{"id":23882,"date":"2012-12-10T12:00:15","date_gmt":"2012-12-10T12:00:15","guid":{"rendered":"http:\/\/www.transcend.org\/tms\/?p=23882"},"modified":"2012-12-09T23:52:54","modified_gmt":"2012-12-09T23:52:54","slug":"diversity-as-a-source-of-conflicts","status":"publish","type":"post","link":"https:\/\/www.transcend.org\/tms\/2012\/12\/diversity-as-a-source-of-conflicts\/","title":{"rendered":"Diversity as a Source of Conflicts"},"content":{"rendered":"

The Simplest Diversity: A Binary Distinction <\/em><\/p>\n

Diversity is unavoidable. It is a preliminary condition of complexity. Could the world be perceived as an amorphous, i.e., totally non-structured mass? Maybe in the eyes of a baby in its first period of life.<\/p>\n

The most elementary form of diversity is the binary distinction. Its apparent simplicity is misleading; it is full of traps attracting the attention of artists, of scientists and of philosophers. Its complexity is suggested by investigations such as that of Bahm (1977), who proposes a typology of polarity involving the categories of opposition, complementarity and tension, which, by their sub-categories, lead to a surprising scrupulous analysis. Antinomy, paradox, antonimy, oxymoron, the linguistic oppositions in structural linguistics, the double bind considered by the Palo Alto school of psychoterapy, the yin-yang distinction in the Chinese philosophy of the fourth and third millenia before Christ are only some of the aspects of binarity. Another surprising fact is the way sometimes the understanding of a binary distinction needs as a preliminary step the understanding of ternarity, under the form of a mediation process. We learn that \u2018hot\u2019 is associated (and in opposition) with \u2018cold\u2019 (and not with \u2018hard\u2019 or with \u2018solid\u2019) when we experience the move from one of them to the other, for instance, when taking a shower.<\/p>\n

Synergy and conflict within a binary distinction\u00a0\u00a0 <\/span><\/p>\n

Most interesting contrasting pairs have this double aspect of both synergy and conflict. According to Galtung (2012), \u201ca conflict is a relation of incompatibility between parties [\u2026]\u201d. We will include \u2018incompatibility\u2019 in a variety of possibilities among which conflicts may be situated. In couples such as <left, right>, <synchrony, diachrony>, <up, down>, <syntagmatic, paradigmatic>, <form, substance>, <form, function>, <competence, performance> each term needs the other, in order to become meaningful, but at the same time each term is opposing the other in some respect, creating a kind of conflict, taking various aspects, from polar difference to rejection or incompatibility. Logic is dealing with contrariness and with contradiction, structural linguistics is essentially based on some antinomic couples, Ferdinand de Saussure\u2019s semiology and Greimas\u2019s semiotics are also using some binary distinctions having this double status of synergy and conflict. Even Peirce\u2019s semiotics, although essentially based on a ternary view, is often using a binary both synergetic and conflictual approach. Mathematical game theory was developed by von Neumann and Morgenstern (1944) basically for games with two players: no of them can avoid the other, so each of them needs the other, but at the same time is opposed to the other in a conflictual way. But just this example calls attention on a feature on which we will focus our attention in the following.<\/p>\n

From explicit to hidden conflicts <\/span><\/p>\n

In all examples considered above, both the synergetic and the conflictual aspects are apparent, visible, of an a priori<\/em> nature. There exist however binary situations where the conflict is not at all visible, being revealed only by a research process whose authors deserve our indebtness. Such a situation appeared in 1927, when Heisenberg stated the so-called uncertainty principle according to which one cannot measure concomitantly the position and the momentum of a quantum object (such as the electron) with an accuracy as high as we want; the accuracy in the measurement of the position will be obtained at the expense of the accuracy in the measurement of the momentum and conversely. In other words, two acts which in the macroscopic world are perfectly compatible, the measurement of both the position and the momentum of a moving object, become incompatible in the quantum universe. The same acts, with the status of an alliance in the macroscopic world, reject the similar alliance in the quantum world. Our expectation, based on the experience acquired in the macroscopic world, is violently deceived: the rules of the quantum world are no longer those of the macroscopic world. Pairs such as Heisenberg\u2019s\u00a0 <position, momentum> are called conjugate pairs<\/em> and are characterized by the a posteriori<\/em> nature of the conflict between their terms, in contrast with antinomies, antonimies, and the double bind, where the conflict ia a priory<\/em> and explicit. Moreover, in a conjugate pair it happens in most cases that the initial expectation is one of cooperation and harmony between the respective requirements, expectation deceived by their investigation.<\/p>\n

