preceding unsigned comment added by 98.223.78.42 (talk) 14:40, 3 June 2015 (UTC)

What's wrong with the definition in the first sentence, and its expansion in the rest of the first paragraph? —David Eppstein (talk) 15:55, 3 June 2015 (UTC)

Lagrange's method

Currently Lagrange's method redirects to Lagrange multiplier, but apparently in the world of PDE's, Lagrange's method means something entirely different. For example, see [3]. Do we cover the PDE meaning of Lagrange's method? Is seems like if it's anywhere it would be in First-order partial differential equation but I didn't see it in recognizable form, though it's not my area of expertise. --RDBury (talk) 12:20, 4 June 2015 (UTC)

It's another name for the method of characteristics, but I don't think it's in very wide use. Sławomir Biały (talk) 13:02, 4 June 2015 (UTC)

I'm confused then because the pdf above gives Lagrange's method in chapter 3 and Method of Characteristics is chapter 4. The WP article on Method of Characteristics seems to follow chapter 3 more than chapter 4 though. My Google search turned up 4 or 5 more PDE books with sections on "Lagrange's Method" and a few stack exchange/mathoverflow posts that reference it, so while it might not be the most common it seems to have enough usage to be noted in the article as an alternate name. Perhaps a DAB page is in order, though the Springer EoM gives a | third meaning for "Lagrange's Method" which diagonalizes a quadratic form. --RDBury (talk) 15:45, 4 June 2015 (UTC)

Mass changes of style

I notice a set of mass changes to some articles. --Ancheta Wis   (talk | contribs) 04:57, 7 June 2015 (UTC)

These are very, very strange. I get the feeling that the anon has very specific epistemological ideas that he (?) would like Wikipedia to adhere to, but I'm not sure if I'm correctly understanding what's going on. Ozob (talk) 05:04, 7 June 2015 (UTC)

I just undid the one at Entscheidungsproblem. It seems to be a mix of style changes and an additional paragraph. The style changes, frankly, suggest that the editor is not a native speaker despite the fact that the IP geolocates to Ohio; but he/she could be a foreign student or something; I saw no reason to keep them. The additional paragraph I have not yet read — that seems to be an orthogonal issue. --Trovatore (talk) 05:14, 7 June 2015 (UTC)

Actually, at a second look, I don't think there is an additional paragraph after all. The editor had put some paragraphs in a different order or a different section. --Trovatore (talk) 05:18, 7 June 2015 (UTC)

Status of new article Complex spacetime

This may be of interest to those on this talk page: as I stated at Wikipedia talk:WikiProject Physics#Status of new article Complex spacetime, this new article might need some broader scrutiny. —Quondum 14:18, 10 June 2015 (UTC)

Orphaned conjugacy-closed subgroup

Conjugacy-closed subgroup is an orphaned article: no other articles link to it. Michael Hardy (talk) 23:22, 10 June 2015 (UTC)

Wikipedia:Articles for deletion/Quaternion rotation biradial

Opine on a proposed deletion at Wikipedia:Articles for deletion/Quaternion rotation biradial. Michael Hardy (talk) 17:34, 14 June 2015 (UTC)

Planetmath changed links format

It seems that some of the links to PM in Wikipedia articles are not working. For example, the following link [4] in the article Hartogs' theorem is not working. It was added using the Template:PlanetMath attribution. New link seems to be [5]. Also the link [6] in Ancient Egyptian multiplication does not work. The new link is [7]. This probably influences many articles (and templates). What ca be done with it? Can this be resolved automatically, or is manual change the only way to go? --Kompik (talk) 08:03, 15 June 2015 (UTC)

By the way, this might be an old problem which I noticed only now. Already this revision of Hartog's theorem from September 2014 contains information that the link is not working. So it is quite probable that this has already been discussed somewhere. --Kompik (talk) 08:07, 15 June 2015 (UTC)

We might be able to change the template for some of the entries. The EgyptianMultiplicationAndDivision could be fixed by changing the line

[http://planetmath.org/encyclopedia/{{{urlname}}}.html

to

[http://planetmath.org/{{{urlname}}}

A bit more testing will be needed to see if this works for all the occurrences. There are 280 transclusions[8] so its not an impossible task.--Salix alba (talk): 14:44, 15 June 2015 (UTC)

It was discussed at Template talk:PlanetMath attribution but it does not look like any action was taken. It seems like the id's need to have 30000 added to them so http://planetmath.org/node/6024 needs to be changed to http://planetmath.org/node/36024.

