# analysis

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## analysis,

branch of mathematics**mathematics,**

deductive study of numbers, geometry, and various abstract constructs, or structures; the latter often "abstract" the features common to several models derived from the empirical, or applied, sciences, although many emerge from purely mathematical or logical

**.....**Click the link for more information. that utilizes the concepts and methods of the calculus

**calculus,**

branch of mathematics that studies continuously changing quantities. The calculus is characterized by the use of infinite processes, involving passage to a limit—the notion of tending toward, or approaching, an ultimate value.

**.....**Click the link for more information. . It includes not only basic calculus, but also advanced calculus, in which such underlying concepts as that of a limit

**limit,**

in mathematics, value approached by a sequence or a function as the index or independent variable approaches some value, possibly infinity. For example, the terms of the sequence 1-2, 1-4, 1-8, 1-16, … are obviously getting smaller and smaller; since, if enough

**.....**Click the link for more information. are subjected to rigorous examination; differential and integral equations, in which the unknowns are functions

**function,**

in mathematics, a relation

*f*that assigns to each member

*x*of some set

*X*a corresponding member

*y*of some set

*Y*;

*y*is said to be a function of

*x,*usually denoted

*f*(

*x*) (read "

*f*of

*x*").

**.....**Click the link for more information. rather than numbers, as in algebraic equations; complex variable analysis, in which the variables are of the form

*z*=

*x*+

*iy,*where

*i*is the imaginary unit; vector

**vector,**

quantity having both magnitude and direction; it may be represented by a directed line segment. Many physical quantities are vectors, e.g., force, velocity, and momentum.

**.....**Click the link for more information. analysis and tensor

**tensor,**

in mathematics, quantity that depends linearly on several vector variables and that varies covariantly with respect to some variables and contravariantly with respect to others when the coordinate axes are rotated (see Cartesian coordinates).

**.....**Click the link for more information. analysis; differential geometry

**differential geometry,**

branch of geometry in which the concepts of the calculus are applied to curves, surfaces, and other geometric entities. The approach in classical differential geometry involves the use of coordinate geometry (see analytic geometry; Cartesian coordinates),

**.....**Click the link for more information. ; and many other fields.

## Analysis

the process of mental and often actual breaking down of an object, phenomenon, or process, its properties, or the relations between objects into parts—their features, properties, and relations. The reverse of analysis is synthesis; often the two are combined in thought and action.

Analytical methods are so widespread in science that the term “analysis” often serves as a synonym for research in general, both in the natural and social sciences—for example, quantitative and qualitative analysis in chemistry, diagnostic analysis in medicine, the decomposition of complex movements into their components in mechanics, functional analysis in sociology, and so forth. The procedures of analysis appear as an organic component in every scientific investigation, usually as its first stage, when the investigator passes from a description of the object under examination as a whole to the exposure of its structure, composition, qualities, and features. Analysis retains its significance in other stages of knowledge as well, although then it acts in conjunction with other processes of investigation. Analytical processes are most important not only in scientific thought but in every activity, since it is related to the solution of cognitive problems. As a cognitive process, analysis is studied by psychology, which views it as a psychological process which goes on at various levels of the reflections of reality in the human and animal brain. The theory of knowledge and the methodology of science study analysis primarily as one method for obtaining new cognitive results.

Analysis is already present also at the sensory level of knowledge and, in particular, is included in the processes of sensation and perception; in its simpler forms it is found in animals. However, analysis and synthesis are directly incorporated into external actions of even the higher species. In man, the sensory and external forms of analysis are superimposed by a higher form of analysis—mental, or abstract logical analysis. This form arose together with the skills in the practical taking apart of material objects in the work process; as the work grew more complex, man acquired the ability to anticipate by mental analysis the physical analysis the work required. The development of productive activity, of thought and language, and of the methods of scientific investigation and proof led to the appearance of various forms of cognitive analysis, in particular to the breaking down of objects into the features, characteristics, and relations that are inseparable from them. As distinguished from sensory and physical analysis, cognitive analysis takes place with the aid of notions and statements, expressed in natural languages and in artificial ones—that is, in the symbolic systems of science. On the other hand, analysis itself, working with other methods, serves as a means for forming concepts about reality.

Several forms of analysis can be used in scientific thought. One is the mental breaking down (often, for example, in an experiment, the real breaking down as well) of the whole into its parts. Such analysis, revealing the structure of the whole, presupposes not only that the parts of which the whole consists will be determined but that the relations between them will be determined as well. This is particularly important when the object to be analyzed is considered representative of some class of objects: here analysis serves to establish an identity of structure for objects in that class based on some relationships, so that knowledge obtained in the study of one object could be transferred to others.

Another form is analysis of the common properties of objects and the relations between objects when that property or relation is broken down into its components; some of them are left out and the others submitted to further analysis; in the next stage those previously left out can be analyzed, and so on. As a result of this type of analysis, common properties and relations can be reduced to more generalized and simplified concepts.

Another type of analysis is the division of classes of objects (sets) into subclasses (nonintersecting subsets) of the given set. Such a form of analysis is called classification. All these and other types of analysis are used both in obtaining new knowledge and in the systematization of already obtained scientific results. Analysis is also widely used in pedagogy.

A more specialized understanding of analysis, related to the type described above, is formal and logical analysis. It is a more precise definition of the logical structure of reasoning, brought about by means of contemporary formal logic. Such a definition can apply to both judgments (logical conclusions, proofs, deductions, etc.) and their component parts (concepts, terminology, and statements) as well as to individual fields of knowledge. The construction of formal systems is the most developed form of logical analysis of concrete fields, concepts, and means of reasoning; these systems are interpreted within these fields or with the help of the given concrete concepts—the so-called formalized languages. Logical analysis is one of the basic cognitive methods of science which has especially grown in importance as a result of the development of mathematical logic, cybernetics, semiotics, and information and logic systems.

Analysis is used in another sense in mathematics: here analysis is reasoning that proceeds from what is to be proved (the unidentified, the unknown) to what has already been proved (the previously identified, known), and synthesis is understood as the reverse of this reasoning process. Analysis, in this sense, is a means for revealing the general idea of the proof but in most instances it is not the proof itself. Synthesis, however, relying on data found in the analysis, demonstrates how the problem subject to the proof follows from previously established assertions and gives the proof of the theorem or the solution of the problem.

### REFERENCES

Mamardashvili, M. K. “Protsessy analiza i sinteza.”*Voprosy filosofii*, 1958, no. 2.

*Problemy myshleniia v sovremennoi nauke*. Moscow, 1964.

Gorskii, D. P.

*Problemy obshchei metodologii nauk i dialekticheskoi logiki*. Moscow, 1966.

Petrov, Iu. A. “Gnoseologicheskaia rol’formalizovannykh iazykov.” In

*lazyk i myshlenie*. Moscow, 1967.

B. V. BIRIUKOV

## analysis

[ə′nal·ə·səs]## analysis

**1.**short for psychoanalysis

**2.**

*Chem*

**a.**the decomposition of a substance into its elements, radicals, or other constituents in order to determine the kinds of constituents present (

**qualitative analysis**) or the amount of each constituent (

**quantitative analysis**)

**b.**the result obtained by such a determination

**3.**

*Maths*the branch of mathematics principally concerned with the properties of functions, largely arising out of calculus

**4.**

*Philosophy*(in the writings of Kant) the separation of a concept from another that contains it