The metaphoric potential of Heisenberg\u2019s principle stimulated many researchers. McCloskey (1994) proposed an \u2018economic uncertainty principle\u2019, under the form of two types of activities, apparently in alliance, giving the impression that the second one is a natural consequence of the other, but proved to be ultimately incompatible, These activities are: to understand a social fact and to get financial advantages from this understanding. After a careful analysis, McCloskey shows that the market has a capacity of rapid reaction, invalidating the act of understanding of the respective social fact. As an\u00a0 example, McCloskey refers to economic statistics, yesterday used to anticipate recession, today used to predict those changes in the policy of the government which will avoid recession.<\/p>\n

Another bold metaphorical use of Heisenberg\u2019s principle was proposed by de Beaugrande (1988): the universe of art is in respect to the ordinary one in a relation similar to that of the quantum world in respect to the world of classical physics. A consequence of this representation is the frequency in art of anti-intuitive aspects, dominated by conjugate pairs of Heisenberg\u2019s type. The gap between art and the everyday life is genuine, so unavoidable.<\/p>\n

The key words: \u2018at the expense of \u2019<\/span><\/p>\n

The zero-sum games are, in von Neumann \u2013 Morgenstern\u2019s theory, the games where two partners are in the situation that one of them gains exactly what the other loses. This is exactly what happens in Heisenberg\u2019s principle, where each of the two considered actions is possible only at the expense of the other and the advantage of one of them is equal to the loss of the other.<\/p>\n

But let us look around; it is what happens in the living universe, where every advantage for some species is a loss for some other species. Each species wants to survive, but the survival of some species is possible only at the expense of the survival of other species. The mineral, the vegetal and the animal realms are also in interactions of the type \u2018at the expense of \u2018. A lot of other examples show that requirements acceptable when each of them is considered independently are incompatible when considered simultaneously.<\/p>\n

Some phenomena of this type are explicit and can be perceived with no difficulty, but others are hidden and require an important effort to reveal them. Let us take, for instance, the identity of the human body. As we have shown (Marcus 2007), this identity has several components: the material, the genetic, the structural, the holographic, the holonomic, the managerial, the computational, the ecological, the dynamic and the semiotic identity (but the list remains open). Among these identities there are complex relations of cooperation and of conflict. For instance, to refer only to the conflictual aspect, there is a conflict among the material and the dynamic (field) identity, the former is deteriorated by the latter (some atoms today in our body leave it tomorrow and determine its instability). There is a conflict among the genetic and the dynamic identity. We inherit some capacities of the body to face situations related to a period far away in the past (because the genetic evolution is very slow), while the interaction (from stimulus-response type to the mental type) with the external world meets situations for which our body is not yet trained. An example in this respect is our linguistic competence; it is limited to the macroscopic world, i. e., that world in which a sharp distinction is assumed between subject and object. On the other hand, the evolution of science in the last hundred years lead to the investigation of phenomena occurring at the microscopic scale, such as the quantum level in physics. Human language was not programmed to cope with such phenomena (Favrholdt 1993).<\/p>\n

The conflictual component of the act of choice\u00a0\u00a0\u00a0\u00a0 <\/span><\/p>\n

According to Paul Val\u00e9ry and to Henri Poincar\u00e9 (see Hadamard (1954)), in any act of invention, two types of operations are involved: combinations and choices, the latter being more important than the former. In the light of this remark belonging to a poet and to a mathematician, we can appreciate the importance of the next results, according to which the act of choice, in two of its fundamental instances, cannot avoid a conflictual situation.<\/p>\n

Both combinations and choices are an expression of the diversity we have to face in our behavior, if we can behave in freedom and not under the pressure of various dictatorships. In this order of ideas, a surprising result was obtained by Arrow (1951) in respect to the aggregation of several individual options into a collective one, making the synthesis. Let us suppose that n (>1) individuals have to make the hierarchy of several (at least 3) actions, for instance to order acording to their preference several \u00a0candidates for the presidency of some institution. Arrow (Nobel prize in economics) formulates two natural requirements for the procedure we use to aggregate the individual hierarchies into a collective one: Pareto optimality (if action a<\/em> is prefered to action b<\/em> in each individual hierarchy, then a<\/em> is prefered to b<\/em> in the aggregated hierarchy too) and independence (if in two diffferent manifestations of individual preferences in respect to the same actions the reciprocal situation of actions a<\/em> and b<\/em> is the same for each individual hierarchies, then it remains the same in the aggregated hierarchy too. Taking into account that these requirements are in agreement with the<\/p>\n

