Nonlinear art

Information Approach in the Humanitarian Science (Taganrog, Russia, 2001) pp.66-73

Pavel B. Ivanov



Art is treated as a nonlinear phenomenon, and three levels of nonlinearity are discussed. The balance of imaging the world and self-expression depends on the level of nonlinearity chosen: on the level of external nonlinearity, the arts produce frames for experiencing the world; on the level of internal nonlinearity, formal transformations of the world are tried; on the level of hierarchical nonlinearity, the practical attitude to both abstract reflection and abstract creation serves to produce a genuine work of art allowing many-level perception.

Traditionally, there are two opposite attitudes to artistic creativity: the "realistic" doctrine insists that every work of art be a reflection of some reality, while the advocates of "pure art" proclaim independence of any reality as the fundamental principle of art. The exaggeration of the imaging side of art leads to naturalism and hypernaturalism, and even to denial of the necessity of any art at all. On the other pole, one finds all kinds of "spontaneous" art, combining sounds, paints or other forms in a random manner. As usual, the polarities meet, and the collections of arbitrarily chosen objects exhibited as examples of free self- expression in "conceptual" art little differ from the well-known Duchamp's ready-mades. Still, there are strong claims from the both camps, and much argument about "authentic" art, with one or another school put forth as its representative. To uncover the truth, one has to consider the differences between various trends in the arts from an integrating standpoint allowing to compare and discriminate among the existing possibilities using definite criteria.

The natural way to come to such a universal view would start from the general schemes of human interaction with nature — activity [1,2]. Any activity assumes a repeated sequence of assimilation of an object by a human actor (the subject) followed by generating another object intended for being assimilated in a particular way (the product):

In the process of interiorization, the product becomes an internal state of the subject rather than a commonly observable thing; in a well-developed activity, the subject is many times acting upon itself before any outer action, and hence any product must bear the signs of that reflexivity. In natural sciences, numerous similar phenomena are known as manifestations of nonlinearity. Since the determinative characteristic of the subject is universality [3, p. 38], one can conclude that the subject's nonlinearity is also a universal feature of every activity — this distinguishes it from nonlinearity in inanimate or living systems. However, the general properties of nonlinear systems known from natural sciences will be present in conscious systems as well, and any kind of activity might be analyzed in respect to nonlinear effects involved.

The essentially nonlinear character of art has been earlier considered in connection to information processing [4]. It has been indicated that creative communication provides a mechanism of generating new information on the basis of a message that did not contain it before, but rather synchronized the addressed person's internal processes and external activities in a specific way ("information laser"); this example also illustrates such an important feature of nonlinear systems as the ability to support various collective modes of motion that can behave as apparently stable particles moving in a quasi-linear medium. There are indications that the very consciousness is one of such collective effects in a social system.

It should also be noted that aesthetic perception can be described in informational terms, which is equivalent to introducing a nonlinear operator of discordance, with its average values measuring the subjective incompatibility of elementary tones [5].

There are different methods of mathematical description of nonlinearity. This work mainly focuses on its dynamic aspect, that is the relation to the equations of motion governing the system's behavior. I do not consider the chaotic modes of motion and statistical structure formation (fractals). From the dynamic viewpoint, three levels of nonlinearity can be generally distinguished, which will be denoted here as "external", "internal" and "hierarchical" nonlinearity.

On the first level, nonlinear behavior arises from a limited way of observing a linear system, which is formally described by various types of clipping the full phase space in the process of observation. The system itself is not influenced by the observer, but the continuity of phase trajectories gets lost; such a behavior can be related to that of the simplest nonlinear function commonly known as unit-step, or Heaviside function:

In physics, similar discontinuities appear as boundary conditions of the wall (mirror) type; in some systems they can lead to chaotic behavior; in other cases, there are continuous solutions (standing waves) with nodes on the boundary. One will always obtain external nonlinearity projecting an infinite phase space onto any finite (at least in one dimension) subspace. A simple example can be provided by a rectangular window opening to the system of interest; depending on the position and orientation of the window, one will observe different patterns of motion. However, there may be other possibilities — for instance, registering only the points of intersection of a continuous (phase) trajectory with a fixed surface; in some cases, this may produce seemingly "discrete" motion, up to complex quasi-stochastic patterns.

The reflexivity of activity associated with that level of nonlinearity can be related to the ways of selecting the size and shape of the aperture clipping the field of view. This a special kind of product that is always accessible to humans as a powerful means of manifesting their universality. Any thing can be included into the sphere of human activity through merely observing it in relation to some accessible product. Thus, a distant star cannot be influenced by the people from the Earth — still, it can be involved in the human practice as an object observed as distinct from the other objects; this star can further be related to other observable or imaginary objects (such as other stars forming a constellation, or the eyes of a girl beloved). When put into an aesthetically saturated environment, any thing can acquire aesthetic value and become a work of art without being actually produced as art.

