for this division is but an abstract representation of the social struggle between mysticism and science.

But he would fight for his own liberty rather than for any abstract principle connected with it -- such as `` cause ''.

The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny quantities has been around for millennia, but these ideas were founded in abstract, philosophical reasoning rather than experimentation and empirical observation.

In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F, the ring of polynomials in the variable x with coefficients in F.

In the context of abstract algebra, for example, a mathematical object is an algebraic structure such as a group, ring, or vector space.

Alaric displayed similar wisdom in political affairs by appointing a commission to prepare an abstract of the Roman laws and imperial decrees, which would form the authoritative code for his Roman subjects.

Pobedonostsev awakened in his pupil little love of abstract study or prolonged intellectual exertion, but instilled into the young man's mind the belief that zeal for Russian Orthodox thought was an essential factor of Russian patriotism to be cultivated by every right-minded emperor.

If we take in our hand any volume ; of divinity or school metaphysics, for instance ; let us ask, Does it contain any abstract reasoning concerning quantity or number?

He is also noted for his mastery of abstract approaches to mathematics and his perfectionism in matters of formulation and presentation.

In computer science, an abstract data type ( ADT ) is a mathematical model for a certain class of data structures that have similar behavior ; or for certain data types of one or more programming languages that have similar semantics.

The notion of abstract data types is related to the concept of data abstraction, important in object-oriented programming and design by contract methodologies for software development.

xiii, " Cum ad Sacrosanctae Romanae Ecclesiae ") prescribes their work, determines how much they may charge for their labour, fixes a certain tax for an abstract or abridgment of twenty-five words, or their equivalent, 150 letters, forbids them to charge more, even though the abstract goes over twenty-five words but less than fifty words, enacts that the basis of the tax is the labour employed in writing, expediting, etc., the Bulls, and by no means the emoluments accruing to the recipient of the favour or benefice conferred by the Bull, and declares that whoever shall charge more than the tax fixed by him shall be suspended for six months from office, and upon a second violation of the law, shall be deprived of it altogether, and if the delinquent be an abbreviator, he shall be excommunicated.

The oldest known physical representations identified as symbols for abstract concepts are abstract engravings found on two pieces of ochre in Blombos Cave, South Africa, in 2001.

This is true for all verbal / abstract communication.

An abstraction can be seen as a process of mapping multiple different pieces of constituent data to a single piece of abstract data based on similarities in the constituent data, for example many different physical cats map to the abstraction " CAT ".

A recent meta-analysis suggests that the verbal system has greater engagement for abstract concepts when the perceptual system is more engaged for processing of concrete concepts.

Part of this is a very abstract technical problem, faced by researchers in distributed artificial intelligence, in getting, say, 20 different surveillance robotic vehicles to talk to each other and compare notes-which is in itself a hard problem.

Thus we can talk of " a job " as an abstract function that could be filled by anybody with the required skills.

After a year in a studio on Sheridan Square, having married an American girl who was a native of Virginia, Helion moved to a village in the Blue Ridge mountains, where he produced some of the most imposing of his abstract canvases.

What I have observed time and time again is a process of integration, integration that begins as abstract design and gradually takes on recognizable form ; ;

Cubism, in its 1911-1912 phase ( which the French, with justice, call `` hermetic '' ) was on the verge of abstract art.

but he believed in the Jew-as-enemy in a kind of abstract, theological way, like a member of a cult speculating on the nature of things.

An abstract describes a chain of transfers from owner to owner and any agreements by former owners that are binding on later owners.

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 truth-values of our mathematical assertions depend on facts involving platonic entities that reside in a realm outside of space-time ") Whilst our knowledge of concrete, physical objects is based on our ability to perceive them, and therefore to causally interact with them, there is no parallel account of how mathematicians come to have knowledge of abstract objects.

An abstract data type is defined indirectly, only by the operations that may be performed on it and by mathematical constraints on the effects ( and possibly cost ) of those operations.

For example, an abstract stack data structure could be defined by three operations:, that inserts some data item onto the structure,, that extracts an item from it ( with the constraint that each pop always returns the most recently pushed item that has not been popped yet ), and, that allows data on top of the structure to be examined without removal.

An abstract data type is defined as a mathematical model of the data objects that make up a data type as well as the functions that operate on these objects.

