Wikipedia:WikiProject Mathematics/A-class rating/2007

Archive for Wikipedia:WikiProject Mathematics/A-class rating, which "is a process for discussing and determining whether mathematics articles should be given A-class status."

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Addition edit

Result: not promoted. Seems to be getting close, but still too weak on education and history. -- Jitse Niesen (talk) 00:57, 21 March 2007 (UTC)[reply]

Ooh, first crack at a new process!
Ironically, I wrote most of this article, ~~but I have to oppose any real recognition of quality at this point.~~ Not because of the items at the top of Talk:Addition, which is a bit of a wish list. I oppose because of the "Interpretations" section. I strongly believe that it is important to describe the various interrelated meanings of addition in an educationally and mathematically enlightened manner, and to do so in an understandable way. I also believe that I've done a terrible job in moving toward that goal, and I don't know how to continue! Melchoir 08:43, 9 March 2007 (UTC)[reply]

…In fact, most of "Extending a measure" should probably be scrapped and rewritten. I don't even have records of all the sources I was reading back when I wrote that stuff. Maybe we should create a sub-article titled Interpretations of addition or similar, and it could contain all the material on the deeper conceptual issues — believe me, they exist and I have seen them — while the summary in the main article could afford to be a little more naive. Melchoir 09:38, 9 March 2007 (UTC)[reply]

Yes I take your point and there is an under construction tag. I've been reading a bit on Piaget research on how children lean mathematics. Piaget views addition as an operation of classes and also an operation on numbers. It might be worth adding something on educational aspects. Whats the Geary ref mentioned in the todo list? --Salix alba (talk) 09:57, 9 March 2007 (UTC)[reply]

Currently most of the educational stuff is reflected in "Performing addition", although not at a deep level. I suppose if Piaget had something to say about addition, it ought to be here somewhere, although he doesn't have the last word. For example, I think I read that Wynn's 1992 experiment was initially taken to be something of a surprise, because Piaget's findings would seem to suggest that if young children can't conserve number, then they have no hope of adding.

Geary… I have no idea! This is why I've since learned to keep better records. Melchoir 10:06, 9 March 2007 (UTC)[reply]

Is there another WP article that covers the history of addition (or the history of arithmetic in general)? Even the arithmetic article seems pretty short on history. The section on "Extending a measure" seemed too technical for the intended reader - eyes would glaze at the phrase "natural identification of sets of functions". So I took the initiative to edit it. CMummert · talk 13:26, 9 March 2007 (UTC)[reply]
- Thanks! There's some historical information scattered throughout the article, but it doesn't tell a coherent story, and I doubt that it could. Questions like "When was addition invented?" don't really make much sense as separate from the history of mathematics, and even some of the present material kind of treads that line. Melchoir 19:23, 9 March 2007 (UTC)[reply]

I was away for a little while, but there were no new comments. Melchoir, is your concern about the Interpretations section satisfied? I have two areas of concern:

~~The generalizations section needs to include a discussion of addition in linear algebra.~~ I have since fixed this.
There is little historical information.

I can work on the first of these, but the second could use a more knowledgable editor. CMummert · talk 19:08, 13 March 2007 (UTC)[reply]

Oh! Sure, Interpretations is good enough for a general article like this one.

I suppose linear algebra could be tacked right onto the end of "In algebra", which currently ends with the words "abelian groups". Of course, linear algebra and vector spaces have a lot more going on than addition, but they're certainly worth a mention, maybe even a diagram. Melchoir 19:50, 13 March 2007 (UTC)[reply]

It might be worth mentioning the Successor function in the interpretations. Looking at the addition of natural numbers section, its very technical writing, which I would not expect a lay reader to be able to follow.

There is some discussion on my talk page, where a couple of non maths people are looking at the lead, both found it very hard to follow.--Salix alba (talk) 00:03, 17 March 2007 (UTC)[reply]

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Peano axioms edit

Support. I don't see much wrong with it. It would be nice if a logician had a look at it, especially since I have a couple of issues which I can't resolve myself.
1. Is there a subtle reason why the articles uses both N and N; specially, in the "Binary operations and ordering" section.
2. "The main source of difficulty is the second-order induction axiom." — is there another source of difficulty?
3. "This is one reason that the first-order axioms of PA are generally considered to be weaker than the second-order Peano axioms." — in my happy world as an applied mathematician who never has to bother about foundational issues, the fact that there is more than one model proves that PA is weaker. So why the weaselish "generally considered to be weaker"? Any references to contrary opinions?
Jitse Niesen (talk) 12:10, 26 March 2007 (UTC)[reply]

