The Order-Disorder Interplay - in Mathematics and Beyond
ORDER VERSUS WHAT? LOOKING FOR CONCEPTUAL CLARIFICATIONS
Solomon Marcus Simion Stoilow Institute of Mathematics Romanian Academy [email protected]
Order as a trap Order has many faces and the right strategy to understand it is not to try to define it (weak hope to be successful), but to look for its possible opposites: disorder, randomness, arbitrariness, obscurity, ambiguity, high entropy, high complexity, amorphousness, vagueness, fuzziness, roughness, lack of cohesion, absence of coherency, chaos, imprevisibility, imprecision, approximation, absence of any structure, absence of any rule. The choice among them is contextually dependent.
The chaos in order The same situation may fit with some types of disorder, but not with the other types. So, order in some respects may no longer be order in other respects. This fact motivates to refer, at least metaphorically, to chaos in order. The simple fact that order is such an ambiguous thing, it may mean so many different things and motivates us to refer to its disorder. Mathematics enters the scene With respect to so many faces of disorder (they could be semiotically distinguished in syntactic, semantic and pragmatic), it is important to mention the fact that today for most of them there are some hypothetical- explanatory mathematical models. Assuming that any such model is a kind of order just in view of its mathematical nature, we infer that most types of disorder are captured by a second, higher order. In other words – and we reach here the essential point – the opposite of order can be perceived only to the extent to which we discover, we identify behind it a new kind of order.
Order in chaos This slogan is not a way to play with words; it reflects the deep nature of things. Similarly, “order in randomness” is perfectly meaningful and a lot of mathematics has been developed to explain it. Order is everywhere; but in some places it is directly or almost directly visible, while in other places it is more or less hidden, sometimes very hidden. Looking for the hidden order of things The whole human enterprise, be it science , art, technology, philosophy, theology etc is to look for the hidden order of things; and it happens that the most interesting and the most surprising aspects of this order are most hidden. It is a game the nature and the universe play with us, repeating a similar game we used to play in our childhood.
“Total disorder does not exist” There is a chapter of Mathematics, more exactly of Graph Theory, called “Ramsey theory” and having the slogan Total disorder does not exist; we can adopt it for the whole human experience, but we have to add: If, however, total disorder exists, it cannot be captured by humans, it is beyond human capacities to approach the world.
Order versus disorder in ten different ways Let us consider some aspects of order vs disorder: 1.structure vs amorphousness; 2.information vs entropy; 3.predictability vs randomness; 4.symmetry vs fractalness; 5.simplicity (low complexity) vs high complexity; 6.stability vs instability, for instance, deterministic chaos; 7.organization vs anarchy; 8.presence vs absence of a pattern; 9. existence of a rule vs absence of any rule. 10.motivated vs arbitrary. We observe that no example is available of a specific case which displays disorder with respect to all these ten aspects.
A first, empirical-intuitive look We may try to understand different ways to interpret order vs disorder by reference to the dictionary meaning of the respective words. We find in Wikipedia that arbitrary is defined as based on random choice and has unpredictable as one of its synonyms; randomness means lack of a pattern; chaos is explained as randomness and by lack of a pattern; symmetry is explained by means of rule; pattern is explained by means of predictability. So, remaining at this level, we no longer are able to clearly separate these terms, to a large extent they seem to be synonymous, they remain in a state of confusion making impossible any attempt to give them a rigorous status.
Looking at order via modern scientific concepts So, the next step of human semiosis was to invent concepts giving a clear status to various types of order such as information, structure, symmetry, predictability, and rule. In various previous works I described this process.
Order as information For instance, the emergence of the information concept followed two distinct lines, one predominantly quantitative, coming from thermodynamics, the other qualitative, stressing the idea of information as form, according to the Latin etymology in-formare (to give a form), starting from Darwinian biology, both in the second half of the 19th century; in a further step, telegraphic communication became a third line of development and it was connected with the quantitative approach, culminating with Shannon’s 1948 work, giving the possibility to measure information by means of its logarithmic formula, the unit of information being the bit (binary digit), defined as the information you get when you precise one of the two possibilities in a binary choice where the respective two possibilities have the same chance of appearance. The need to bridge these two lines of development is still waiting for a satisfactory answer. Itnis connected to another delicate task: the need to bridge information and sign.
