Order Versus Chaos or the Ghost of Indeterminacy
Truth, certainty, precision are the highest criteria for judgement on any statement concerning structures and processes of a universe within the reach of the human mind. People learned that the mythological heritage cannot satisfy all spiritual needs and, consequently, the mind activity extended from the perceptual explanation of the world to the conception of means to record and disseminate the resulting cognitions. Thinking became gradually a researching activity with a lasting educational component and was able to develop a twofold advance: philosophy dealing with general retrospective analyses and prospective outlooks, and science focussing the attention on particular actual problems approached by specific means. In spite of obvious differences, both philosophical and scientifical thoughts are to submit their statements to he above criteria for assuming the noblest tasks of Paideia.
At the turning of our century the science of the inert world, i.e. physics and chemistry, discovered phenomena that compelled the scientists to revise old deterministic patterns of explanation wich became controversial, and to look for new ones. During our century concepts like natural law, order, certainty became a matter of doubt for both theoretical and experimental scientists. Almost concomitantly biologists discovered that life phenomena had to be approached as chains of changes, so that the concept of creation was to be redefined together with the concept of order. Similar changes were recorded in the social sciences wich are dealing with animal and human collectivities. Finally, the uncontroversial model of exactness, mathematics, had refine its tools in order to tackle the problems issued from empirical sciences and to use efficiently the amazing facilities provided by electronic computational devices.
All these changes of pattern in science have entailed many ideatic changes in the philosophical camp, where were working not only specialized philosophers but also scientists who used to expound sometimes polemically their own theses on topics from their domains. A significant result of this interpenetration is the increasing number of studies and papers under titles that join the term "philosophy" with the name of particular sciences. It sounds like a rejoinder to works with titles that remind a scientific model of some philosophers (e.g. Kant's "Prolegomena zu einer jeden künftigen Metaphysik die als Wissenschaft wird auftreten können", Fichte's "Wissenschaftslehre", Husserl's "Philpsophie als strenge Wissenschaft").
Taking into account that philosophy is looking backwards and forwards claiming to be perennial, while science focusses the actuality, the field of the former extends on several interconnected levels and the latter unfolds on different contiguous domains divided in macroscopic and microscopic subdomaines. Traditionally, philosophy counts its main levels beginning with the universe (ontology) and the human knowledge (epistemology), continuing with the discourse on universe and knowledge (logic), and ending with the evaluation of all previous levels (axiology). Philosophy intends to educate humanity by answering questions about existence, report on existence and about the value of both existence and report on it.
As it may be presumed from the double title of this paper, one of the major question for both philosophy and science of our days is whether order and chaos are on equal foot, whether indeterminacy is an authentic face of the Being or is merely a transient companion like a ghost haunting the consciouness of scientific researchers, reminding the mythological inception of the universe. The Christian mediaeval thinkers concluded with arguments from the Holy Writ and from the classical philosophy that "esse est ordo". This fundamental statement belongs to those truths which in the modern era Leibniz declared as possible as long as their impossibility cannot be proved . If thinking is an activity of the mind and its object is referential or autoreferential, then there are three alternatives: first, order is intrinsically inherent to the world, secondly, order is extrinsically imposed by the immanent mind, or thirdly, order is imposed by a transcendental factor. Each from these hypotheses has representatives in the history of thinking, but nowadays it seems that the mediaeval option is prevailing , , .
In contrast with order, chaos is known from mythological cosmologies and from the Bible under the name of "tohu-bohu". Chaos meant a state of things before they got an order and, consequently, before they began to exist and could be perceived through senses or, alternatively, organized by the conceptual activity of the mind. It results that chaos could not be perceived but only presumed or, rather, conceived as something quite opposed to order. Linguisticaly, the opposition to order may be expressed by disorder and this fact gives the key to understand the "birth" of chaos: it is, indeed, a conceptual birth, not an existentional one and, therefore, chaos has nothing to do with existence, since it is produced by logic and linguistic routines favoured, of course, by the binarity of our mental framework. This paper is intended to show that existence means intrinsic order, structural organization and functional processes while disorder, chaos or randomness, are only "une façon de parler" to express a sudden change or an unexpected loss of pattern followed by an interlude before the perception of a new order or structural organization or a new process. For these reasons one can say that chaos is rather an epistemological topic as it is related to describability and predictibility difficulties. However, scientists had to cope with phenomena that seemed to behave "chaotically" beyond any epistemological interpretation, as they belong to the ontological level.