I went through the article and made an effort to address these. Except for the section "Existence and uniqueness", which is working with things that clearly are not natural numbers, the bold N should be used. The second two bullets have been addressed by rewording the sentences. I will go through and reformat the references this week. CMummert · talk 13:13, 26 March 2007 (UTC)[reply]

I'm not comfortable that no specialists have commented, so I asked a couple of editors who know more about logic than me to comment. I hope that this discussion can be kept open for a few more days to give them a chance to have a look. -- Jitse Niesen (talk) 08:20, 29 March 2007 (UTC)[reply]

Support. This article seems to be well written, and provides a good overview of the Peano axioms. My only reservation is that the first-order vs. second-order discussion seems to involve a fair amount of hand waving. it's also worth noting that the Peano axioms are generally (sop far as I, a nonspecialist, know) considered in a first order context, and they're not really expected to be categorical. Greg Woodhouse 15:50, 26 March 2007 (UTC)[reply]

Support. I only have two concerns (aside from N vs. N).
1. It is not obvious, and possibly needs to be stated, that induction does imply recursion or inductive definition.^[1] (The reason I have that book is left as an exercise for the reader.)
2. I also don't recall reading that "there is no countable nonstandard model of PA in which either the addition or multiplication operation is computable." I assume it's in one of the references....
Everything else falls within a logician's "common knowledge" or can easily be derived from such. (Well, my knowledge, anyway.) — Arthur Rubin | (talk) 13:48, 29 March 2007 (UTC)[reply]

Re number 2, this is called "Tennenbaum's theorem" and is proved in Kaye's book. I added a reference and the point. I am not sure what you are proposing for #1. CMummert · talk 14:08, 29 March 2007 (UTC)[reply]

In Peano axioms#Binary operations and ordering, the text reads: "To define the addition operation + recursively in terms of successor and 0,...." What I'm saying is that this definition requires set theory or second-order logic. Once addition and multiplication are defined, it's possible to formalize definitions of exponentiation and sequences (or finite set theory) within first-order PA, which then allows the formalization of recursive definitions. — Arthur Rubin | (talk) 01:45, 30 March 2007 (UTC)[reply]

I added a few sentences, which I hope will resolve your concerns. CMummert · talk 02:20, 30 March 2007 (UTC)[reply]

Weak oppose.
- (+) I didn't spot any glowing errors, and I think the coverage is pretty good.
- (+) I am glad the authors have taken pains to inline short definitions of all jargon. It is a style that I wish would be adopted more generally in WP articles.
- (–) For an article on logic, there is a general feeling of imprecision in the text. To start with, I would greatly encourage a syntactic separation of object and meta levels. There is a lot of repetition and redundancy.
- (–) I don't see any reason to present the axioms based off 1. Peano's reasons for leaving 0 out are obscure, and it would be better to state that Peano's original formulation deviates from the standard formulation off 0 rather than adopt his idiosyncracies.
- (–) I am not happy with the Construction of the natural numbers in set theory section. It spends too much time on the definition and not enought time discussing the relation of Peano#9 to the axiom of infinity from ZF.
- (–) I wish the article would use the <math> tags and depend on texvc for the formatting, instead of homebrewing with wiki syntax.
- Overall, I think the article is almost, but not quite feature quality. Congratulations to the authors. — Kaustuv Chaudhuri 15:01, 30 March 2007 (UTC)[reply]
I'm not sure how to resolve these comments. Would you be willing to make some changes yourself? There isn't a lot of metamathematical study here, so I don't see how to mark "meta" syntax. The axiom of infinity is mentioned; how would you like to see its description expanded? The issue of texvc versus wiki formatting surely isn't a serious issue. In general, the standard for an A-class article should not be "perfection". CMummert · talk 17:02, 30 March 2007 (UTC)[reply]
Sorry for the lack of specificty in my evaluation. There are several changes I intend to make. It is a matter of finding an hour of uninterrupted editing time. Perhaps over the weekend. — Kaustuv Chaudhuri 17:08, 30 March 2007 (UTC)[reply]
One of the specific points is whether to start at 0 or 1. If the axioms are nowadays usually based off 0, then that would seem to be a good reason to do the same here, especially as this is already done later in the article (sections on addition/multiplication and Kaye's formulation).