common sense, we expect that their fulfilment will raise no problem. But Arrow shows that the price we have to pay for this is the acceptance of a dictator, i.e., of an individual imposing his hierarchy of preferences as the synthetic hierarchy, irrespective\u00a0 the options expressed by the other individuals. Arrow\u2019s result is called the impossibility theorem<\/em> or the dictatorship theorem<\/em>, because it shows that two natural requirements, both of them very reasonable, get in conflict with another reasonable requirement, the absence of a dictator. However, it was shown that \u2018Arrow\u2019s paradox\u2019 disappears as soon as the preferences are expressed not in the binary classical logic, but in a logic with several values, for instance, a fuzzy logic (Skala 1978).<\/p>\n

Another \u2018impossibility\u2019 result is due to Paun (1983) and is concerned with the aggregation of social indicators. Using an informal style,we can say that he proved that three requirements, each of them very reasonable, to be sensible<\/em> (improvement of the individual indicators yields a corresponding improvement of the aggregated indicator), to be anticatastrophic<\/em> (small modifications of the individual indicators cannot lead to huge modification of the aggregated indicator), and to be non-compensatory<\/em> (deterioration of some indicators cannot be compensated by improvement of other indicators: for instance, bad food cannot be compensated by good dwelling) get in conflict when they are imposed simultaneously. This means that two of them can be satisfied only at the expense of the third one, which will not be satisfied.<\/p>\n

Sensibility and clarity may get in conflict \u00a0<\/span><\/p>\n

In his Introduction to the French \u201cEncyclop\u00e9die\u201d, D\u2019Alembert observes that as soon as we approach the sensible properties of objects, obscurity prevails. In other words, sensibility and clarity may get in conflict and this fact explains the importance of vagueness, of the diffuse and of the obscure in poetry, in art in general, because poetry and art are mainly concerned with the sensible aspects of life. The art criticism is directly concerned by this siuation. Georges Braque observes that in art only what cannot be said is really relevant.<\/p>\n

Latent conflicts between truth and clarity, between exactness and truth\u00a0 <\/span><\/p>\n

Niels Bohr observes (Favrholdt 1993) that truth may get in conflict with clarity. Near to this possible conflict is another one, between exactness and truth (\u2018clarity\u2019 and \u2018exactness\u2019 like each other). There is a French saying: Presque <\/em>et quasiment<\/em> emp\u00eachent de mentir\u201d. Imagine I tell you that I arrived yesterday at University at 9 hours, 2 minutes and 7 seconds; this exactness is very risky, I may be wrong, i.e., in conflict with the truth. In order to avoid this risk, I prefer to say \u201cI arrived [\u2026] around nine o\u2019clock\u201d.<\/p>\n

Conflictual couples: <certainty, reality>, <rigor, meaning>, <reality, rigor>\u00a0\u00a0\u00a0\u00a0 <\/span><\/p>\n

All these three couples may be conflictual. Einstein observes (see Rosen 1978) that as soon as mathematical statements are certain they do not refer to reality and as soon as they refer to reality they are no longer certain. In the same semantic field with certainty<\/em> is rigor<\/em>; Thom (1983) observes that rigor is obtained at the expense of meaning, while som authors credit Socrates with the idea according to which the price we have to pay in order to reach an acceptable level of rigor is to replace the real world by a fictional one (R\u00e9nyi 1965). Euclid\u2019s \u2018Elements\u2019 and Homer\u2019s stories are equally fictional. If for Homer things are obvious, let us recall that Euclid begins with \u201cI call point what has no parts\u201d and this is his way to introduce the fictional object called \u2018point\u2019.<\/p>\n

An irreductible conflict between consistency and completeness<\/span><\/p>\n

We have in view G\u00f6del\u2019s incompleteness theorem (1931), asserting that in some formal systems (which include \u201cenough arithmetics\u201d) two natural requirements, non-contradiction<\/em> and completeness<\/em>, cannot be both satisfied; the price we have to pay for having one of them is the sacrifice of the other.<\/p>\n

It is interesting to observe that 12 years after this event Hjelmslev (1943) launches a new theory of language, aiming to stress the need in linguistics of a high level of formalization and of a treatment of an algebraic nature; in this respect, he formulates three requirements for any linguistic description: coherence<\/em> (by which he understands non-contradiction), exhaustiveness<\/em> (equivalent to completeness) and simplicity<\/em> (i.e., in modern terms, low complexity). Taking into account the general algebraic and formal orientation of Hjelmslev\u2019s doctrine, we may ask whether the impossibility to fulfil his requirements is not a consequence of G\u00f6del\u2019s result.<\/p>\n