Imperative ADT definitions often depend on the concept of an abstract variable, which may be regarded as the simplest non-trivial ADT.

Although the description sitting-on ( graph 1 ) is more abstract than the graphic image of a cat sitting on a mat ( picture 1 ), the delineation of abstract things from concrete things is somewhat ambiguous ; this ambiguity or vagueness is characteristic of abstraction.

Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.

I put abstract feelings on the same level as abstract thoughts.

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order ( the axiom of commutativity ).

The album features his interaction with students at Fresno State College ( now California State University, Fresno ) on such topics as " sensitivity training ," " humanitarianism ," " abstract art " ( Capp hated it ), and of course, " student protest.

A more abstract definition, which is equivalent but more easily generalized to infinite-dimensional spaces, is to say that bras are linear functionals on kets, i. e. operators that input a ket and output a complex number.

The differential operator must be understood to be an abstract operator, acting on kets, that has the effect of differentiating wavefunctions once the expression is projected into the position basis:

Certain government bureaucracies, such as a department of motor vehicles will issue an abstract of a completed transaction or an updated record intended to serve as a proof of compliance with some administrative requirement.

Intuitionistic logic substitutes constructability for abstract truth and is associated with a transition from the proof to model theory of abstract truth in modern mathematics.

Mathematical theorems, on the other hand, are purely abstract formal statements: the proof of a theorem cannot involve experiments or other empirical evidence in the same way such evidence is used to support scientific theories.

Dehn's proof is an instance in which abstract algebra is used to prove an impossibility result in geometry.

In computational complexity theory, an interactive proof system is an abstract machine that models computation as the exchange of messages between two parties.

They vary in the amount of abstract algebraic notions required to understand the proof.

Meanwhile, in order to make the proof of the Riemann hypothesis for curves over finite fields that he had announced in 1940 work, he had to introduce the notion of an abstract variety and to rewrite the foundations of algebraic geometry to work with varieties without projective embeddings ( see also the history section in the Algebraic Geometry article ).

The verification of these systems is done by providing a formal proof on an abstract mathematical model of the system, the correspondence between the mathematical model and the nature of the system being otherwise known by construction.

One of his first contributions was the proof of the Sylow theorems for abstract groups.

Weil's interest was in putting an abstract variety theory in place, to support the use of the Jacobian variety in his proof of the Riemann hypothesis for curves over finite fields, a direction rather oblique to Zariski's interests.

A book focusing on elliptic curves, beginning at an undergraduate level ( at least for those who have had a course on abstract algebra ), and progressing into much more advanced topics, even at the end touching on Andrew Wiles ' proof of the Taniyama – Shimura conjecture which led to the proof of Fermat's last theorem.

The occasion of the proof by Hassler Whitney of the embedding theorem for smooth manifolds is said ( rather surprisingly ) to have been the first complete exposition of the manifold concept precisely because it brought together and unified the differing concepts of manifolds at the time: no longer was there any confusion as to whether abstract manifolds, intrinsically defined via charts, were any more or less general than manifold extrinsically defined as submanifolds of Euclidean space.

* The abstract concept of an Interactive proof system

All aspects of law are fair game on APDA, including constitutional law ( e. g. whether a Supreme Court case was wrongly decided ), procedural law ( e. g. whether standards of proof should differ for criminal and civil law ) and abstract legal theory ( e. g. whether retributive justice is a moral justification for the criminal justice system ).

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.

Bolzano's philosophical work encouraged a more abstract reading of when a demonstration could be regarded as analytic, where a proof is analytic if it does not go beyond its subject matter ( Sebastik 2007 ).

The proof just given indicates the scope of the identity in abstract algebra: it will hold in any commutative ring R.

This connection between complexity and the logic of finite structures allows results to be transferred easily from one area to the other, facilitating new proof methods and providing additional evidence that the main complexity classes are somehow " natural " and not tied to the specific abstract machines used to define them.

The easier classes often include basic proof techniques, number theory, graph theory, and combinatorial game theory, while the more difficult classes cover advanced topics in abstract algebra, topology, theoretical computer science, category theory, and mathematical analysis.

A cleaner version of this proof can be given using the statement from abstract algebra that a polynomial ring over an integral domain is again an integral domain.