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a. of or relating to a highly desirable and possible state of affairs
b. of or relating to idealism



(1) Something existing not in reality but only in consciousness; the mode of existence of an object reflected in the consciousness. In this sense the ideal is usually contrasted to the real.

(2) The result of the process of idealization—an abstract entity that cannot be experienced, for example, an ideal gas, a point, or an absolutely black body.

(3) Something perfect that corresponds to an ideal.

The diverse conceptions of the ideal tend toward one of two opposite poles: materialism or idealism. Idealism treats the ideal as a self-sufficient principle, existing apart from the material world, that cannot be deduced from matter or explained by way of it., Different systems based on an idealist world view interpret the ideal in various ways: as primordial nonmaterial essences, or “ideas,” the archetypes of all things (as in objective idealism of the Platonic type); as the activity of an absolute spirit or universal reason (as in the objective idealism of the Hegelian type); as a special substance existing alongside material substance (as in the dualism of the Cartesian type); as the immediate data of individual consciousness, represented as something primary, from which all else arises (as in subjective idealism); or as a special world of values and meanings (as in contemporary critical realism and phenomenology).

Pre-Marxist materialism, which criticized the notion of the ideal as a special substance opposed to the material world, regarded the ideal as a function of matter organized in a special way. The weak point of pre-Marxist materialism’s treatment of the ideal was that it regarded the ideal as a product of passive contemplation and not as the result of and a means for human activity.

In Marx’ definition, “the ideal is nothing else than the material world reflected by the human mind, and translated into forms of thought” (K. Marx and F. Engels, Soch., 2nd ed., vol. 23, p. 21). Dialectical materialism proceeds from an understanding of the ideal as phenomena that are sociohistorical in their nature and origin, and it regards the ideal above all as reflections of the material world of objects in the consciousness of human beings, as subjective images of objective reality mediated through social praxis. The ideal, then, represents a definite aspect of human consciousness, which characterizes the specific mode of existence of consciousness that cannot be reduced to material processes and phenomena, such as physical and physiological ones.



(mathematics) one of the primary algebraic concepts. Having arisen originally in connection with the study of algebraic irrational numbers, ideals subsequently found numerous applications in other branches of mathematics.

Any (rational) integer may be factored into the product of prime factors; for example, 60 = 2 x 2 x 3 x 5. In this case the factorization is unique to within one order and to the sign of the factors:

60 = 2 X 5 X 3 X 2 = (-2) X 2 X (-3) X 5 …

In the 19th century, mathematicians encountered the necessity of factoring numbers of a more general nature. If, for example, we examine numbers of the type Ideal (mathematics), where m and n are any (rational) integers, then just as for ordinary integers each number here can always be factored into the product of nonfactorable factors. However, in this case the uniqueness of factorization is violated. For example, the number 9 (which results if we assume that m = 9, n = 0) here allows two different factorizations:

9 = 3 X 3 and Ideal (mathematics)

and none of the factors Ideal (mathematics) can be further factored into a product of numbers of the type m + nIdeal (mathematics) There will be no violation of the ordinary laws of the uniqueness of factorization if the property of divisibility is linked not to the numbers but to the ideals. In modern algebra ideals are introduced in arbitrary rings. In the case of number rings (for example, the set of numbers of the type Ideal (mathematics) examined above), ideals are also called ideal numbers. An ideal is the set of numbers belonging to a given number ring (and in the case of an arbitrary ring, the set of its elements) which possesses the following properties: (1) the sum and difference of two numbers (elements) of the set belong to this set and (2) the product of a number (element) from this set and any other number (any other element) of the ring also belongs to this set. Then we examine, instead of the numbers, the ideals that correspond to them; for example, the ideal p = (9), which consists of all numbers divisible by 9, corresponds to the number 9.

The numerical concepts connected with the divisibility of numbers are transferred to ideals: one ideal may be divided by another if any element of the former also lies in the latter (for numbers this is equivalent to saying that any number of the first ideal may be divided by at least one number of the second). The product of ideals is defined as the least ideal containing all possible paired products of elements from both ideal-factors. The greatest common divisor of two ideals is the least ideal that contains elements of both the first and the second ideal. In the set of integers, any ideal consists of multiples of some fixed number: any ideal is a principal ideal. In the general case, for algebraic irrational numbers, not any ideal is a principal ideal. Divisibility by a principal ideal is equivalent to divisibility by the number corresponding to this ideal. Because of the existence of ideals that are not principal ideals, the theorem that any ideal may be factored uniquely into the product of subsequently non- factorable ideals remains valid for algebraic integers. These non-factorable ideals, also called prime ideals, perform the role of prime numbers and are characterized by the fact that without fail they contain at least one of the factors if they contain their product. Thus, in the example considered above, (3) = P1P2Ideal (mathematics) where P1 = (3, Ideal (mathematics), are nre ideals; for example, the ideal p1, which is the largest common divisor of the ideals Ideal (mathematics), consists of all numbers of the type 3kIdeal (mathematics), where k and l are any rational integers.

