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The following discussion is an archived debate of the proposed deletion of the article below. Please do not modify it. Subsequent comments should be made on the appropriate discussion page (such as the article's talk page or in a deletion review). No further edits should be made to this page.

The result was Keep, WP:SNOW close. Tim Vickers (talk) 17:13, 25 July 2008 (UTC)[reply]

Duality (mathematics)

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Duality (mathematics) (edit | talk | history | protect | delete | links | watch | logs | views) (delete) – (View log)

This entire article is completely vacuous and artificial.

The definition (if it can even be called one) given states that:

Generally speaking, dualities translate concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion.

This could describe any of a number of different mathematical operations. An attempt is made to be more specific by characterizing dualities as involutions, but some of the items listed below are not involutions (the dual of a linear space for example). It seems like the only thing that the examples have in common is that they contain the word "dual".

I have never seen the word "dual" or the concept of duality itself defined or discussed in mathematical literature. This article certainly doesn't cite any instances of such discussions. In fact, the word is actually used in many different ways in mathematics. Thus to try to define duality as a single mathematical concept is misleading and incorrect.

I would like to suggest deleting this article or changing the title to "Mathematical terms containing the word dual". 68.89.168.74 (talk · contribs) Text copied from article talk page. ➨ ЯEDVEЯS has wasted eight of nine lives 11:50, 23 July 2008 (UTC)[reply]