The homebrewed wiki syntax is perfectly fine for me. It's widely used here and it often gives better results that the <math> tag. -- Jitse Niesen (talk) 14:59, 31 March 2007 (UTC)[reply]

I started making small changes today and ended up rewriting significant portions of it. My goal was to be precise, avoid needless repetition, and adhere to the WP:MOS as best I can. I won't alter the main article until this nomination finishes, but here is my proposal: User:Kaustuv/ip/Peano axioms. — Kaustuv Chaudhuri 02:44, 1 April 2007 (UTC)[reply]

The rewriting seems very nice, although I have a few quibbles that could be discussed on the talk page. The content is essentially identical to that in the current article in terms of scope and depth. The reading level is slightly higher than the current version. I hope you will implement your changes in the main article. CMummert · talk 03:18, 1 April 2007 (UTC)[reply]

Comments I don't think I'd ever seen the distinction "Peano axioms" = 2nd order, "Peano arithmetic" = 1st order, prior to Wikipedia. Is this really standard? (Confusing matters even further is that "2nd order Peano arithmetic" is a first-order (two-sorted) theory, at least in my usage; do we not have an article on PA2 somewhere?) --Trovatore 08:17, 31 March 2007 (UTC)[reply]

I think that when nonlogicians say "Peano's axioms" they usually mean the unformalized ones, which correspond most closely to the second order way of thinking (like all of informal mathematics). On the other hand, the article needs to have something to call the second order axioms, and "Peano arithmetic" is right out.

There is indeed an article on Second order arithmetic - I'll make sure it is in the see also section. The article on second order arithmetic also has to explain that there are both first and second order semantics and that the first order ones are more commonly used. CMummert · talk 14:14, 31 March 2007 (UTC)[reply]

References edit

^ Rubin, Jean E. (1990). Mathematical Logic: Applications and Theory. Saunders College Publishing. p. 311. ISBN 0-03-012808-0. In an inductive definition, the proof that the function F is unique is an easy proof by mathematical induction and will be left as an exercise. However, the proof that the function F exists uses set theoretical techniques that we shall not discuss here.

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Knot theory edit

Decision: Promoted to A class since there are no objections. -- Jitse Niesen (talk) 05:04, 26 April 2007 (UTC)[reply]

Support article has received a major rewite reciently and is in good shape. --Salix alba (talk) 14:03, 15 April 2007 (UTC)[reply]

I'm not in a position to comment seriously on the content. I did some trivial copyediting this morning. The only remaining stylistic issue I see is that WP:SCG encourages attributions of historical motivations and eponymous concepts to have more explicit references. The history section in particular could use additional references. If there is a survey paper on the history of knot theory, that would serve well, otherwise some refs to the original papers by a few of the researchers would do. CMummert · talk 12:32, 16 April 2007 (UTC)[reply]

- Ok, so you are saying that there should be refs to the papers by jones, Witten, etc.? Also, I don't really any know any general histories of knot theory (although I know of articles on specific episodes or time periods); besides the Silver reference for the 19th century stuff, I mainly used the stuff that appears in more historically-oriented knot theory books. In general, some comments may not be easily referenced; it's remarkably hard to find that one reference that says "the Jones revolution changed the landscape of knot theory", even if it is undisputed knowledge. --C S (Talk) 00:51, 22 April 2007 (UTC)[reply]

- - I think that a few references to the original papers by Jones, Witten, etc. would be plenty. To be fair, these references are included in the linked articles described lower down in this article, but this isn't apparent from the lead section. I see there is a graduate texts in mathematics reference - is it suitable as a general reference for these things? CMummert · talk 01:34, 22 April 2007 (UTC)[reply]

- - - The GTM by Lickorish is a good reference for some of the basics of the theory of the Jones polynomial, extension to a 3-manifold invariant, and the quantum group invariants (in a single most basic case). But I don't think there is one general reference for all the work inspired by the Fields Medalists mentioned in the article. But I added some of the original reference, with a couple good "further reading" type refs. --C S (Talk) 23:26, 22 April 2007 (UTC)[reply]

- - - - It looks fine now. I don't see any other important issues. CMummert · talk 12:51, 23 April 2007 (UTC)[reply]

Support, with the same qualification of not actually knowing anything as CMummert. I found two sentences rather confusing:
1. "Note that if we believe that the Alexander-Conway polynomial is actually a knot invariant, this shows that the trefoil is not equivalent to the unknot." — This suggests to me that the Alexander-Conway polynomial may actually not be a knot invariant.
2. "Conway found a number of omissions but only duplication in the Tait-Little tables" — I don't know what's meant here.