The concept of “ideal” (or of “ideal number” in the original terminology) was introduced in 1847 for one particular case of number fields by the German mathematician E. Rummer. Strict and complete substantiation of the theory of ideals for any number fields was given independently by the German mathematician R. Dedekind in 1871 and the Russian mathematician E. I. Zolotarev in 1877. The theory of ideals gained new content in the middle of the 20th century in connection with the development of the general theory of rings.


Van der Waerden, B. L. Sovremennaia algebra, 2nd ed., parts 1–2. Moscow-Leningrad, 1947. (Translated from German.)



an ideal image that determines the manner of thinking and acting of an individual or a social class. The formation of nature in conformity with the ideal is a characteristically human form of activity, since it presupposes the specific creation of an image of a goal of activity before the goal is realized.

The problem of the ideal was developed in detail by German classical philosophy. It was posed most acutely by I. Kant in connection with the problem of “inner purpose.” According to Kant, phenomena without goals that could be represented in terms of images could not have ideal forms either. Man, as a representative of the race, is the only being acting according to “inner purpose.” For animals, inner purpose is realized unconsciously and therefore does not acquire the form of an ideal, of a specific kind of a goal. According to Kant, the ideal as imagined perfection of the human race (that is, perfection attained through imagination) is characterized by the complete and absolute surmounting of all contradictions between the individual and society, that is, between the individuals constituting the “race.” Therefore, realization of the ideal would coincide with the end of history. Because of this, the ideal, according to Kant, is in principle unattainable and represents only an “idea” of a regulative type. It points out a direction toward the goal rather than providing an image of the goal itself, and therefore it guides man more as a sense of proper direction than as a clear image of the end result. It is only in art that the ideal can and must be represented as an image in the form of the beautiful. The ideal of science, that is, of “pure reason,” is given in the form of the law of contradiction; and the moral ideal, that is, the ideal of “practical reason,” in the form of the categorical imperative. Nowhere can a condition corresponding to the ideal be graphically represented; it is impossible, since such a condition is not realizable in the course of finite time, no matter how long. Therefore, the ideal and the “beautiful” are synonymous, and the existence of the ideal is assumed only in art. These ideas of Kant were developed in the works of F. von Schiller, J. G. Fichte, and F. W. von Schelling, as well as by the German romanticists.

G. Hegel, who acutely understood the impotence of the Kantian conception of the ideal, dethroned it as an abstraction that actually expressed one of the stages of the developing reality of the “spirit” (that is, of the history of the intellectual culture of humanity) and that was opposed to another such abstraction, that of “empirical reality.” The latter was allegedly antagonistic in principle to the ideal and incompatible with it, according to Kant. For Hegel the ideal becomes an aspect of reality, an image of the human mind that eternally develops through its own immanent contradictions and that surmounts its own results and its own “alienated” conditions, rather than a primordially external and hostile “empirical reality.” The ideal of science (scientific thinking) therefore can and must be set as a logical system, and the ideal of practical reason must be set as an image of a rationally organized state, rather than as formal and in principle unrealizable abstract imperative demands addressed to the individual. Therefore, the ideal as such is always concrete, and it is gradually realized in history. Any attained level of development appears from this point of view as a partially realized ideal, as a phase of the subordination of empirical reality to the power of thought, the force of the idea, and the creative strength of the concept, that is, the collective reason of all people united by an idea. The image of the concrete goal of the acitivity of the “species,” that is, of humanity at a given stage of its intellectual and moral development, is always drawn up in the form of the ideal. The most acute and urgent universal contradictions are presented as actually resolved within the framework of the ideal. The “spirit” always implements existing tasks rather than the abstract-formal goal of “absolute perfection,” seen as a condition that is stagnant and devoid of life (and therefore of contradictions as well).

Insofar as the ideal is defined by Hegel in the vein of German classical philosophy as a visually contemplated image of a goal, further elaboration of the problem of the ideal shifts for Hegel to aesthetics, to a system of definitions of “the beautiful.” However, realization of the ideal as “the beautiful” is related by Hegel to the past—to the age of the classically antique “kingdom of beautiful individuality.” This is because Hegel considered bourgeois cultural development (idealized by him) to be the completion of man’s social history. For Hegel, whose theory immortalized the capitalist division of labor, the idea of comprehensive and integral development of the individual was a romantic dream, that is, a reactionary ideal. But without such a development the idea of “beautiful individuality” becomes unthinkable even in a purely theoretical sense. “The beautiful” (and thereby the ideal as such) thus appears for Hegel more as an image of the past of human culture than of the future of human culture.