It seems like {{PlanetMath reference}} and {{PlanetMath}} were already fixed. I think I've now fixed {{PlanetMath attribution}}.--Salix alba (talk): 15:35, 15 June 2015 (UTC)

Quotient structure

We seem to have no article on quotient structures in general. We have Quotient (disambiguation), which lists a bunch of things as if they were different things known by the same name, with no idea common to all of them. Should we have a quotient structure article? Michael Hardy (talk) 18:18, 15 June 2015 (UTC)

However, we have "substructure", and maybe indeed "quotient structure" could be similar. There I see: "Substructures as subobjects"; and in "Subobject" I see: "The dual concept to a subobject is a quotient object". Boris Tsirelson (talk) 20:28, 15 June 2015 (UTC)

Quotient structure is discussed a little in Congruence relation. I guess that quotient structure is more of a universal algebra thing and quotient object is more of a categorical thing. Both could usefully be discussed in one article, along with concrete examples. --Mark viking (talk) 20:49, 15 June 2015 (UTC)

Quotient space redirects to Equivalence relation, in which the subject is sketched. Epimorphism is also (or should be) related. It is not clear to me what deserves to be added. So, for the moment, I suggest to redirect quotient structure to Equivalence relation, and to add to this article the few things that are lacking (which?). D.Lazard (talk) 21:18, 15 June 2015 (UTC)

Deletion sorting for Math articles?

Is there deletion sorting for mathematics-related articles or do they just end up under Wikipedia:WikiProject Deletion sorting/Science?

Anyway there are three articles at AfD about algorithms for voting systems, not sorted into anything technical:

• Wikipedia:Articles for deletion/Proportional approval voting (3rd nomination)
• Wikipedia:Articles for deletion/Sequential proportional approval voting (2nd nomination)
• Wikipedia:Articles for deletion/Satisfaction approval voting

Thanks for looking at these. StarryGrandma (talk) 23:05, 19 June 2015 (UTC)

No new articles

At Wikipedia:WikiProject Mathematics/Current activity, we find no new articles since May 25. Jitse's bot, run by Jitse Niesen, which edits that page, has done so more recently, and mathbot, which, among other things, lists new articles, whose lists are used by Jitse's bot, has been active more recently. Are there no new articles? Michael Hardy (talk) 18:44, 31 May 2015 (UTC)

Are you looking at the same current activity that I am? Because I see 14 new articles on May 26 (starting with Alicia Dickenstein), 6 new articles on May 28, and a large number of new articles (possibly caused by moving some category involving tilework into the ones the bot lists) on May 29. —David Eppstein (talk) 19:17, 31 May 2015 (UTC)

Yes, there a many entries dated after May 25. A diff shows them. I see "This page was last modified on 31 May 2015" at the bottom of Wikipedia:WikiProject Mathematics/Current activity. If you se an older date then try to bypass your cache. PrimeHunter (talk) 19:26, 31 May 2015 (UTC)

• Speaking of which, we have several mathematical article submissions such as Draft:Loewner order or Draft:Contributions on developping measuring tools on surfaces up for review at the moment. Care to offer your thoughts? Cheers, FoCuSandLeArN (talk) 21:10, 2 June 2015 (UTC)

The second article is not suitable I would say, as it duplicates content that is already in a much more readable form elsewhere in the encyclopedia. The first is certainly an encyclopedic topic, but the name "Loewner order" is not very standard. Most mathematicians would not attribute this to Loewner, just calling it the partial ordering of hermitian matrices. Sławomir Biały (talk) 19:51, 5 June 2015 (UTC)