This latter point is very important in assessing many phenomena in modern art using ready-made things that originally had no relation to the arts. The Duchamp's urinal might well be a piece of art if it were regarded as such. This not the quality of the thing itself that makes it an artwork, but rather the way of its observation and perception — and using in a universal way, as a part of the culture. This is a fundamental property of any product at all, but it is in the arts that it first becomes essential and determinative. Things in the arts are never the same as in any other field of activity, or in nature. There is no absolute necessity to create something unusual or unnatural to call it art. Thus, a good photograph can be a work of art, while a most abstract painting can remain nothing but waste of paint. On the other hand, no naturalist art can reproduce nature as it is, as long as it tries to reproduce it in a different material — mere projection of a three-dimensional thing onto a plane is enough to make it somewhat quite different, and judged by different criteria.

This is where the universal reflexivity enters the game. Something produced by a person can be called art if and only if it effectuates the humanity's self-reflection, representing a scheme of activity in another activity. It does not matter what kind of materials have been used, or how heavily they have been processed. It is only the way of looking at the world that matters.

This disclosure of the peculiar faces of reality could be regarded as a definition of art. The other levels of creative spirituality (science and philosophy) are base on this ability too, and that is why they have to be a kind of art in their own development.

The second level of nonlinearity (internal nonlinearity) is characterized by the presence of nonlinear terms in the equations of motion [6]. The model of a forced anharmonic oscillator can illustrate the possibilities met here:

The presence of the nonlinear term proportional to the third power of x makes the dynamics of this system most complex and sensitive to initial conditions and the values of the coefficients a, b, c. This explicit nonlinearity has been paid much attention in the literature. However, in this model, nonlinearity can also be introduced in an implicit way. Thus, one could assume that the force f(t) is not analytical in t, e.g. has jumps or other singularities. In this case, the behavior of the system near the singular point of the force will be different in different regions of the phase space, which is equivalent to an effective nonlinear dependence on x and ; this occurs even without the cubic nonlinear terms, that is, with c = 0. Such systems can manifest a complex autocorrelated behavior, like in the well-known school example of parametric resonance. Another type of implicit nonlinearity is associated with singularities in the coefficients a(t) and b(t); it is only recently that such nonlinear systems has attracted significant attention, mainly in relation to stochastic dynamics [6].

The systems with explicit nonlinearity are a classical model of self-action — that is why they can be immediately related to essentially reflexive human activity, including the arts. Implicit nonlinearity can model various kinds of cultural influences on individual activity.

Applied to the arts, the concept of internal nonlinearity refers to the transformation of reality in the process of its reflecting in a work of art. Communication between the people is the basic mechanism of this transformation — when this is communication between individuals folded into self-communication, one comes to explicit nonlinearity; when one deals with communication between different levels of the subject (individual and group), implicit nonlinearity should be considered.

In practice, this level is present in almost every work of art, since the very notions of subjectivity and creativity imply the artists' repeated interaction with themselves, and hence explicit nonlinearity; there can be no figurative art without transforming reality in certain aspects, and any naturalism can be nothing but imitation of nature, rather than its reproduction. Few people would assert that the aesthetic value of artwork could be measured solely by the accurateness of imitation. There is more controversy about the social position of art, related to implicit nonlinearity. Paradoxically, denial of any sociality in the arts is typical for many branches of art entirely based on explicit nonlinearity (impressionism, abstractionism, etc.); here, they agree with the advocates of "pure art" from the "realistic" camp. However, cultural dependence of art can never be discarded, and implicit nonlinearity might be considered as primary in most arts. Once again, the assessment of a product as a work of art cannot be based on formal criteria only, demanding universal reflexivity. Obviously, this is equivalent to the demand that every work of art reflect the major trends in the contemporary cultural development.

The difference of external and internal nonlinearity is reflected in the millenniums-old argument about the primacy of imaging or imagination in the arts. Dominance of external nonlinearity leads to the conception of art picturing the world as close to reality as possible; inversely, exaggeration of internal nonlinearity is evident in aesthetic theories proclaiming free self-expression the final goal of art. However, no reality can be pictured without transforming it, and no transformation is possible without anything to transform. Internal and external nonlinearity will always coexist in the arts, and it is their mutual saturation that results in the uniqueness of individual artwork, which requires considering the synthetic level of hierarchical nonlinearity.