2. The ontological status of indeterminacy
If an imprecise knowledge of real things or events is not entirely imputable to the observer, it may be presumed that the structure of the existent entity under observation is an entangled intricacy and undiscernable mixture of changing elements, wich may be called indeterminacy. Examples of indeterminacy were known in the 19th century from the Brownian movement explained by an incalculable sequence of collisions of particles and from thermodynamics whose laws stated that in isolated systems entropy, i.e. the measure of transformation of ordered movement into disorder, is increasing. In our century Heisenberg formulated the relations of uncertainty concerning the simultaneous precise measure of position and velocity of microscopic particles and, consequently, the impredictibily of their behaviour. Later on Edward Lorenz described the "deterministic chaos" starting from climatological phenomena like the movements of wings of a butterfly in South Asia and a cyclone in Atlantic. B. Mandelbrot studied degrees of irregularities in geometrical shapes and in natural objects and developed a theory of such occurrences, called fractals, with applications in anatomy and turbulence. Ilya Prigogine led researches on mixing processes and dissipative structures and established that in states far from quilibrium fluctuations are emerging with points of bifurcations generating new forms of order instead of old ones. Such processes happen also in the living world without intervention from outside and Francisco Varela called such processes autopoiesis.
The Brownian movement, Boltzmann's entropy, Heisenberg's uncertainty relations, Edward Lorenz' deterministic chaos, Mandelbrot's fractals, Prigogine's dissipative structures, Varela's autopoiesis are paradigms demonstrating the penetration of indeterminacy in the scientific thinking wich is not prepared to deal with it smoothly. On the order side the scientific thinking could not reject indeterminacy because it violates the alleged lawfulness of the nature. Causal as well as final determinism was at stake and epistemologists had to answer whether order is inherent to existence or it is imposed from outside. A plea for indeterminism seems to be no plausible solution for the ontological status of indeterminacy .
3. The logical status of indeterminacy
In the universe of discourse some objects or processes "resisted" to any deterministic approach and therefore they were epitomized indeterminate entities. On the other side they were credited as belonging to an inherent order or structure on the ontological level. Now the discourse itself involves a conceptual and linguistic structure which facilitates the intercommunication and, thus, is ipso facto bound to certain rules, i.e. to a kind of determinism. However, the ghost of indeterminacy haunts on the logical level as it haunts on the ontological one. Firts we met the logical indeterminacy in Organon when Aristotle dealt with statement on uncertain future events meaning that they are unpredictible. In this case statement convey indeterminacy wich lies outside them, in a conjectural future. In our century Kurt Gödel discovered that axiomatic systems possess undecidable sentences like the adiabatic systems in thermodynamics that possess entropy. In both cases indeterminacy means apparently loss of "order" if this is understood as something "nec varietur" in an Eleatic version of the universe. In fact order cannot be lost as it cannot be acquired. Order is intrinsic to existing entities and "disorder", means in fact, a change of a previous order, thus, another kind of order. If we fail to get rid from the bias of the deterministic pattern of thought, there is no chance to understand that indeterminacy, disorder, chaos, randomness, vagueness are terms wich serve merely as "une façon de parler" without meaning exactly what they prima facie may suggest: a diametrically clear-cut opposite to order.
The polysemous use of negation is a controversial matter since the Antiquity, but scientists chose exclusively the logical interpretation of negation according to the principles of non-contradiction and tertium non datur and built up the modern science. While the latter principle began to be criticized only later by intuitionist mathematicians because of its unlimited use, the principle of non-contradiction remained uncontroversial. This fact explains the misunderstandings in the contemporary discussions on indeterminacy and uncertainty, or chaos, in science.
The semanitic analysis of negation calls for an essential revision of the dichotomic division of the logical discourse in affirmative and negative statements and, consequently, replacing the latter by statements different from the affirmative ones. Negation becomes in this case one kind among several kinds of statements different from the affirmative statements. On this way the fundamental binarity of our thinking is not violated and the referential diversity may be more adequately described by our discourse. If logicians agree with this new outlook, then indeterminacy and its synonims are quite compatible with order and the Spinozian saying "omnis determinatio est negatio" seems obsolete.