I'm experiencing the same problem described by user Michael Hardy... Currently, the last date I can see on the "New articles:" section on the table is "5 Jun". But there were indeed mathematical articles created after 5 Jun (Yair Minsky (6 Jun) and Richard Canary (7 Jun) are proof of this). — Preceding unsigned comment added by 189.6.195.23 (talk) 13:24, 8 June 2015 (UTC)

The bot has been having problems on and off for a couple of months. See the last two sections of User talk:Jitse's bot. I've tried pinging and e-mailing Jitse Niesen but have had no response. I don't know if there's anything else that can be done, such as someone else take over maintaining them.--JohnBlackburne^words_deeds 13:32, 8 June 2015 (UTC)

There are two bots involved, mathbot and Jitse's bot. But in this case, mathbot appears to be picking up the new articles; it's the updates to the current activity page by Jitse's bot that have gone missing. —David Eppstein (talk) 17:45, 8 June 2015 (UTC)

Jitse Niesen has not made any contributions since 10:54, 20 October 2014, so I am afraid that we must assume that he is unable or unwilling to participate further in the English Wikipedia. JRSpriggs (talk) 04:03, 9 June 2015 (UTC)

If Jitse Niesen has left us, does someone inherit his bot? Will the Mathematics Wikiproject collapse permanently if his bot dies? Michael Hardy (talk) 23:48, 10 June 2015 (UTC)

What does it take to run a bot? Do we or can we use a sever at the wikimedia foundation? It would be nice, long term, if we can rely on an institution instead of an individual editor. (Right now, we are in the post-apocalyptic world, I suppose.) -- Taku (talk) 23:14, 19 June 2015 (UTC)

Relations

Are Dependence relation and Dependency relation the same thing? WhatamIdoing (talk) 03:13, 22 June 2015 (UTC)

No. Completely different. And probably, both are not widely used. Boris Tsirelson (talk) 05:51, 22 June 2015 (UTC)

Rfd for Category of graded vector spaces

Hi there.

Over at Wikipedia:Redirects_for_discussion/Log/2015_June_20#Category_of_graded_vector_spaces this has been listed. I think it is redundant and perhaps harmful since it is not a Wikipedia catgegory but a redirect to an article: surely the grading/graduation is itself the way to categorise them so to categorise the grading would be some higher order function, which I'm sure is not intended here. (To categorise the categorisation, i.e. in formal logic of some kind, Bertie Russell turning in his grave).

I only have maths up to basic graduate degree level, and mostly on the engineering side of it rather than theory, so I'd appreciate your opinion on whether this is useful. Si Trew (talk) 07:22, 22 June 2015 (UTC) see my userpage on 3d slope formula here — Preceding undated comment added 00:34, 23 June 2015 (UTC)

What formula is this?

I am in the process of renaming two-letter filenames on Commons, to provide less ambiguous titles. Today I came across this one:

What is this equation? Or, more directly, what would be an appropriately unambiguous, but concise, name for this image file? Cheers! bd2412 T 19:55, 24 June 2015 (UTC)

A good question to the person that uploaded it. Boris Tsirelson (talk) 21:10, 24 June 2015 (UTC)

The person who uploaded it seems to be a ghost. They appeared on Commons a month ago, uploaded a dozen and a half images like this, and disappeared. Are these anything at all? bd2412 T 21:24, 24 June 2015 (UTC)

Perhaps it's time to exorcise the ghost? — Arthur Rubin (talk) 21:50, 24 June 2015 (UTC)

If these images are useful, we should keep and properly describe them. If they are not, they should be deleted. I don't have the background to determine whether these are at all meaningful. bd2412 T 22:19, 24 June 2015 (UTC)

These appear to have something to do with magnetism and perhaps magnetic flux. Equations are better written in the text and I don't mean to be harsh, but the illustrations are small enough that they probably aren't too useful. --Mark viking (talk) 22:55, 24 June 2015 (UTC)

It does not seem to be used anywhere. We should delete it. JRSpriggs (talk) 09:17, 25 June 2015 (UTC)

The Assayer

The book The Assayer by Galileo is one of the works that ushered in the scientific method as well as the method of indivisibles closely related to Galileo's atomism. Recently I added sourced material on this aspect of the book. Input would be appreciated. Tkuvho (talk) 07:36, 25 June 2015 (UTC)