This kind of self-action is related to the development of a nonlinear system through interaction with its environment. The stages of the system's development become the levels if its hierarchy, and the reflection of one level in another produces a special kind of nonlinearity, combining the features of external and internal nonlinearity.

Development is difficult to formally model, and hierarchical nonlinearity is usually formalized in a static way, fixing one of the possible unfoldings of the system's hierarchy. Statistical description is a common example: the complete description of the system's dynamics is being replaced with a few average characteristics, with their evolution treated as a higher-level process, with its own regularities. In certain cases, this higher-level dynamics may be governed by simple laws resembling those at a lower level (adiabatic processes); in other cases, kinetic description may be possible, with the evolution of the macroscopic state of the system governed by "master equations", usually obtained through averaging the lower- level equations of motion (physical kinetics) or utilizing the natural structuredness of the lower level (e.g. in chemical reactions).

For the higher level, the representation of the whole phase space with a few average values resembles clipping of the phase space on the level of external nonlinearity, the very way of introducing statistical distributions involving segmentation of the phase space into a number of unit cells [7]. For the lower level, the influence of higher-level constraints is analogous to implicit nonlinearity introduced through the dependence of the equations of motion on external parameters.

There may be other models of hierarchical nonlinearity, for instance, using the methods of Hamiltonian dynamics, involving canonical transformations with a few of the new coordinates corresponding to the integrals of motion of the system; the existence of relatively slowly varying coordinates allows to treat them as higher-level parameters while considering the evolution of the other (rapidly changing) coordinates. Reducing the motion of a number of material points to the motion of their center of mass is among the most well-known examples. In general, splitting the whole system into relatively isolated subsystems is a nontrivial problem, which may be especially difficult when the system's evolution may involve structural changes, or phase transitions.

The idea of the many-level perception of a work of art has long since become a banality. However, there are few models of aesthetic perception and creativity accounting for their hierarchical nature. One of them uses an informational measure of discordance and quantum mechanical considerations to construct a hierarchy of musical scales and predict their properties and evolution [8, 5]. Still, a few qualitative conclusions could be drawn without developing any detailed models. First of all, merging external and internal nonlinearity implies communicating the schemes of activity along with communicating its products, which makes any communication act an at least two-level hierarchy. That is, the ways of viewing a work of art must be controlled along with the viewer's impression of it. This can only be possible through socially organizing the process of aesthetic perception, so that every aesthetic act would be considered in a certain cultural context. For instance, the very atmosphere of an exhibition hall assumes quite particular ways of behavior, all deviations being socially discouraged. Art criticism and other mechanisms produce the same effect on the artists themselves. As an immediate consequence, some kinds of art may acquire irreproducibility, with every aesthetic act becoming unique because of the impossibility of reproducing the cultural conditions required for treating it as art. Thus, a joke cannot be repeated with the same success, and Duchamp's ready-mades would not produce any noticeable impression on a modern connoisseur of art; similarly, most samples of Neanderthal art can only have historical rather than aesthetic value — though some of them could be renewed as art through including to a quite different cultural context, thus adding external nonlinearity. That is, one has to account for cultural realities to be able to produce art — albeit not entirely accepted by the contemporaries.

To summarize, an artist can produce forms as well as fill them with a culturally significant content, which will ensure the universality of the product and its hierarchical perception. The proportion of external and internal nonlinearity may vary depending on the aesthetic task, but the author has to recur to the both, seeking for a peculiar balance, which would eventually deserve the name of art.


  1. A N Leontiev Activity, Consciousness and Personality (Englewood Cliffs, NJ: Prentice Hall, 1978)
  2. P B Ivanov "A hierarchical theory of aesthetic perception: Scales in the visual arts" Leonardo Music Journal, 5, 49-55 (1995)
  3. E V Ilienkov Dialectical Logic (Moscow: Politizdat, 1984)
  4. P B Ivanov "Art as creative communication" Interaction between Man and Culture: Information Standpoint Proceedings of International Symposium (Taganrog, Russia, 1998) pp. 95-100
  5. P B Ivanov "Discreteness, continuity and hierarchical scaling in the arts" Information Approach in Empirical Aesthetics Proceedings of International Symposium (Taganrog, Russia, 1998) pp. 66-77
  6. H M Hubey Diagonal Infinity (N.Y.: World Scientific, 1999)
  7. L D Landau and E M Lifshitz Statistical Physics, part 1 (Moscow: Nauka, 1976)
  8. L V Avdeev and P B Ivanov "A mathematical model of scale perception" J. Moscow Phys. Soc., 3, 331-353 (1993)

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