Abstract nonsense
Encyclopedia
In mathematics
, abstract nonsense, general abstract nonsense, and general nonsense are terms used facetiously by some mathematician
s to describe certain kinds of arguments and methods related to category theory
. (Very) roughly speaking, category theory is the study of the general form of mathematical theories, without regard to their content. As a result, a proof
that relies on category theoretic ideas often seems slightly out of context to those who are not used to such abstraction, sometimes to the extent that it resembles a comical non sequitur. Such proofs are sometimes dubbed “abstract nonsense” as a light-hearted way of alerting people to their abstract nature.
More generally, “abstract nonsense” may refer to any proof (humorous or not) that uses primarily category theoretic methods, or even to the study of category theory itself. Note that referring to an argument as "abstract nonsense" is not supposed to be a derogatory expression, and is actually often a compliment regarding the sophistication of the argument.
that introduced the notion of a "category
" in 1942, Saunders Mac Lane
wrote the subject was 'then called "general abstract nonsense"'. The term is often used to describe the application of category theory and its techniques to less abstract domains.
The term is believed to have been coined by the mathematician Norman Steenrod
, himself one of the developers of the categorical point of view. This term is used by practitioners as an indication of mathematical sophistication (or possession of a deeper perspective) rather than as a derogatory designation.
Certain ideas and constructions in mathematics display a uniformity throughout many domains. The unifying theme is category theory. When their audience can be assumed to be familiar with the general form of such arguments, mathematicians will use the expression Such and such is true by abstract nonsense rather than provide an elaborate explanation of particulars.
, definition of natural transformation
s between functor
s, use of the Yoneda lemma
, arguments exploiting classifying space
s, and so on.
To spell out a concrete example, consider a 3-manifold
M with positive Betti number
. One would like to show that M admits a map
to the 2-sphere which is "non-trivial", i.e. non-homotopic to the constant map. By a general nonsense argument, there is a map
to the Eilenberg-MacLane space
, corresponding to a non-trivial element in H2(M). Since K(Z,2) is a complex projective space and the latter admits a skeleton structure with no cells in odd dimensions, we can apply the cellular approximation theorem
to conclude that the map f can be pushed down to the 2-skeleton, which happens to be the 2-sphere.
Though this proof establishes the truth of the statement in question, the proof technique has little to do with the topology
or geometry
of the 2-sphere, let alone 3-manifolds. The result is that the proof offers little geometric insight into the nature of such a map. On the other hand, the proof is surprisingly short and clean, and a “hands-on” approach involving the physical construction of such a map would be potentially laborious. A reader expecting a long, difficult proof might be surprised—or even delighted—by this bit of general nonsense.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, abstract nonsense, general abstract nonsense, and general nonsense are terms used facetiously by some mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
s to describe certain kinds of arguments and methods related to category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
. (Very) roughly speaking, category theory is the study of the general form of mathematical theories, without regard to their content. As a result, a proof
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
that relies on category theoretic ideas often seems slightly out of context to those who are not used to such abstraction, sometimes to the extent that it resembles a comical non sequitur. Such proofs are sometimes dubbed “abstract nonsense” as a light-hearted way of alerting people to their abstract nature.
More generally, “abstract nonsense” may refer to any proof (humorous or not) that uses primarily category theoretic methods, or even to the study of category theory itself. Note that referring to an argument as "abstract nonsense" is not supposed to be a derogatory expression, and is actually often a compliment regarding the sophistication of the argument.
History
The term predates the foundation of category theory as a subject itself. Referring to a joint paper with Samuel EilenbergSamuel Eilenberg
Samuel Eilenberg was a Polish and American mathematician of Jewish descent. He was born in Warsaw, Russian Empire and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.He earned his Ph.D. from University of Warsaw in 1936. His thesis advisor...
that introduced the notion of a "category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
" in 1942, Saunders Mac Lane
Saunders Mac Lane
Saunders Mac Lane was an American mathematician who cofounded category theory with Samuel Eilenberg.-Career:...
wrote the subject was 'then called "general abstract nonsense"'. The term is often used to describe the application of category theory and its techniques to less abstract domains.
The term is believed to have been coined by the mathematician Norman Steenrod
Norman Steenrod
Norman Earl Steenrod was a preeminent mathematician most widely known for his contributions to the field of algebraic topology.-Life:...
, himself one of the developers of the categorical point of view. This term is used by practitioners as an indication of mathematical sophistication (or possession of a deeper perspective) rather than as a derogatory designation.
Certain ideas and constructions in mathematics display a uniformity throughout many domains. The unifying theme is category theory. When their audience can be assumed to be familiar with the general form of such arguments, mathematicians will use the expression Such and such is true by abstract nonsense rather than provide an elaborate explanation of particulars.
Examples
Typical instances are arguments involving diagram chasing, application of the definition of universal propertyUniversal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment...
, definition of natural transformation
Natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...
s between functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
s, use of the Yoneda lemma
Yoneda lemma
In mathematics, specifically in category theory, the Yoneda lemma is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory...
, arguments exploiting classifying space
Classifying space
In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG by a free action of G...
s, and so on.
To spell out a concrete example, consider a 3-manifold
3-manifold
In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...
M with positive Betti number
Betti number
In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....
. One would like to show that M admits a map
Map (mathematics)
In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...
to the 2-sphere which is "non-trivial", i.e. non-homotopic to the constant map. By a general nonsense argument, there is a map
to the Eilenberg-MacLane space
Eilenberg-MacLane space
In mathematics, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" , and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. In mathematics, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without...
, corresponding to a non-trivial element in H2(M). Since K(Z,2) is a complex projective space and the latter admits a skeleton structure with no cells in odd dimensions, we can apply the cellular approximation theorem
Cellular approximation
In algebraic topology, in the cellular approximation theorem, a map between CW-complexes can always be taken to be of a specific type. Concretely, if X and Y are CW-complexes, and f : X → Y is a continuous map, then f is said to be cellular, if f takes the n-skeleton of X to the n-skeleton of Y for...
to conclude that the map f can be pushed down to the 2-skeleton, which happens to be the 2-sphere.
Though this proof establishes the truth of the statement in question, the proof technique has little to do with the topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
or geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
of the 2-sphere, let alone 3-manifolds. The result is that the proof offers little geometric insight into the nature of such a map. On the other hand, the proof is surprisingly short and clean, and a “hands-on” approach involving the physical construction of such a map would be potentially laborious. A reader expecting a long, difficult proof might be surprised—or even delighted—by this bit of general nonsense.