Context: Tkuvho is engaged in a content-based edit war over the relevant material; he and User:William M. Connolley have both been blocked for 24 hours as a result. There are no obvious signs of consensus-forming. Probably, the input of some neutral editors would improve the situation. --JBL (talk) 13:12, 25 June 2015 (UTC)

Dualizing sheaf listed at Redirects for discussion

An editor has asked for a discussion to address the redirect Dualizing sheaf. Please participate in the redirect discussion if you have not already done so. Ivanvector (talk) 14:48, 25 June 2015 (UTC)

Copyright Violation Detection - EranBot Project

A new copy-paste detection bot is now in general use on English Wikipedia. Come check it out at the EranBot reporting page. This bot utilizes the Turnitin software (ithenticate), unlike User:CorenSearchBot that relies on a web search API from Yahoo. It checks individual edits rather than just new articles. Please take 15 seconds to visit the EranBot reporting page and check a few of the flagged concerns. Comments welcome regarding potential improvements. These likely copyright violations can be searched by WikiProject categories. Use "control-f" to jump to your area of interest.--Lucas559 (talk) 22:25, 25 June 2015 (UTC)

Another terminology problem: what is a metric?

The word "metric" is used differently in (1) the context of a metric space, and (2) such context as "the metric of space-time".

A manifold with a metric tensor is called a pseudo-Riemannian manifold. (Quoted from "Metric (mathematics)#Important cases of generalized metrics".)

A special case of great importance to general relativity is a Lorentzian manifold, in which one dimension has a sign opposite to that of the rest. (Quoted from the lead of "Pseudo-Riemannian manifold".)

Thus, sometimes a space endowed with a metric is not a metric space. Regretfully, this confusing situation is not explained enough in our articles. See the ongoing dispute: Talk:Metric space#Questions to User:Verdana Bold about a recent attempt to emphasize the "more physical" usage of the word "metric". Indeed, that usage is notable and should be mentioned. However, this should be made carefully enough. Boris Tsirelson (talk) 17:35, 27 June 2015 (UTC)

Ceci N'est Pas Un Cercle

There is a silly argument going on between myself and two other editors at the new article real projective line about whether $\mathbb {R} P^{1}$ "really" is a circle. At first, it was argued that it must be homeomorphic but not diffeomorphic to a circle, because the universal cover of $S^{1}$ is $S^{1}$, and that of $\mathbb {R} P^{1}$ is $S^{1}$, and the property of being one's own universal cover is a diffeomorphism invariant - fortunately this argument was dropped. Then it was argued that it may be topologically equivalent to a circle, but have a different geometric structure, but later it was accepted that the natural metric makes $\mathbb {R} P^{1}$ isometric to a circle. Presently it is argued (I think) that there is some intermediate structure, stronger than the topological structure but weaker than the metric space structure, wherein cross-ratios of four points can be defined without reference to any distance function. I believe this is impossible; defining the cross-ratio (and proving that it is well-defined) requires using the metric. The reason this is significant is that I believe it is important to state clearly in the lede that the object is a circle, without reference to terminology that may confuse laymen, such as "homeomorphic," "isometric," or "birationally equivalent." This is the text I want:

In mathematics, the real projective line is a circle. It is a trivial example of a more general class of topological spaces known as projective spaces. The real projective line is formed from lines through the origin in two-dimensional space. One considers these lines to be the "points" of the real projective line, and the angle between them to be the "distance" between these "points." Formally, the lines through the origin form equivalence classes, allowing for the real projective line to be defined as a quotient space.