Human communication under conflictual requirements<\/span><\/p>\n

As we have shown in Marcus (1983), the communication process has at least eleven components and functions (and not only six, as in the wellknown Jakobson\u2019s scheme): the expressive function, directed towards the source, the codification function, directed towards the sender, the decodification f., dir. tds the receiver, the conative f., dir. tds the destination, the phatic f., dir. tds the channel, the metalinguistic f., dir. tds the code, the referential f., dir. tds the context (in Jakobson\u2019s terminology), the poetic f., dir. tds the message, the perturbation f., dir. tds the noise (both physical and semantic) and the therapeutic f., dir. tds the observer of the communication process. The referential function may be in its turn decomposed in the extensional function, directed towards the object\u00a0 (approximated by Frege\u2019s \u2018Bedeutung\u2019), and the intensional function, directed towards the interpretant (approximated by Frege\u2019s \u2018Sinn\u2019). Among these communication functions there are interactions of various types: a function may favor another function, it may deteriorate it or it may be neuter in respect to it. When taking into account the whole system of these interactions, we are faced with a complex calculus of influences (Marcus-Tataram 1987 a, b), where we distinguish between synergetic (cooperative) and conflictual aspects. For instance, there is a genuine conflict between the expressive function, on the one hand, and the codification and decodification functions, on the other hand: stronger is the former, weaker will be the latter. So, we cannot raise the problem of a simultaneous optimization of all communication functions, we can only look for a convenient compromise, in respect to the particular aspects of the case we are considering. For instance, it is generally accepted that in diplomatic communication as well as in communication with children the phatic function has priority, while in scientific communication the referential and the metalinguistic functions are the most important.<\/p>\n

Grice\u2019s cooperative principle includes a conflictual component too\u00a0\u00a0 \u00a0<\/span><\/p>\n

In the same order of ideas we may refer to Grice\u2019s cooperative principle (Grice 1975) and show that his conversation\u2019s postulates are only in some respects cooperative; in some other respects they are conflictual. To give only one example, his requirement to be brief is in conflict with his requirement to avoid ambiguity, because the latter needs to supplement the information part of a message with the control part, able to make possible to detect and to correct the possible errors produced by various types of noise. Redundancy becomes unavoidable and, according to what is known from information theory, 50% of human communication is redundant (see, for more, Marcus 2001).<\/p>\n

A hidden conflict between to see<\/em> and to understand<\/em><\/span><\/p>\n

According to Thom (1993: 85), who attributes the idea to a mathematician of the XIX-th century whose name he no longer remembers, mathematics mirrors the need of the human brain to see – and we can see only what is continuous \u2013 and its need to understand \u2013 and we can understand only the finite, i.e., the discrete. So, when we see we don\u2019t understand and when we understand we no longer see. This fact may seem at a first glance in conflict with the everyday experience, but after a little reflection we realize that any continuous entity is perceived and conceptualized by means of a finite number of parameters defining it. On the other hand, we see points and lines only because, in the way we represent them, we use continuous approximations of them. The pair <to see, to understand>\u00a0 has a status of conjugate pair essentially of the same type as Heisenberg\u2019s conjugate pair. By means of this conflictual relation between seeing and understanding, we get new reasons for the emergence of the discrete representations in fields such as mathematics, physics and linguistics of the XX-th century; as a matter of fact, the label \u2018discrete mathematics\u2019 begun to be used only in the second half of the XX-th century, perhaps under the pressure of information sciences. But the discrete-continuous distinction was used already in the Greek antiquity, although the respective terminology was missing (the word \u2018discontinuous\u2019 was usually used for what today we call \u2018discrete\u2019).<\/p>\n

Putnam – Lakoff\u2019s conjugate pair<\/span><\/p>\n

In his critical attitude against objectivist semantics, Putnam (1981) has developed an argument against what he calls metaphysical realism. <\/em>Using it, Lakoff (1987: chapter 15) proposed an argument claiming that objectivist semantics is internally inconsistent. Specifically, the following two statements, apparently not only consistent, but supporting each other, are in conflict, claims Lakoff, following Putnam:<\/p>\n

Semantics characterizes the way that symbols are related to entities in the world.<\/em> \u00a0\u00a0Semantics characterizes meaning<\/em>.<\/p>\n

