Editing Analogy (section)

====Mathematics====
Some types of analogies can have a precise [[mathematical]] formulation through the concept of [[isomorphism]]. In detail, this means that if two mathematical structures are of the same type, an analogy between them can be thought of as a [[bijection]] which preserves some or all of the relevant structure. For example, <math> \mathbb{R}^2 </math> and <math> \mathbb{C} </math> are isomorphic as vector spaces, but the [[complex numbers]], <math> \mathbb{C} </math>, have more structure than <math> \mathbb{R}^2 </math> does: <math> \mathbb{C} </math> is a [[Field (mathematics)|field]] as well as a [[vector space]].

[[Category theory]] takes the idea of mathematical analogy much further with the concept of [[functor]]s. Given two categories C and D, a functor ''f'' from C to D can be thought of as an analogy between C and D, because ''f'' has to map objects of C to objects of D and arrows of C to arrows of D in such a way that the structure of their respective parts is preserved. This is similar to the [[#Shared structure|structure mapping theory of analogy]] of Dedre Gentner, because it formalises the idea of analogy as a function which makes certain conditions true.