Having this context in mind may give the reader a better chance of understanding the technical details that follow. I think the importance of stating clearly that we are talking about nothing other than a complicated way to define a circle is shown by the talk page itself. Even a relatively mathematically-sophisticated reader might see "a space homeomorphic to a circle" and think that it has some other geometric structure, that it is a double cover of a circle, or whatever. This is what led a normally quite competent editor to the logical and factual fallacies in the "homeomorphic but not diffeomorphic" argument above; he was trying too hard to rationalize a wrong idea, when the truth was just too simple. On a related note, the article is also suffering from the idea that the group $PSL_{2}(\mathbb {R} )$ is the "automorphism group" of the real projective line. I think that someone just made this up. It is now the central principle in the platform of the "not a circle" party. --Sammy1339 (talk) 13:49, 26 June 2015 (UTC)

For understanding the context of this post, I recommend to the reader to consult Talk:Real projective line § Is the projective line a circle?. I have nothing to add to my posts in this discussion, especially the last ones. D.Lazard (talk) 14:09, 26 June 2015 (UTC)

"Obviously" a projective line is not a circle (real or otherwise). There can be an isomorphism between the two depending on a category in the context; say, one is working in the category of topological spaces, but such an isomorphism is not the equality ("is"); a morphism ceases to make sense once you step out of the category. -- Taku (talk) 17:04, 26 June 2015 (UTC)

For example, saying "a projective line a twisted cubic curve" is at worst wrong or at best very misleading; you can't replace an isomorphism by an identity. -- Taku (talk) 17:22, 26 June 2015 (UTC)

But (correct me if I'm wrong) there's no category in which they are not isomorphic. --Sammy1339 (talk) 19:09, 26 June 2015 (UTC)

I don't know if it would ever occur to me to read the word "circle" as referring to an intrinsic abstract manifold, even one with a metric. To me the word implies an embedded circle, the set of all points in some larger (usually Euclidean) metric 2-manifold equidistant from a given point. Is there some natural way of identifying such an embedding space in the case of the real projective line? --Trovatore (talk) 19:16, 26 June 2015 (UTC)

Well, I think of "circle" as meaning an intrinsic manifold, but if you want to look at it that way, then yes, there is: take $\mathbb {C} ^{2}$ and form the complex projective line (Riemann sphere), and look at what happened to the two real axes. They became a great circle, formed by the real line and the point at infinity. --Sammy1339 (talk) 19:33, 26 June 2015 (UTC)

But this doesn't seem to be exactly inherent in the notion of the real projective line. I think this is the basic problem. The real projective line is intrinsic, whereas the circle is embedded. Therefore they are not the same. --Trovatore (talk) 19:40, 26 June 2015 (UTC)

So would it be appropriate to say it's an "abstract circle?" --Sammy1339 (talk) 19:42, 26 June 2015 (UTC)

Possibly. --Trovatore (talk) 19:43, 26 June 2015 (UTC)

Just so we understand each other though, is the Riemann sphere not a sphere? --Sammy1339 (talk) 19:47, 26 June 2015 (UTC)

No, for the same reason; in fact, our very own Riemann sphere doesn't say so in the first paragraph (it defines it as a one-point compactification of the complex plane, which is correct; in fact, even more correctly it uses the term "model"). Interestingly, complex projective line redirects to "Riemann sphere", which is probably ok. By analogy, should real projective line redirect to a "circle"? Am I the only one who thinks that's crazy? Having separate articles already suggests the two concepts (RP¹, S¹) are distinct. -- Taku (talk) 22:47, 26 June 2015 (UTC)

Of course the article Riemann sphere calls the Riemann sphere a sphere -- any time you have a construction "the [adjective] [noun]," you can expect the reader to understand that you are speaking of a kind of [noun] unless you say otherwise, and this does not require special mention. (In mathematics sometimes it is the case that an [adjective] [noun] is not actually a [noun], and in this situation one can expect to find somewhere an explanation of this situation, e.g., "The terminology is somewhat confusing: every topological manifold is a topological manifold with boundary, but not vice versa.")