We are in the presence of a clear conjugate pair. As a matter of fact, \u201cthe relationship between symbols and the world does not characterize meaning\u201d (Lakoff 1987: 229). Lakoff points out the weak point of objectivist semantics, according to which \u201cmeaning is defined in terms of truth for whole sentences and in terms of reference for parts of sentences\u201d (Lakoff 1987: 230). This shortcoming, claims Lakoff, is in conflict with the requirement that \u201cthe meanings of the parts cannot be changed without changing the meaning of the whole\u201d (Lakoff 1987:230).<\/p>\n

Semiotic potential conjugate pairs:<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <syntactics, semantics>, <object, interpretant>\u00a0\u00a0\u00a0 <\/span><\/p>\n

Syntactics and semantics may get in conflict when a gap is created between them, either in favor of syntactics and at the expense of semantics or conversely. Syntactics is based on rules and behind rules there is always a potential game that may have an attractive capacity. Once started, this game may become an aim in itself and the meaning is vorgotten. Computers may stimulate this trend. Textbooks of mathematics are full of exercises calling for the application of procedures, devices, formulas, algorithms and operations (what French authors call math\u00e9matiques r\u00e9cettes de cuisine<\/em>); natural language, ideas, concepts, motivations, relevance almost disappear in such texts and we are near to the schizophrenia in which children are involved by crazy programs and textbooks. On the other hand, semantics in absence of syntactics is the other danger, that may transform semantics in a purely speculative activity, having no operational value. The whole history of mathematics has been an alternation of periods dominated by the fight for concepts and periods dominated by the fight for operations. Semantics in absence of syntactics is powerless, syntactics in absence of semantics is blind.<\/p>\n

In a different order of ideas, semantics dominates syntactics in a formal system, under conditions in which the true statements form a non-countable set, while the statements that can be proved form a countable set. According to G\u00f6del\u2019s incompleteness theorem, the existence of true, but non-provable statements cannot be avoided in some formal systems. On the other hand, if we take semantics as the interpretative part of a formal system, i.e., as a specific mapping of the formal system into another system, then the syntactic component of a formal system has a kind of autonomy, it may be conceived in absence of a semantic component. Syntactics is generative, semantics is interpretative.<\/p>\n

Object and interpretant are usually in a tension, due to the unavoidable gap existing between them. For children, this gap is in favor of the object; for researchers in speculative domains such as mathematics, theoretical physics, theoretical linguistics or philosophy, this gap is usually in favor of the interpretant.<\/p>\n

More conjugate pairs: \u00a0<love, marriage>,<\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<fidelity, beauty>,\u00a0\u00a0 \u00a0<knowledge, health> <\/span><\/p>\n

In each of the above couples, terms seem to be in harmony, almost needing each other. However, behind each of these couples there are some dangers we will point out in the following.<\/p>\n

A sociological investigation of the love relations between men and women in the Romanian society, two hundred years ago (Vintila-Ghitulescu 2006), reveals the unexpected fact that, at that time, love, instead to lead to marriage, was in most cases, in conflict if not incompatible with it (Avramescu 2006).<\/p>\n

Two natural requirements concerning the translation of poetry from one language into another language, fidelity in respect to the original version and aesthetic beauty of the resulting piece of poetry in the language in which translation is made, prove to be in most cases in quarrel, if not in a relation of incompatibility. This situation made an author to call metaphorically some literary translations les belles infid\u00e8les. <\/em><\/p>\n

An interesting situation in psychotherapy was told to us by Vianu (2006) and refers to the couple <knowledge, health>, the first term being accepted as favoring the second one (more we know our state of health, more we are able to improve it). However, in psychotherapy something strange happens: beyond some step, more knowledge about our self no longer helps the cure of our illness, the effect being the opposite, we are thrown in misfortune.<\/p>\n

From each direction, some surprising conjugate pairs may emerge, they are everywhere around us. May be the ten biblical commandments should also be reconsidered from this point of view: are they all cooperative ? Apparent alliances may always hide some latent conflicts.<\/p>\n

Numerical approximations under the sign of a conjugate pair<\/span><\/p>\n

The main conjugate pair in this respect is <empirical efficiency, theoretical convergence>. Physicists and engineers know that just for the most empirically efficient processes of approximation it is very hard to prove their convergence in the mathematical sense of this word. Such processes are usually known only by their first terms. Conversely, in most cases in which convergence can be easily established, it is so slow that it is practically useless. A classical example in this respect is the convergence to \u03c0\/4 of the series 1 \u2013 1\/3 + 1\/5 \u2013 1\/7 + \u2026(\u03c0 is here the ratio between the length of the circumference of a circle and the length of its diameter; this ratio is the same for any circle and for this reason it is considered the symbol of circularity and has the status of a universal constant).<\/p>\n