  • Keep Making sense of sometimes vague notions of "duality" has been an important impetus to research, going back to the discovery of projective geometry at least; often category theory can it precise meaning nowadays, but it takes time and understanding to convert vague ideas into precise mathematics- this dates back to the original Eilenberg Maclane paper on the General Theory of Natural Equivalences and the vector space example among others. An example of a somewhat loose duality correspondence similar to this article's usage, is Eckman-Hilton duality cf the book in the Peter Hilton article. There's plenty of other ways to extend it too, Grothendieck dualities in all sorts of sheaf cohomologies, a concept of duality in analytic number theory, etc. The point is that the ways the word is used are generally not so different. They bear some kind of "family resemblance."-and the article points out an instance where the word is used in a totally unrelated way - "dual numbers." Although there may not be perfect overarching concepts, there generally is some way to relate one usage to another on this list.John Z (talk) 12:48, 23 July 2008 (UTC)[reply]
  • Keep notable concept (or meta-conept) in mathematics. This article is more accessible than dual (category theory) and dual object. It just needs some references. Gandalf61 (talk) 15:36, 23 July 2008 (UTC)[reply]
  • Strong keep. Projective duality, planar graph duality, linear algebra polarity, linear programming duality, etc., are all very important concepts within their fields (enough for separate Wikipedia articles) but also related in important ways to each other: e.g., polarity is one way of forming the projective dual, and if a planar graph is represented as the skeleton of a polyhedron, its planar dual will be the skeleton of the polyhedron's polar dual; the linear algebra duality between vectors and linear functions restricts to a projective duality between projective points (represented by equivalence classes of vectors) and hyperplanes (represented by zero-sets of linear functions), etc. The article could stand to be fleshed out to describe these connections more carefully rather than (as it is now) almost being a {{mathdab}} page, but that isn't an argument for deletion. —David Eppstein (talk) 21:36, 23 July 2008 (UTC)[reply]
  • Keep- The article needs a lot of work, but it's an inherently encyclopedic topic. Reyk YO! 22:32, 23 July 2008 (UTC)[reply]
  • Note: This debate has been included in the list of Science-related deletion discussions. -- Fabrictramp | talk to me 22:41, 23 July 2008 (UTC)[reply]
  • Delete The article has no sources and appears to be improper synthesis or original thought. Colonel Warden (talk) 12:57, 24 July 2008 (UTC)[reply]
    • Did you make any attempt to look for sources? There are numerous books directly on the topic of duality. The nominator appears to be worried about one sentence in the lede, rather than about the topic of duality. — Carl (CBM · talk) 13:21, 24 July 2008 (UTC)[reply]
      • Arithmetic Duality Theorems by J. S. Milne (Academic Press) [1]
      • Duality Principles in Nonconvex Systems by David Yang Gao (Springer) [2]
      • Duality in Analytic Number Theory by Peter Elliot (Cambridge Univ. Press) [3]
      • Duality in Measure Theory by C. Constantinescu (Springer) [4]
      • Introduction to Grothendieck Duality Theory by Allen Altman and Steven Kleiman (Springer) [5]
      • Theory of Duality in Mathematical Programming by Manfred Walk (Springer) [6]
      • Duality in Optimization and Variational Inequalities by C.j. Goh (Taylor & Francis) [7]
      • Natural Dualities for the Working Algebraist by David M. Clark and Brian A. Davey (Cambridge Univ. Press) [8]
    • "Colonel Warden"'s complaint that duality (projective geometry) is opaquely written caused me to look at that article. I would say it's written for a more advanced level of reader than necessary: one could introduce the concept in such a way that most undergraduates would understand what is being said, with a variety of examples, and then later in the article look at it from a more advanced point of view. That would improve the article. That said, I don't think it's opaquely written; it's actually written in such a way that any mathematician would understand it. Certainly references could be added; there are dozens of books that present the material. As for the question "Why are the maths articles all so bad?", that can only cause one to take "Colonel Warden" less seriously. They vary in quality but are generally good. The "original research" complaint is silly since the concept is standard and the examples are the usual ones. Michael Hardy (talk) 15:19, 24 July 2008 (UTC)[reply]
  • Mathematics articles on Wikipedia are not generally good - there are only sixteen which pass muster. One likely consequence is that, as selective editions like Wikipedia 1.0 are published, explicitly selecting good material for inclusion, mathematics will be largely excluded owing to the poor quality of its articles. Colonel Warden (talk) 15:53, 24 July 2008 (UTC)[reply]
  • That is nonsense. There are lots and lots of good math articles. That only sixteen have been officially labeled "Good Articles" means nothing. How many have even been considered? And if they're not officially Good Articles, that doesn't mean they're not good articles. Here's a fact: a mathematician with decades of experience but little familiarity with the workings of Wikipedia recently recommended Wikipedia to my attention as a source of information on mathematics; he was apparently unaware that I'd done quite a lot of work on it. And other formidable professional mathematicians without much familiarity with the Wikipedia community or it practices have repeatedly said similar things to me. And Colonel Warden, I haven't seen you around Wikipedia's math article or their talk pages much, if at all, so it's not clear that you know much about this. I, on the other hand, do. Michael Hardy (talk) 17:19, 24 July 2008 (UTC)[reply]