Weak points which in my opinion should be addressed to make it perfect are: the lead section is rather long, and there is not much discussion about applications. -- Jitse Niesen (talk) 04:32, 24 April 2007 (UTC)[reply]

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Poincaré conjecture edit

Result: Promoted to A class. The only outstanding criticism is the rather vague "the prose could also be made a bit tighter". No comments have been made for quite some time, so I have to make a decision, and this discussion leads me to the conclusion that the article is not quite FA class but close. -- Jitse Niesen (talk) 11:27, 18 May 2007 (UTC)[reply]

This article has made it through a period of tumultuous change and currently reads very nicely. The lede has been much improved, and there is a well-written section on Perelman and Ricci flow worked on by R.e.b. and others. I believe concerns about readability and accessibility have been addressed as much as can be reasonably expected. Perhaps some improvements to the exposition could be made, e.g. analogy to heat equation in the Ricci flow section, and so forth, but I figure people can use this review to make such improvements.

Since accessibility has been a frequent complaint, let me made one more remark: the statement of the Poincare conjecture is actually much harder to understand than problems like P vs. NP (probably the easiest of the Millennium problems to explain). Even compared to something like the Riemann hypothesis – after all, more people are familiar with basic complex analysis (or advanced calculus) than manifolds and geometric topology. So I hope this discussion doesn't focus on accessibility, although I will agree that this is an issue we need to be concerned with in relation to this article. --C S (Talk) 01:57, 4 May 2007 (UTC)[reply]

oppose while good I don't think its A-class yet. I think the section on higher dimensions could be expanded a little as it adds context to the 3 dimensional case. The section Hamilton's Program and Perelman's solution has a lot of full references in the text, these could be better presented with a shorter title in the text and a reference/footnote. The prose could also be made a bit tighter. Reading through the Ricci flow section I came across ... as all constant curvature manifolds are well understood, which provoced my interest wanting to find out more about this, but alas no link. --Salix alba (talk) 08:00, 5 May 2007 (UTC)[reply]
- I expanded the higher dimensions part and made a couple other changes like the caption for the lede image. Also, that interesting sentence you quote isn't so good. Hyperbolic 3-manifolds aren't nowhere nearly as well understood as spherical 3-manifold or flat 3-manifolds (see these links to satisfy your curiosity ^.^). In this context where you have some restriction on the fundamental group (simple connectivity, and later in the section, free products of finite groups and infinite cyclic groups), the hyperbolic case doesn't have to be worried about. Anyway, I think the point is that in this context, the only constant curvature guys will be spherical ones, and the only simply-connected spherical 3-manifold is a 3-sphere; I made the appropriate modification.--C S (Talk) 11:32, 6 May 2007 (UTC)[reply]

I had a go at the formatting of the references and the "External links" section, which seemed a bit excessive. I agree with Salix alba that the section on higher dimensions should have a bit more. I'm also wondering whether the term "three-dimensional sphere" in the first paragraph should be explained: I fear that most people would think that it refers to the 2-sphere. And perhaps the article needs more motivation. Why is the conjecture so important? -- Jitse Niesen (talk) 14:16, 5 May 2007 (UTC)[reply]

- Hopefully the new caption should help. As for your question, that's hard to answer. For a long time, like Fermat's Last Theorem, it wasn't really important, just tricky, so gained a big reputation. To a large degree, topologists just worked around it and managed just fine. Later on, the conjecture fit into Thurston's picture of what a classification should look like, but even then, if that part of the geometrization conjecture had been shown false, I don't think it would have been too devastating. Right now the lede says it is considered one of the "most important" conjectures in topology; I've never been too happy with that. I think it's always been more famous than important. Thurston said once in one of his papers that his hope for the geometrization conjecture is that it would be a more productive conjecture to tackle than the Poincare conjecture. So I would say its importance, such as it is, comes from other important conjectures, rather than on its own. Another take on this is that the Poincare conjecture really is a truly basic question about 3-manifold topology. Can there be other simply-connected 3-manifolds other than the 3-sphere? It's an intriguing question, which is undoubtedly why so many have gotten "Poincaritis". --C S (Talk) 11:32, 6 May 2007 (UTC)[reply]

Hey guys, I am on a unassessed article journey and I am going to give the article a B until you decided on this issue.--Cronholm144 01:02, 12 May 2007 (UTC)[reply]

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Area of a disk edit

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The consensus is to promote to A class. — Carl (CBM · talk) 02:13, 5 June 2007 (UTC)[reply]

Minor point in the Archimedes proof section. I think it need to restate which triangle we are talking about, just to make the section self consistant. As ever reference, if it wants to become an FA it will need some references inline. While reading it I did feel the need for references for each section. --Salix alba (talk) 11:36, 22 May 2007 (UTC)[reply]

Triangle is now explicit so section is self-contained.