Discrepancy between \u2018general\u2019 and \u2018individual\u2019 in the field of randomness<\/span><\/p>\n

Let us consider words on a given finite non-empty alphabet A. Emile Borel had the idea to consider the infinite word x on A random if each element in A has the same probability of appearance in x (i.e., if n is the cardinal of A, then this probability will be 1\/n) and, more than this, all words of the same finite length have the same probability of appearance in x. He applied this idea to real numbers, conceived as infinite words on the alphabet {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and called normal<\/em> any random real number. The motivation of this convention is that it takes into account a basic aspect of the intuitive idea of randomness: equal preference in x for any element of the alphabet A and, more generally, equal preference for all words of the same finite length. Borel proves that almost all real numbers are normal, i.e., the numbers that are not normal form a negligible set: it is of measure zero in the sense of Lebesgue (it may be covered by an infinite sequence of intervals of total length as small as we want). In respect to such a result, we expect that it will be very easy to give non-trivial examples of normal numbers. By \u2018non-trivial\u2019 we mean numbers which are not rational, because it is obvious that rational numbers don\u2019t fulfil Borel\u2019s condition of normality. But surprisingly, this does not happen: for the most individual irrational numbers we are ignorant about their possible normality. For instance, this is the situation with square root of 2 or of 3, with the number \u03c0 and with the famous golden number. So, two requirements giving the impression that they favor each other, the general and the individual knowledge of the status of normality of real numbers, prove to have completely different status: the former is possible, while the latter is impossible. Let us observe that this contrast does not have the status of a conjugate pair, because the difficulty to check the normality of an individual irrational number is not dependent on Borel\u2019s theorem. So, we cannot claim that the price we have to pay for the knowledge of the global behavior of real numbers in respect to normality is the incapacity, in most cases, to establish whether a given irrational number is normal. This incapacity is mirrored in the simple fact that the set Z of measure zero occurring in Borel\u2019s theorem has a non-constructive existence, it does not permit to decide, for a given irrational number, whether it belongs or not to Z. There are investigations trying to make constructive the negligible sets occurring in some theorems like Borel\u2019s, but even in these cases the cost (complexity) of the obtained effectiveness is not known, so the difficulty to decide the normality or non-normality of various irrational numbers remains sofar very high.<\/p>\n

The gap between collective and individual in mathematics\u00a0\u00a0 \u00a0\u00a0\u00a0<\/span><\/p>\n

What happens with normal numbers is typical for many other situations in mathematics. It is much easier to obtain information about collective aspects than about individual (local) aspects; but the price we have to pay for this privileged situation of the collective is the non-constructive existence of a negligible, exceptional set of elements, a set whose most elements remain unknown. \u2018Non-constructive\u2019 means here the fact that, given an arbitrary element, we cannot decide whether it belongs or not to the exceptional set.<\/p>\n

Here are some examples.<\/p>\n

A classical theorem states that all real numbers, excepting a countable set of them, are transcendental (i.e., they are not roots of an algebraic equation).<\/p>\n

But despite this information, telling us that, in some sense, most real numbers are transcendental, it is very difficult to produce examples of individual transcendental numbers. The first examples of such numbers were discovered only in the XIX-th century and they include the number \u03c0 too. This discrepancy between the richness and the elegance of collective descriptions, on the one hand, and the scarceness and difficulty of individual examples, on the other hand, is strongly related to the way human mind proceeds when approaching global behaviors: the price we have to pay for simplicity and elegance is the non-constructiveness of our procedure, making unavoidable the difficulty to decide individually the behavior under investigation. The exceptional set of algebraic numbers can be defined constructively (Calude 1994-2002), but the cost of this effectiveness is not known.<\/p>\n

Another example: A classical theorem, by Lebesgue, asserts that any monotonous (increasing or decreasing) real function f in the interval [a, b] is differentiable almost everywhere in [a, b]. In other words, this means that the points where f is not differentiable is negligible (of Lebesgue measure equal to zero). But due to the non-constructive existence of the negligible set, it is difficult, for the points of continuity of f, to decide individually whether f is differentiable or not.<\/p>\n

Global(local) randomness co-exist with local(global) non-randomness<\/span><\/p>\n

A successful idea was that of conceiving randomness as high complexity, while complexity was conceived as algorithmic or program-size complexity; roughly speaking, the algorithmic complexity of a finite word x on a finite non-empty alphabet A is the length (measured in bits) of the smallest computer program describing x. Obviously, this length is never larger than the length of x; if these two lengths are equal, then x is said to be random (Solomonoff, Kolmogorov, Chaitin; see, for instance, Calude 1994-2002).\u00a0 This way to understand randomness can be extended to infinite words and it implies Borel randomness.A theorem about Borel random infinite words on the alphabet A states that any such word includes infinitely many occurrences of any finite word over A.<\/p>\n

