Editing Pure mathematics (section)

==Generality and abstraction==
[[File:Banach-Tarski Paradox.svg|thumbnail|right|350px|An illustration of the [[Banach–Tarski paradox]], a famous result in pure mathematics. Although it is proven that it is possible to convert one sphere into two using nothing but cuts and rotations, the transformation involves objects that cannot exist in the physical world.]]
One central concept in pure mathematics is the idea of generality; pure mathematics often exhibits a trend towards increased generality. Uses and advantages of generality include the following:

* Generalizing theorems or mathematical structures can lead to deeper understanding of the original theorems or structures
* Generality can simplify the presentation of material, resulting in shorter proofs or arguments that are easier to follow.
* One can use generality to avoid duplication of effort, proving a general result instead of having to prove separate cases independently, or using results from other areas of mathematics.
* Generality can facilitate connections between different branches of mathematics. [[Category theory]] is one area of mathematics dedicated to exploring this commonality of structure as it plays out in some areas of math.

Generality's impact on [[intuition (knowledge)|intuition]] is both dependent on the subject and a matter of personal preference or learning style.  Often generality is seen as a hindrance to intuition, although it can certainly function as an aid to it, especially when it provides analogies to material for which one already has good intuition.

As a prime example of generality, the [[Erlangen program]] involved an expansion of [[geometry]] to accommodate [[non-Euclidean geometries]] as well as the field of [[topology]], and other forms of geometry, by viewing geometry as the study of a space together with a [[Group (mathematics)|group]] of transformations.  The study of [[number]]s, called [[algebra]] at the beginning undergraduate level, extends to [[abstract algebra]] at a more advanced level; and the study of [[function (mathematics)|function]]s, called [[calculus]] at the college freshman level becomes [[mathematical analysis]] and [[functional analysis]] at a more advanced level.  Each of these branches of more ''abstract'' mathematics have many sub-specialties, and there are in fact many connections between pure mathematics and applied mathematics disciplines. A steep rise in [[abstraction]] was seen mid 20th century.

In practice, however, these developments led to a sharp divergence from [[physics]], particularly from 1950 to 1983. Later this was criticised, for example by [[Vladimir Arnold]], as too much [[David Hilbert|Hilbert]], not enough [[Henri Poincaré|Poincaré]]. The point does not yet seem to be settled: [[string theory]] pulls one way towards abstraction, while [[discrete mathematics]] pulls back towards proof as central.