Criticizing Hegel’s idealism, Marxism-Leninism reworked Hegel’s dialectical ideas relative to the ideal, its structure, its role in the life of society, and its potentials for concrete realization. Understanding the ideal to be an image of the goal of the activity of people united by a common task, K. Marx and F. Engels focused their basic attention on an investigation of the real conditions of existence of the basic classes of contemporary (bourgeois) society and on an analysis of the actual universal needs that motivate these classes to act and that are refracted in their consciousness in the form of the ideal. For the first time, the ideal was understood as a reflection of the contradictions of developing social reality in the minds of people living in the clutches of these contradictions. The ideal always peculiarly reflects a contradictory sociohistorical situation, fraught with the urgent but unsatisfied needs of more or less broad masses of people, social classes, and groups. It is in the form of the ideal that these groups of people create for themselves an image of reality; within the framework of this image, existing oppressive contradictions are viewed as surmounted and “transcended,” and reality is represented as being “purified” and free from these contradictions. This does not mean that the future condition should be seen in the form of an ideal devoid of all developmental contradictions. Existing contradictions, concretely historical in terms of essence and origin, are ideally resolved through the ideal. Therefore, the ideal emerges as an active force that organizes the consciousness of men and unites them for the solutions of concrete, historically urgent tasks.

Classes through which the progress of all of society is realized form correspondingly progressive ideals. All active people seeking a way out of crisis situations are gathered together under the banner of these ideals. Such were, for example, the ideals of the Great French Revolution; in the contemporary age, such are the ideals of the Great October Socialist Revolution. Today the communist world outlook is the only system of ideas that embodies the progressive ideal, because it shows people the only possible road to the future out of the confusion of contradictions unresolved by capitalism; this road is the building of communism, which creates conditions for the free and comprehensive development of the personality.


Marx, K., and F. Engels. “Nemetskaia ideologiia.” Soch., 2nd ed., vol. 3.
Marx, K. “Kritika Gotskoi programmy.” Ibid., vol. 19.
Kant, I. “Kritika esteticheskoi sposobnosti suzhdeniia.” Soch., vol. 5. Moscow, 1965.
Schiller, F. von. “Pis’ma ob esteticheskom vospitanii.” Sobr. soch., vol. 6. Moscow, 1957.
Hegel, G. W. F. Nauka logiki, vols. 1–2. Sobr. soch., vols. 5–6. Moscow, 1937–39.
Hegel, G. W. F. Estetika, vols. 1–3—. Moscow, 1968–72—.
Debol’skii, N. G. “Ob esteticheskom ideale.” Voprosy filosofii i psi-khologii, 1900, book 55, pp. 759–816.
Lifshits, M. A. “I. I. Vinkel’man i tri epokhi burzhuaznogo mirovozreniia.” In the collection Voprosy iskusstva i filosofii. Moscow, 1935.
Murian, V. M. Esteticheskii ideal. Moscow, 1966.
Il’enkov, E. V. Ob idolakh i idealakh. Moscow, 1968.
Schlesinger, A. Der Begriff des Ideals. Leipzig, 1908.
Tsanoff, R. A. Moral Ideals of Our Civilization. New York, 1942.
Bertin, G. M. L’ideale estetico. Varese-Milan, 1949.



A subset I of a ring R where x-y is in I for every x,y in I and either rx is in I for every r in R and x in I or xr is in I for every r in R and x in I ; in the first case I is called a left ideal, and in the second a right ideal; an ideal is two-sided if it is both a left and a right ideal.


Ideal DEductive Applicative Language. A language by Pier Bosco and Elio Giovannetti combining Miranda and Prolog. Function definitions can have a guard condition (introduced by ":-") which is a conjunction of equalities between arbitrary terms, including functions. These guards are solved by normal Prolog resolution and unification. It was originally compiled into C-Prolog but was eventually to be compiled to K-leaf.


A numerical constraint language written by Van Wyk of Stanford in 1980 for typesetting graphics in documents. It was inspired partly by Metafont and is distributed as part of Troff.

["A High-Level Language for Specifying Pictures", C.J. Van Wyk, ACM Trans Graphics 1(2):163-182 (Apr 1982)].


In domain theory, a non-empty, downward closed subset which is also closed under binary least upper bounds. I.e. anything less than an element is also an element and the least upper bound of any two elements is also an element.
References in classic literature ?
Hence for Aristotle as for Plato, the natural state or the state as such is the ideal state, and the ideal state is the starting-point of political inquiry.
To the beauty of Spenser's imagination, ideal and sensuous, corresponds his magnificent command of rhythm and of sound.
Juxtaposition--or you may call it propinquity--or, if you like, being rather too near for one maintaining the position of an ideal.
For instance, the ideal of the Superman is put forth quite clearly in all his writings during the years
He had chosen his ideal, and he was bound to serve her, and break lances for her, and acknowledge her as the ideal of pure Beauty, whatever she might say or do afterwards.
Their ardor alternated between a vague ideal and the common yearning of womanhood; so that the one was disapproved as extravagance, and the other condemned as a lapse.
The ideal," he admits, [32] "poisons for me all imperfect possession"; and again, "The Buddhist tendency in me blunts the faculty of free self-government, and weakens the power of action.
There is more behind this than a mere university ideal.
The higher virtue, which is identical with knowledge, is an ideal only.
Neither must we forget that the Republic is but the third part of a still larger design which was to have included an ideal history of Athens, as well as a political and physical philosophy.
There is no statue like this living man, with his infinite advantage over all ideal sculpture, of perpetual variety.
They had lived just the life that to Levin seemed the ideal of perfection, and that he had dreamed of beginning with his wife, his family.