Imagine now that A is the alphabet of English. This means that any infinite random word on A will include infinitely many times all works of Shakespeare; in other words, the global randomness of the infinite word x implies the local non-randomness of x. On the other hand, if u is a finite random word, then the infinite word obtained by infinite concatenation of u with itself will be periodic, thus non-random: it is shown in this way that an infinite non-random word can be locally random.<\/p>\n

*******************<\/p>\n

The examples above show the rich variety of conflicts emerging within the simplest diversity phenomena. We have focused our attention on those situations in which the conflict is not directly visible, it comes as a surprise, hidden in apparent alliances. Many of these conflicts were discovered only after deep investigations and the respective authors, such as Heisenberg, G\u00f6del or Arrow, deserve our admiration. It may also be observed that the conflicts were of different types. Some of them were at the empirical-observational level, other were at the logical level; some of them were of a semiotic nature; some of them were sharp, other were a matter of degree. Some of them were of the type of zero-sum games, of the type \u2018at the expense of\u2019, in other conflicts, like some of those occurring in mathematics, one of the terms in conflict had no impact on the other term.<\/p>\n

For other aspects of the problems considered above see Marcus (1998 a, b).<\/p>\n

References:<\/strong><\/p>\n

Arrow, Kenneth (1951\/1961) Social choice and individual value<\/em>. Chicago:<\/p>\n

Wiley.<\/p>\n

Avramescu, Catalin (2006) \u2018Dragostea la romani\u2019. Dilema Veche<\/em> 148, 14.<\/p>\n

Bahm, Archie J. (1977) Polarity, dialectic, and organicity.<\/em>Albuquerque,<\/p>\n

New Mexico: World Books.<\/p>\n

Beaugrande, Robert de (1988) \u2018 Quantum aspects of artistic perception\u2019.<\/p>\n

Spiel<\/em> 7, 1, 1-37.<\/p>\n

Calude, Cristian S. (1994-2002) Information and randomness. An <\/em><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 algorithmic perspective<\/em>. Berlin et al.: Springer.<\/p>\n

Favrholdt, D. (1993) \u2018Niels Bohr\u2019s view concerning language\u2019. Semiotica<\/em><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em>94, 5-34.<\/p>\n

Galtung, Johan (2012) Rational conflict resolution: What stands in the<\/p>\n

way? World Peace Academy. Transcend Media Science, Basel,<\/p>\n

Switzerland.<\/p>\n

G\u00f6del, Kurt (1931) \u2018Uber formal unentscheidbare S\u00e4tze der Principia<\/p>\n

Mathematica und verwandter Systeme\u2019. Monatshefte f\u00fcr<\/em><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Mathematik und Physik<\/em> 38, 173-198.<\/p>\n

Grice, H. Paul (1975) \u2018Logic and conversation\u2019, in Peter Cole and Jerry<\/p>\n

Morgan (eds.) Syntax and Semantics 3: Speech Acts<\/em>. New York:<\/p>\n

Academic Press, 41-58.<\/p>\n

Hadamard, Jacques (1954) The psychology of invention in the<\/em><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 mathematical field.<\/em> New York: Dover.<\/p>\n

Hjelmslev, Louis (1943) Prolegomena to a theory of language.<\/em>Madison:<\/p>\n

University of Wisconsin Press.<\/em><\/p>\n

\u00a0 <\/em>Lakoff, George (1987) Women, fire, and dangerous things.<\/em>Chicago: The<\/p>\n

University of Chicago Press.<\/em><\/p>\n

\u00a0 <\/em>Marcus, Solomon (1983) \u2018Diplomatic communication\u2019, in L. Vaina, J.<\/p>\n

Hintikka ( eds.) Cognitive constraints on communication. <\/em>Reidel,<\/p>\n

19-31.<\/p>\n

Marcus, Solomon (1998 a) \u2018No system can be improved in all respects\u2019.<\/p>\n

In G. Altmann, W.A. Koch (eds.) Systems. New paradigms for<\/em><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 human sciences.<\/em> Berlin, New York: Walter de Gruyter, 143-164.<\/p>\n

Marcus, Solomon (1998 b) \u2018Imprecision, between variety and uniformity:<\/p>\n

the conjugate pairs\u2019, in J.J.Jadacki, W. Strawinski (eds.) In the<\/em><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 world of signs. Essays in honour of Professor Jerzy Pelc.<\/em> Poznan<\/p>\n

