Van Emde Boas observes " even if we base complexity theory on abstract instead of concrete machines, arbitrariness of the choice of a model remains.

The development of abstract algebra brought with itself group theory, rings and fields, Galois theory.

The most general setting in which these words have meaning is an abstract branch of mathematics called category theory.

Category theory deals with abstract objects and morphisms between those objects.

This is a very abstract definition since, in category theory, morphisms aren't necessarily functions and objects aren't necessarily sets.

On a more abstract level, model theoretic arguments hold that a given set of symbols in a theory can be mapped onto any number of sets of real-world objects — each set being a " model " of the theory — providing the interrelationships between the objects are the same.

This point of view is useful in abstract algebraic dimension theory due to Gabriel and Rentschler.

Some, such as computational complexity theory, which studies fundamental properties of computational problems, are highly abstract, while others, such as computer graphics, emphasize real-world applications.

Some fields, such as computational complexity theory ( which explores the fundamental properties of computational problems ), are highly abstract, whilst fields such as computer graphics emphasise real-world applications.

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 ( also called morphisms, although this term also has a specific, non category-theoretical meaning ), where these collections satisfy some basic conditions.

Homological algebra is category theory in its aspect of organising and suggesting manipulations in abstract algebra.

Topos theory is a form of abstract sheaf theory, with geometric origins, and leads to ideas such as pointless topology.

Category theory is also, in some sense, a continuation of the work of Emmy Noether ( one of Mac Lane's teachers ) in formalizing abstract processes ; Noether realized that in order to understand a type of mathematical structure, one needs to understand the processes preserving that structure.

Modern European conservatives such as Edmund Burke have found the extreme idealism of either democracy may endanger broader liberties, and similarly reject " abstract reason " as a guide for political theory.

In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is invariant under all automorphisms of the parent group.

In computational complexity theory, a problem refers to the abstract question to be solved.

The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra.

Dialectics as represented in dialectical materialism is a theory of the general and abstract factors that govern the development of nature, society, and thought.

Directed sets also give rise to direct limits in abstract algebra and ( more generally ) category theory.

His work laid the foundations for Galois theory and group theory, two major branches of abstract algebra, and the subfield of Galois connections.

In mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain ( also called a Euclidean ring ) is a ring that can be endowed with a certain structure – namely a Euclidean function, to be described in detail below – which allows a suitable generalization of the Euclidean division of the integers.

From the age of fifteen he was deeply interested in the theory of polytopes, which later became his main research interest.

In the general theory of polyhedra and polytopes, two conflicting meanings are currently jostling for acceptability:

The most long standing and popularised theory has been the attempts to link Assyrian ancestry to the ancient Germans.

Axiomatic set theory was developed in response to these early attempts to study set theory, with the goal of determining precisely what operations were allowed and when.

For example, in the 19th century, the Sun appeared to be no more than 20 million years old, but the Earth appeared to be no less than 300 million years ( resolved by the discovery of nuclear fusion and radioactivity, and the theory of quantum mechanics ); or current attempts to resolve theoretical differences between quantum mechanics and general relativity.

* Natural deduction, an approach to proof theory that attempts to provide a formal model of logical reasoning as it " naturally " occurs

Much research in graph theory was motivated by attempts to prove that all maps, like this one, could be graph coloring | colored with four color theorem | only four colors.

In graph theory, much research was motivated by attempts to prove the four color theorem, first stated in 1852, but not proved until 1976 ( by Kenneth Appel and Wolfgang Haken, using substantial computer assistance ).

Applied ethics is a discipline of philosophy that attempts to apply ethical theory to real-life situations.

The endosymbiosis theory attempts to explain the origins of organelles such as mitochondria and chloroplasts in eukaryotic cells.

Initial attempts in the United States in the early 1970s were generally based on sociological theory and focused on the function of women characters in particular film narratives or genres and of stereotypes as a reflection of a society's view of women.

He attempts to reconcile the two perspectives numerically, calculating the effect of the stretching of space-time, based on Einstein's theory of general relativity.

Interest theory argues that the principal function of human rights is to protect and promote certain essential human interests, while will theory attempts to establish the validity of human rights based on the unique human capacity for freedom.

To oversimplify its concepts somewhat, natural law theory attempts to identify a moral compass to guide the lawmaking power of the state and to promote ' the good '.

On the basis of this theory of distributive justice, Nozick argues that all attempts to redistribute goods according to an ideal pattern, without the consent of their owners, are theft.

Social interactionist theory is a claim that language development occurs in the context of social interaction between the developing child and knowledgeable adults who model language usage and " scaffold " the child's attempts to master language.

The theory attempts to explain how what we call intelligence could be a product of the interaction of non-intelligent parts.

Moore's arguments may rule out attempts to define ' goodness ' in such obviously naturalistic terms as ' happiness ' while at the same time do not preclude similar attempts to define the good in terms of God's will — in other words, divine command theory.

The last part of the theory attempts to explain how we experience spatiotemporal consistency in our interaction with environmental stimuli.

The predictions by Heaviside, combined with Planck's radiation theory, probably discouraged further attempts to detect radio waves from the Sun and other astronomical objects.

Many attempts have been made to derive its value ( currently measured at about 1 / 137. 035999 ) from theory, but so far none have succeeded.

A third theory of perception, enactivism, attempts to find a middle path between realist and anti-realist theories, positing that cognition arises as a result of the dynamic interplay between an organism's sensory-motor capabilities and its environment.

The mathematical theory of probability has its roots in attempts to analyze games of chance by Gerolamo Cardano in the sixteenth century, and by Pierre de Fermat and Blaise Pascal in the seventeenth century ( for example the " problem of points ").