It does not want to become FA. Our mathematics citation guidelines do not require in-line references, but I agree that here a few could be a convenience for the reader. The citations are collected at the end, and (thanks to COGDEN's {{Citation}} template) we can now hyperlink Harvard-style; will do so. --KSmrq^T 23:42, 22 May 2007 (UTC)[reply]

I reformatted the two tables using CSS. Here are some more comments about other sections. I am assuming that this article is aimed at an audience with little more than a United States high school education in geometry. The second person is used very frequently in this article; I will try to rephrase some sentences to reduce it.

Archimedes proof. An image showing the circle and triangle would be nice.
Onion proof. It would be nice if the image used Δt instead of dt and the article explained (maybe vaguely) why this works. That would make this section less stubby at the same time. Readers are likely to be confused when they run into a diagram labeled with dt if they took high school calc in the US.
Fast approximation. The derivation is probably written over the head of the audience.
Dart throwing. I would be interested for a source for the rule of thumb about how many samples are needed, since I don't know much about Monte Carlo methods.

CMummert · talk 23:55, 25 May 2007 (UTC)[reply]

Ad 4) I'm overseas, so can't help with a reference (which the article does need), but I can confirm the rule of thumb. The precise statement is that the standard deviation of the result you get by throwing darts is proportional to 1/sqrt(N) when N is number of darts. This is basically the central limit theorem.

PS: Google Books finds Computational Physics by J. M. (Joseph Marie) Thijssen, p. 273 [1]. -- Jitse Niesen (talk) 18:40, 29 May 2007 (UTC)[reply]

Looks good overall. I'm somewhat unhappy with the "generalizations" section. So I fixed it to be more intuitive and descriptive (fixing a missing π in the process). --C S (Talk) 06:13, 28 May 2007 (UTC)[reply]

I don't have any outstanding issues. Time to promote? CMummert · talk 17:42, 1 June 2007 (UTC)[reply]

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Fermat's last theorem edit

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The consensus is to remove the A-class rating. — Carl (CBM · talk) 03:04, 19 June 2007 (UTC)[reply]

I've been going through the articles which were assessed as A-Class before this review process was instigated, checking if they still meet the standard. This one caught my eye, because in September 2006, it was listed as a candidate for Mathematics Collaboration of the Week with the comment "The history of this conundrum is really fascinating, but the article hardly does it justice". It doesn't meet the criteria for MATHCOTM collaboration, so I removed it from the list, but it may be worthy of an A-Class Review.

It was listed as A-Class in October 2006, although there were no substantial changes since September. There have not been many changes since then: a little more on early attempts at the proof, some additional references, and more pop-culture.

It is surely close to A-Class (unless we are going to worry about citation and WP:SCG), but I wonder if it has really nailed it on the completeness front: there are opportunities here to add more mathematical content (rather than history and pop-culture). For instance, the contributions of Germain and Kummer could be expanded: the latter in particular inspired the development of ring theory, ideals, and the ideal class group. And surely more could be said about the proof — perhaps we should separate the proof from the history of the proof. At present the text seems to imply that Wiles proved the Taniyama–Shimura conjecture — whereas my understanding is that he only proved a special case (which was sufficient for FLT) — and links with the Langlands program go unmentioned.

So I think it might benefit from the keen eye (and keyboard) of a few project members! Geometry guy 19:12, 11 June 2007 (UTC)[reply]

PS. It is currently listed as Bplus-Class, but this is partly an accident: the page was moved from "Fermat's Last Theorem", but the Comments weren't, so I thought the rating was unsigned, and didn't feel confident to sign it off as A-Class, so I downgraded it to Bplus. In other words, this is more of a review than a promotion question.