I do not have strong feelings about the underlying debate, but I certainly would not have thought to criticise or correct someone who referred to RP^1 as a circle. --JBL (talk) 23:07, 26 June 2015 (UTC)

I should have phrased it differently. Is $S^{2}$ with the spherical metric a sphere? --Sammy1339 (talk) 01:47, 27 June 2015 (UTC)

@Sammy : If one defines distance between two points on a circle to be the length of the chord between them, then that is topologically but not metrically equivalent to the circle intrinsic metric in which the distance is the arc length. Might that be the circle that is not isomorphic to the projective line, or is the chord-length metric a natural thing to use for that? As far as cross-ratios go, linear fractional transformations on the complex numbers take circles to other circles while changing distances but leaving cross-ratios intact. Therefore the function assigning cross-ratios to quadruples of points on the circle does not determine the function assigning distances between them. In that sense one can say that the cross-ratio function amounts to less structure than the metric but more than the topology. Michael Hardy (talk) 23:41, 26 June 2015 (UTC)

(Late response to original post.) To play devil's advocate: If RP^1 is a circle, then what is the radius of the circle? The average reader of Wikipedia thinks that circles have radii. They do not implicitly understand that you have omitted the words "smooth manifold diffeomorphic to". The one-point compactification construction suggests that the radius is infinite. The fact that the unit circle in R^2 double-covers RP^1 suggests that its radius is 1 / 2.

The average reader is not familiar with notions of space that lack notions of distance. And do we expect such readers to read this article? Or should this article expect that the reader is coming from some background in topology, algebraic geometry, etc.? Mgnbar (talk) 00:20, 27 June 2015 (UTC)

To put it another way: If RP^1 is just a circle, then why does it have any special name at all? (I have been asked such questions about Riemann spheres being "just" spheres.) Mgnbar (talk) 00:27, 27 June 2015 (UTC)

@Michael Hardy:@Mgnbar: There is a natural metric on the real projective line and it is the same as the metric of a circle with radius one-half. I explained this on the talk page. About why $\mathbb {R} P^{1}$ has a name, it assuredly wouldn't, were it not for $\mathbb {R} P^{n}$ - in fact I am having trouble finding sources that even talk about it in any detail, presumably because it is just a circle. --Sammy1339 (talk) 01:57, 27 June 2015 (UTC)

It is not "just" a circle. Topologically it is a circle, but it may differ in some other respects from what is normally called a circle. Michael Hardy (talk) 04:01, 27 June 2015 (UTC)

Under your metric, the real line does not isometrically embed into RP^1. So readers may be confused by statements about how RP^1 is just the real line with one point at infinity added. That's the point. There are several ways to think about projective spaces. Mgnbar (talk) 03:30, 27 June 2015 (UTC)

The euclidean real line cannot be isometrically imbedded in $\mathbb {R} P^{1}$. --Sammy1339 (talk) 03:56, 27 June 2015 (UTC)

I think that our disagreement comes from different viewpoints, which exist because RP^1 appears in various areas of mathematics. Your viewpoint seems to be differential geometry. I mentioned the real line because its topological embedding into RP^1 is an important example in topology (the one-point compactification). Your talk of metrics and circles seems beside the point to me, because in elementary algebraic geometry, which uses projective spaces heavily, there is often no mention of circles and metrics.

So maybe we should stick to precise (and verifiable) statements about how RP^1 is topologically a circle, can be endowed with a metric under which it is isometric to a Euclidean plane circle with radius 1/2, etc.? Mgnbar (talk) 12:35, 27 June 2015 (UTC)

I would also like to suggest that at some point someone makes an attempt to find sources supporting their position, rather than relying on appeals to pure reason only. (And that is the last from me, probably.) --JBL (talk) 01:15, 27 June 2015 (UTC)

@Joel B. Lewis: Yes, I share that concern. But we also have to use common sense and write the article in a way laypeople can understand. We're supposed to paraphrase sources, not copy from them. I am bothered by some of the uncited claims in the article though, particularly the stuff about the "automorphism group." --Sammy1339 (talk) 01:57, 27 June 2015 (UTC)

Yes, sources determine what we can and cannot write here. But keep in mind that different sources may use terms such as "circle" differently. In fact, a topology textbook and a geometry textbook may disagree on what that term means. And maybe neither of those books is written for this article's audience. Mgnbar (talk) 03:30, 27 June 2015 (UTC)

Contrary to what was indicated by Sammy above, the universal cover of $\mathbb {R} P^{1}$ and of $S^{1}$ (which are the same thing) is $\mathbb {R}$. Remember that a universal cover must be simply connected. JRSpriggs (talk) 11:00, 28 June 2015 (UTC)