The coherence requirements compel us to outline a revised logic with two principles: one of the identity and the other of the alterity, wich rule single statements, and the principle of sufficient reason caring for the meaningfulness of any concatenation of statements referring to the knowledge of the universe. The propositional calculus needs two binary operators: "idem" (identity), "aliter" (alterity), and a diadic operator "neque-neque" with the following truth tables:
The diadic operator, like Sheffer's stroke, has two entries:
This logic, which may be called "aoristic", is apt to maintain the binarity and at the same time to express all grades of the possibility from the virtuality and probability to reality and necessity. As long as order is acnowledgedly inherent to any form of reality, disorder, indeterminacy, uncertainty or chaos are neither identical nor opposite, but different from order, as we learned from the ontological status of indeterminacy. The principle of alterity saves the logicality of indeterminacy which can be analysed in detail.
The universe of the aoristic logic consists of indeterminate components, conventionally called alternatives, enjoying full equiprobabilty, stability, and homogeneity. Owing to the aoristic principles, any area of indeterminacy can be identified, the equiprobable alternatives can be discerned, and the chain of valid statements can be secured by the sufficient reason to avoid any confusion between intrinsic order and induced order from outside. Because the alternatives are quiprobable, each has the value 1/n in the interval (0,1), where n is the number of alternatives, 0 means non-existence, i.e. unutterable, and 1 is certainty. Indeterminacy or uncertainty are not incompatible with dynamics and, therefore, changes or transformations are not ecluded, but they mean loss of stability and homogeneity inside the area of indeterminacy or uncertainty. This may occur if one or more alternatives get other values than 1/n. It may happen that a "dissident" alternative divides in equiprobable subalternatives and, beside the order, now poorer, uncertain entity, there emerges a new uncertain entity enjoying stability and homogeneity. The process of the engendering uncertain entities may continue until complete determination and loss of indeterminacy. Each moment of this process is a level of compossibility which may serve as a measure of indeterminacy or uncertainty: the greater the number of levels, the nearer the certainty or determination. The process itself occurs on the ontological level, but its accurate description is available by means of the aoristic logic.
4. The axiological status of indeterminacy
As an independent philosophical discipline axiology is much younger than ontology, logic, and epistemology. For a long time axiology was associated with ethical, aesthetical, and economical issues. In the 19th century axiology got a relative autonomy with Lotze, Windelband, and Rickert, who grounded their valuational insights on the idea of a realm of values. In our century axiology unfolded in the context of the phenomenological movement. The concept of value, which is the core of the axiological thinking, has its roots in the category of quality and in the use of the term "axiom" in logical arguments and in mathematical demonstrations grounded on its etymology.
Based on such undeniable roots, the concept of value interferes with indeterminacy in both theoretical and pragmatical maters. This fact became explicit only in 1965, when L.A. Zadeh  published a paper under the title "Fuzzy sets" introducing the concept of fuzziness or vagueness, which soon got various philosophic, scientific, and engineering interpretations. Zadeh illustrated fuzziness in set theory by stressing the gradualty of membership relation between elements and sets in some cases.
Fuzzy experts extended the researches to many properties enjoying gradualty and are trying to approach the valuation of such properties by optimal standards. Engineers and software specialists are coping with complex phenomena which need an approach of this kind, as complexity of higher degree attains to indeterminacy. On the other side logicians and mathematicians are approaching to the foundation of fuzziness using different tools of philosophic and linguistic relevance. The inconclusive results of these efforts seem to have the origin in the fact that the referentials of the statements on fuzziness are neither ontological, nor logical, but axiological, i.e. not arbitrary, but more or less contextual or conventional , , , .
Indeterminacy and related concepts are equally pervading existence and becoming on ontological, logical, and axiological levels. In spite of strenuous endeavours to attain to truth, certainty, and precision both in theoretical and pragmatical matters, the mental activity is openly or subreptitiously challenged by indeterminacy, that seems to threaten the pattern of order inherent to the world.
The semantical analysis of logical negation discloses the roots of the ominous power of indeterminacy. Aiming to elucidate such matters an aoristic logic was to be outlined purposing to make possible the harmless coexistence of order and indeterminacy in the framework of the fundamental binarity of true and false.
The message of the philosophy at the end of this millenium is to strengthen the hope that ghosts of any kind can lessen, if not lose for good and all, their impact on the more and more clear-minded mankind.
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