PPS. It has now been reassessed as B-Class by Arcfrk, suggesting it needs our attention even more! Geometry guy 09:13, 12 June 2007 (UTC)[reply]

I agree that the article could use more content about the actual mathematics of the proof and related areas. Unfortunately this will take some time to do correctly, and a strong knowledge of number theory to do well, as people have noted on the article talk page. It might be worth canvassing some number theorists, and maybe hanging an outline on the article to give people easy places to put in a little time. I would guess that most of the claims about the history were taken from one of the popularizations listed at the end. — Carl (CBM · talk) 14:24, 12 June 2007 (UTC)[reply]

I presume most of this was taken from Singh, and that's why he's a Reference. Most of the history looks good; but someone should check that it's from Singh, and so indicate in the references. Septentrionalis PMAnderson 02:26, 15 June 2007 (UTC)[reply]

I have to say the writing in this article is truly atrocious. I'll help a bit, but this really needs a whole overhaul, mathematical issues aside. --C S (Talk) 02:58, 14 June 2007 (UTC)[reply]

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I don't think there is consensus to promote to A-class, but the article seems close. Perhaps another nomination in a month or two, after the edits User:Turgidson proposed]]. — Carl (CBM · talk) 16:19, 11 December 2007 (UTC)[reply]

Homotopy groups of spheres edit

Homotopy groups of spheres (edit | talk | history | links | watch | logs) review
I believe that this article now meets the standard for A-Class. In particular, I believe it is comprehensive, factually accurate and verifiable, and would like experts to check this and point out areas for improvement. However, I am mainly nominating here to ask for input from members of the Mathematics WikiProject (and others) on the presentation, balance, and accessibility of the article. Is the technical material accessible enough for an article of this nature? Is there too much technical material? In what ways could the presentation could be improved? How broad is the target audience? I hope this will stimulate some useful discussion.
Nominated by: Geometry guy 18:43, 5 November 2007 (UTC)[reply]

I like the article a lot, but I think it can still be improved, especially by adding material about some other current approaches, not yet mentioned in the article. If there is some interest in that, I can give it a shot, and see how it goes. (I'm not a member of MWP, so I'm not quite sure how one gets involved in these discussions...) Turgidson 03:40, 15 November 2007 (UTC)[reply]

Thanks for commenting. Comments from everyone are most welcome here, on the mathematics project talk page, and on the article talk page — there is certainly no membership requirement! What material do you think is missing? Geometry guy 19:53, 15 November 2007 (UTC)[reply]

Computing the homotopy groups of S^2 via braid groups. I could add a section on that -- it would tie up this subject to a whole slew of others ones (including links in S^3 and combinatorial group theory). It's just that I need some time to do a more-or-less reasonable job, and it would need some pictures to make it look nice, so I'm a bit nervous to invest too much time and effort without knowing whether there would be some interest in this (I'm in an especially funky mood after some upsetting experiences at certain recent CFD/DRVs). Any thoughts? Turgidson 20:25, 15 November 2007 (UTC)[reply]

This sounds great to me. In particular, we desperately need a good image to associate with π₃(S²). I appreciate your mood: believe me, Homotopy groups of spheres is much more fun than certain CfD/DRV's that we don't need to mention! And it has great editors (KSmrq, R.e.b., Jakob Scholbach etc.) working on it too. Geometry guy 20:35, 15 November 2007 (UTC)[reply]

Thanks for the encouragement. OK, I'll give it a shot, then, just please have a bit of patience, it may not be completely polished right away, as I will need to do it in between other things I'm doing at the same time, both on and off WP. Let's continue the discussion on the talk page, once I get off the ground. Turgidson 20:42, 15 November 2007 (UTC)[reply]

I read through the first several sections in detail, and skimmed that later ones. It's a good article, although (perhaps unavoidably) only accessible to someone with a decent ability to handle blue links and notation. Concrete areas of possible improvement:

Add some images to the "n-sphere" section near the beginning. For example suspensions should be able to be illustrated without too much effort.
In the section "general theory", be more clear that the patterns are actually provable and continue indefinitely. This is implied but maybe not clear to a naive reader.
The "applications" section is likely to be of interest; specific references for more of the bullet points would be nice.

For someone who already has an undergrad background, or who is willing to accept undefined terms to get a bigger picture, this is a very nice article to read. — Carl (CBM · talk) 03:51, 17 November 2007 (UTC)[reply]

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[1] Rubin, Jean E. (1990). Mathematical Logic: Applications and Theory. Saunders College Publishing. p. 311. ISBN 0-03-012808-0. In an inductive definition, the proof that the function F is unique is an easy proof by mathematical induction and will be left as an exercise. However, the proof that the function F exists uses set theoretical techniques that we shall not discuss here.

[1]