  • PS Could I ask you to be specific about your complaints about Wikipedia's math articles? Maybe some specific examples of what you find not good? Michael Hardy (talk) 17:20, 24 July 2008 (UTC)[reply]
Is this an invitation to "step outside"? My money's on Michael Hardy. --C S (talk) 17:51, 24 July 2008 (UTC)[reply]
  • Keep. This is basically a disambiguation article, and an important one. The nominator seems to have trouble reading and also understanding deletion process. The article does not in fact define a notion of duality as s/he described. In fact, it starts out by saying "duality has numerous meanings". The second sentence is merely an attempt to give an overview statement of the notions of duality as they appear in mathematics. Calling the article completely "vacuous and artificial" reflects a lack of understanding. If the nominator wants to rewrite the beginning of it, there's no need to nominate this for deletion. --C S (talk) 14:06, 24 July 2008 (UTC)[reply]
  • Keep per John Z, C S, and Gandalf. CRGreathouse (t | c) 14:13, 24 July 2008 (UTC)[reply]
  • Keep I really don't see my vote is needed, but still. -- Taku (talk) 14:40, 24 July 2008 (UTC)[reply]
  • Keep I don't think the arguments for deletion are accurate. Chillum 14:41, 24 July 2008 (UTC)[reply]
  • Keep As a general rule anyone wishing to delete an article could first discuss the matter with the main contributors to the article, at the article talk page. This may spare us some unnecessary votes. Katzmik (talk) 14:57, 24 July 2008 (UTC)[reply]
  • Keep The article in its present form is mainly a list of examples, and they're pretty standard examples. The idea is a standard one, if not very precisely defined, and is noteworthy. If there are imperfections or deficiencies in the article, that's a reason to improve it, not to delete it. Michael Hardy (talk) 15:05, 24 July 2008 (UTC)[reply]
  • Keep The narrative is nebulous in many spots, and I think it never explains why duality is important, but the article is otherwise good. --Uncia (talk) 15:23, 24 July 2008 (UTC)[reply]
  • Keep. This is a WP:SNOW if ever there was one. If the article is crappy, it should be fixed. The stated reason for deletion, "I have never seen the word "dual" or the concept of duality itself defined or discussed in mathematical literature" is a phenomenal admission of both ignorance and indolence, as any trivial web or literature search would have turned up dozens of examples. (Search for "dual" on MathWorld.) I would hope that nominator recognizes that this nomination was a serious mistake. -- Dominus (talk) 15:35, 24 July 2008 (UTC)[reply]
    • If he means he's never seens a general definition of the concept in the mathematical literature, that may only mean he's like everyone else. I don't think it's a precisely defined concept (yet?). But that doesn't mean the various ways it's used are unrelated and disparate concepts. Michael Hardy (talk) 15:48, 24 July 2008 (UTC)[reply]
      • The article is not (as the nominating IP thought) attempting to define duality. Rather, this is something like a list, or a series of short summary style sections on different important examples of duality. User:C S above called it a disambiguation article, and he is correct in spirit. The text at the top is simply a brief introduction to the list, and could be rewritten if anyone had a serious argument (on the article talk page) that it was original research. Although I don't have the book at hand, it looks like the Oxford user's Guide to mathematics does try to give a general definition. [9] There is also a Britannica article [10], but it is substandard as usual. — Carl (CBM · talk) 16:02, 24 July 2008 (UTC)[reply]
        • I'm not too happy either with those links, but I found a somewhat better article on duality in the Springer Encyclopedia of Mathematics. [11] (The Britannica article seems to be in the spirit of the Springer EoM article on duality principle.) It's a hodge-podge of duality in different areas, but one section begins "Duality is a very pervasive and important concept in (modern) mathematics". That section's purpose seems to be to link to all the different duality articles on that encyclopedia. What a novel concept. Maybe we should try and delete the Springer article too (oh wait, we can't). --C S (talk) 17:41, 24 July 2008 (UTC)[reply]
  • Keep. Consider renaming this to List of duality concepts in mathematics (or similar). This article serves an encyclopedic purpose as a well-annotated list for navigating related concepts. siℓℓy rabbit (talk) 16:34, 24 July 2008 (UTC)[reply]
  • Keep per John Z, Eppstein, C S, etc. Richard Pinch (talk) 19:47, 24 July 2008 (UTC)[reply]
  • Keep Because the argument to delete has been been mercilessly repudiated. On a side note, I'm talented on running monte-carlo analysis of combinations of words and the differential meanings of each group. I give you... Duality (in the mathematics) which I think disambiguates everything mentioned by both sides of the delete discussion and is much clearer than the proposed alternatives yet most satisfactorily succinct. Afterall, I'm never taken a math class beyond Calc 1 since I was 18, so I can sympathize to Colonel's conclusion that there aren't resources which focus on the single issue of duality itself. Duality is a recurring theme which presents itself in different ways in many diverse branches. Sentriclecub (talk) 20:34, 24 July 2008 (UTC)[reply]
The above discussion is preserved as an archive of the debate. Please do not modify it. Subsequent comments should be made on the appropriate discussion page (such as the article's talk page or in a deletion review). No further edits should be made to this page.
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