Wikipedia:WikiProject Mathematics/A-class rating

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Contents 1 Goals 2 Process 2.1 Nominating an article 2.2 Leaving a comment 2.3 Closing the discussion 3 Criteria 4 List of A-class math articles 5 Current discussions 5.1 Peano axioms 5.2 References 6 Archived discussions

This is a somewhat formal system for promoting mathematics articles to A-class status.

[edit] Goals

The goals of the A-class rating and the review process are:

1. Recognize quality articles, and make the assignment of A-class status less subjective.
2. Encourage articles under review to improve, rather than expecting them to be perfect before they are nominated.
3. Encourage a collegial discussion process rather than an adversarial approval process.

[edit] Process

[edit] Nominating an article

Any editor may nominate an article for A-class rating discussion. To do so:

1. Add the argument |ACD=yes to the end of the {{math rating}} template near the top of the article's talk page. Capitalization is important. If there is no maths rating template, you may add one. Save the page.
2. The sentence "A discussion on promoting this article to A-class status is underway." will appear in the math project banner. Click the link "discussion".
3. Enter {{subst:maths acd}} ~~~~ into the empty page and save. If the article has previously been nominated for an A-class rating, please archive the previous discussion by moving (not copying) it to an archive page, which will leave a redirect. Replace this redirect with the code above.
4. Add the line {{Wikipedia:WikiProject Mathematics/A-class rating/Name of nominated article}} to the top of the section Current discussions below.

[edit] Leaving a comment

All editors are welcome to comment on the discussion page. The criteria should be kept in mind, but editors are free to use their own judgement about the quality of an article.

In light of the goals above, reviewers should make an effort to improve the article rather than making comments about minor flaws. Thus comments should be limited to whether a reviewer supports the A-class rating or has a serious objection to it. Objections over relatively minor issues of writing style or formatting should be avoided; a comprehensive, accurate, well sourced, and decently written article should qualify for A-class status even if it could use further copyediting.

[edit] Closing the discussion

Any editor may close the discussion after seven days, although common sense should be used not to close a discussion that is still receiving many comments.

To close a discussion:

1. Remove the |ACD=yes parameter from the {{maths rating}} template in the article's talk page.
2. If there are no remaining substantial objections, change the class= parameter in the {{maths rating}} template to read class=A-class. If there are substantial objections and the article is not already rated A-class, leave the rating unchanged. If it is rated A-class, assign it another rating using the criteria.
3. Move the line {{Wikipedia:WikiProject Mathematics/A-class rating/Name of nominated article}} from the list of requests below to the current archive page.
4. Add {{subst:archive top}} and {{subst:archive bottom}} to the top and bottom of the discussion subpage, respectively.

[edit] Criteria

Readers should keep the following description in mind when deciding whether to support the A-class rating:

An A-class mathematics article is very useful to the intended reader and gives a correct and reasonably complete treatment of its subject. It is a model for other mathematical articles.

The article contains sufficient motivation for the topic. It places the topic in context by providing an both an appropriate historical account as well as links to related areas of contemporary mathematics.

The article's breadth, completeness, and balance have no flaws apparent to a nonexpert reader. A nonmathematician will typically find nothing wanting, although the article may omit a few minor points. Minor edits and adjustments might improve the article, however, particularly if made by an expert in the area.

The article meets the spirit of the Wikipedia style manual, the manual of style for mathematics, and the scientific citation guidelines.

Other sets of critera for A-class articles have been developed by WikiProject Military History and the WP 1.0 Editorial team.

[edit] List of A-class math articles

A-class mathematics articles are automatically added to the category A-class mathematics articles by the {{maths rating}} template.

[edit] Current discussions

Please add new discussions at the top of this section.

[edit] Peano axioms

Peano axioms (edit|talk|history|links|watch|logs) review
Nominated by: CMummert · talk 00:07, 23 March 2007 (UTC)

• 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)

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)

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)

• 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)

• 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)

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)

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)

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

• 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)

  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)

  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)

  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)

  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)

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)

• 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)

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)

[edit] References

[edit] Archived discussions

• Wikipedia:WikiProject Mathematics/A-class rating/2007