Talk:Duality (mathematics)

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[edit] 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) 05:26, 7 March 2008 (UTC)

[edit] Explanation of edit

So I put in this explanation -- "Usually duality is associated with some sort of general operation, where finding the "dual" of an object twice retrieves the original object (hence the "duality"). This does not preclude the possibilty that an object is its own dual, but there should be at least some objects which are distinct from their duals."

I'm not sure this is the best wording, but I think it conveys the "feeling" behind duality which transcends the categories in place.

Demorgan's dual: objects - binary operators, operation - swapping and and or and negating. Dual polyhedron: objects - polyhedra, operation: faces to vertices Dual spaces: objects - vector spaces, operation: taking linear functionals

[edit] older stuff

Further dualities include;

Charles Matthews 09:43, 21 Apr 2004 (UTC)

Actually I take the dual in dual numbers to imply double, rather than any duality.

Charles Matthews 07:03, 28 Apr 2004 (UTC)

What does the phrase "is dual to" mean? helohe 21:05, 7 Jun 2005 (UTC)

[edit] category theory

Shouldn't there be a reference to the notion of 'dual' from category theory? If someone with more expertise does not add it in the next days, I will give it a go. Ringan 22:50, 16 October 2005 (UTC)Erik Douglas

[edit] "Geometric" duality?

The planar graph example in the "geometric" section isn't particularly geometric: it's topological. I nuked the word 'geometric' from the link for that reason, but now it doesn't fit in its section. Bhudson 22:45, 8 December 2006 (UTC)

[edit] duality as involution

I have noticed that in several of the most important examples of mathematical 'duality', an involution operation is involved. In Lp spaces, for example, the Riesz representation theorem may be thought of as an involution: "take some function f \in L_{p} and form a functional by integrating it against g \in L_{q} where 1 / p + 1 / q = 1." When this operation is done twice it returns us to f. Planar graph duality has a similar property: exchanging vertices and faces twice returns us to the original graph, so duality is an involution on the class of planar graphs. De Morgan duality may be thought of in this way as well, etc. etc.

It seems to me that duality follows this pattern sufficiently often that it should be framed as such in the introduction. My recent edits reflect that. Hopefully you all will find it a valuable direction to take the article. 69.215.17.209 17:44, 22 April 2007 (UTC)

[edit] Duality not always an involution

It may be worth noting that the dual operator is not always an involution. For example, for finite dimensional vector spaces it holds true that (V * ) * = V. However, this is not true for infinite dimensional vector spaces(seeDual_space#The_infinite_dimensional_case). Alexander.fairley 08:30, 19 May 2007 (UTC)