Concept
Duality (mathematics)
Overview
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. In other cases the dual of the dual – the double dual or bidual – is not necessarily identical to the original (also called primal). Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry.
In mathematical contexts, duality has numerous meanings. It has been described as "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics".
Many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of the second type to some family of scalars. For instance, linear algebra duality corresponds in this way to bilinear maps from pairs of vector spaces to scalars, the duality between distributions and the associated test functions corresponds to the pairing in which one integrates a distribution against a test function, and Poincaré duality corresponds similarly to intersection number, viewed as a pairing between submanifolds of a given manifold.
From a category theory viewpoint, duality can also be seen as a functor, at least in the realm of vector spaces. This functor assigns to each space its dual space, and the pullback construction assigns to each arrow f: V → W its dual f∗: W∗ → V∗.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
In the original texts
Passages from public-domain books where Duality (mathematics) is discussed.
This duality suggests that any definition of points must be effected by means of the straight line, and any definition of the straight line must be effected by means of points. When we take the third dimension into account, it is true, the duality is no longer so simple; we have now to take account also of the plane, but this only introduces a circle of three terms, which is scarcely preferable to a circle of two…
Bertrand Russell, An essay on the foundations of geometry
states what it is; names the idea; opens on it
We have thus, as regards the opposition of positive and negative, three different sorts of duality, according as we are dealing with facts, image-propositions, or word-propositions. We have, namely: (1) Positive and negative facts; (2) Image-propositions, which may be believed or disbelieved, but do not allow any duality of content corresponding to positive and negative facts; (3) Word-propositions, which are always…
Bertrand Russell, The Analysis of Mind
names the idea; opens on it
The third and fourth images, where Yin constitutes the basis, are imperfect matter, which has again the two determinations of youth and age, strength and weakness. These lines are further united in sets of three, and thus eight figures result, which are called Kua, ☰, ☱, ☲, ☳, ☴, ☵, ☶, ☷.
Georg Wilhelm Friedrich Hegel, Hegel's Lectures on the History of Philosophy
states what it is; names the idea
Thus there arises a classification of modes of formation, or in the language commonly used by mathematicians, a classification of Functions. Any number, considered as formed from any other number, is called a function of it; and there are as many kinds of functions as there are modes of formation. The simple functions are by no means numerous, most functions being formed by the combination of several of the…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
states what it is
With such a fact before us we may safely infer that the first clear numerical conception was that of duality as contrasted with unity. And this notion of duality must necessarily have grown up side by side with those of likeness and equality; seeing that it is impossible to recognise the likeness of two things without also perceiving that there are two. From the very beginning the conception of number must have…
Herbert Spencer, Illustrations of Universal Progress
names the idea
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Place in the tradition
Part of Intuitionistic logic.