Studies in the Philosophy of the Sciences and the Humanities 62,<\/p>\n

59-72.<\/p>\n

Marcus, Solomon (2001) \u2018The conflictual aspect of Grice\u2019s cooperative<\/p>\n

principle\u2019, in Giovanna Cosenza (ed.) Paul Grice\u2019s Heritage,<\/em><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em>Semiotic and Cogniive Studies IX, Brepols (Belgium): Turnhout,<\/p>\n

235-246.<\/p>\n

Marcus, Solomon (2007) \u2018Towards a typology of identities of the human<\/p>\n

body\u2019, in Adela Rogojinaru (coord.) Comunicare, Relatii Publice si<\/em><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Globalizare. <\/em>Bucuresti: Tritonic, 17-28.<\/p>\n

McCloskey, D. N. (1994) \u2018An economic uncertainty principle\u2019. Scientific<\/em><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 American<\/em>, November 1994, p.107.<\/p>\n

Paun, Gheorghe (1983) \u2018An impossibility theorem for social indicators<\/p>\n

aggregation\u2019. Fuzzy Sets and Systems<\/em> 9, 205-210.<\/p>\n

Putnam, Hilary (1981) Reason, truth, and history<\/em>. Cambridge: Cambridge<\/p>\n

University Press.<\/p>\n

R\u00e9nyi, Alfred (1965) Dialogues on mathematics <\/em>(in Hungarian). Budapest.<\/p>\n

Rosen, R. (1978) \u2018Complementarity in social structures\u2019. Journal of<\/em><\/p>\n

\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Social and Biological Structures<\/em> 1, 191-200.<\/p>\n

Skala, Heinz J. (1978) \u2018On many-valued logics, fuzzy sets, fuzzy logics<\/p>\n

and their applications\u2019. Fuzzy Sets and Systems<\/em> 1, 129-140.\u00a0 <\/em><\/p>\n

\u00a0\u00a0<\/em>Thom, Ren\u00e9 (1983) Paraboles et catastrophes<\/em>. Paris: Flammarion.<\/p>\n

Thom, Ren\u00e9 (1991) Pr\u00e9dire n\u2019est pas expliquer<\/em>. Paris: Flammarion.<\/p>\n

Vianu, Ion (2006) e-mail message, October 2006.<\/p>\n

Vintila-Ghitulescu, Constanta (2006) Focul amorului<\/em>. Bucuresti:<\/p>\n

Humanitas.<\/p>\n

von Neumann, John; Morgenstern, Oskar (1944) Game theory<\/em><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0and economic behavior. <\/em>Princeton: Princeton University Press.<\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 economic behavior.<\/em><\/p>\n

 <\/p>\n

_____________________________<\/em><\/p>\n

Solomon Marcus is Professor of Mathematics at the University of Bucharest and a Member of the Romanian Academy. His research concerns Mathematics, Computer Science, Linguistics, Poetics, Semiotics, Anthropology, History and Philosophy of Science and Applications of Mathematics in Social Sciences and the Humanities. In these fields, he published several tens of books\u00a0and several hundreds of articles. He is recognized as one of the initiators of Mathematical Linguistics and of Mathematical Poetics. He was\u00a0the chief of the Roamnian team in the United Nation University Project “Goals, Processes, and Indicators of Development”, directed by Johan Galtung.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"

The Simplest Diversity: A Binary Distinction – Diversity is unavoidable. It is a preliminary condition of complexity. Could the world be perceived as an amorphous, i.e., totally non-structured mass? Maybe in the eyes of a baby in its first period of life.<\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[200],"tags":[],"class_list":["post-23882","post","type-post","status-publish","format-standard","hentry","category-academia-knowledge-scholarship"],"_links":{"self":[{"href":"https:\/\/www.transcend.org\/tms\/wp-json\/wp\/v2\/posts\/23882","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.transcend.org\/tms\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.transcend.org\/tms\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.transcend.org\/tms\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/www.transcend.org\/tms\/wp-json\/wp\/v2\/comments?post=23882"}],"version-history":[{"count":0,"href":"https:\/\/www.transcend.org\/tms\/wp-json\/wp\/v2\/posts\/23882\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.transcend.org\/tms\/wp-json\/wp\/v2\/media?parent=23882"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.transcend.org\/tms\/wp-json\/wp\/v2\/categories?post=23882"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.transcend.org\/tms\/wp-json\/wp\/v2\/tags?post=23882"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}