Order as structure Dominated longtime by quantitative problems, mathematics (and science in general) became step by step equally interested in qualitative questions and so the idea to look for some fundamental structures giving the possibility to move from atomistic to global aspects, from individual problems to global ones became more and more central. In the 19th century, this trend became very active in mathematics, with Galois’s notion of an algebraic group; with Mendeleev’s periodic table and with chemical isomerism (see, in this respect, the book by Roald Hoffmann, Nobel laureate in chemistry: The same and not the same (New York: Columbia University Press, 1995), just the slogan of chemical isomerism); with the emergence of structuralism in linguistics and psychology; with the rise of the energy paradigm in physics; with the emergence of the evolutionism in biology. In the 20th century, this trend became stronger and stronger, with the Russian formalism in literature, Prague Linguistic Circle and the great structuralist project initiated in mathematics by the group Nicolas Bourbaki, pointing out three basic types of structures: order, algebraic and topological structures. To them, the emergence of computer science added as a fundamental structure various types of rewriting systems.
Order as rule We have in view the syntactic dimension not only in linguistics, but in any type of system. For instance, in a formal system of arithmetic like that of Russell and Whitehead there is a syntactic dimension dominated by formal rules which decide the correctness of various assertions and a semantic dimension, related to their truth value. In the field of generative linguistics, a set of rules is always a basic component of a generative grammar; the same situation in the grammars involved in the study of programming languages and in the general theory of formal grammars and languages. The Von Neumann – Morgenstern’s theory of strategic games is also essentially based on the application of some rules, like many other games, from chess and go to tennis and foot-ball. Rules are the most popular form of order in the social life and the basic term of reference when a decision should be adopted about a possible mistake that deserves to be punished. This mentality dominates the way education is conceived today. Many rules consist of formulas or algorithms. Learning the rules of social life, the rules of various fields of knowledge is the main task for those who have to succeed in examinations at school, at university and beyond them. However, a life aiming to be creative should go beyond existing rules.
Order as symmetry We read in Wikipedia that “in everyday language symmetry refers to a sense of harmonious and beautiful proportion and balance”. It comes from the aesthetic ideal of Greek antiquity, aiming for simplicity, harmony, regularity, symmetry and right proportion. This ideal became also valid for Renaissance and remained valid till our days. As a matter of fact, it corresponds to the mentality promoted by Euclidean geometry, having under its main attention the regular geometric forms: the straight lines, the circle, the regular polygons, the regular polyhedrons, the sphere, the ellipse, the parabola, the hyperbola etc.
Total order does it exist? This question, symmetric to another one, formulated at the beginning of this presentation, is unavoidable. We adopted the conjecture of a negative answer to the question Total disorder does it exist? By symmetry, our conjecture is that the question in the title above has a negative answer, so the symmetry between order and disorder is very plausible. But why this conjecture has no chance to be proved mathematically and transformed into a theorem? For the same reason for which its symmetric cannot be proved mathematically. Neither order, nor disorder, in their most comprehensive version, have a finite formal definition, so the expected proofs could never be obtained in a finite time, by a finite number of steps. There are however some mathematical results convergent with the conjecture of non-existence of total order. For instance, it is well- known that there is no computer program to check all possible computer programs. The meta- language of algorithmic thinking cannot be the same algorithmic thinking. Both “computer program” and “algorithmic thinking” are examples of order in its highest possible version.
Generality at the expense of precision As soon as we remain at the general, vague, imprecise description of symmetry, proposed in Wikipedia as well as in other famous dictionaries, no rigorous investigation of symmetry is possible, because the border between symmetry and its negation is not at all clear and it may depend very much on the personal taste of the subject. But as soon as we move to a precise definition of symmetry, for instance as the set of invariants under the action of a specific group of transformations, we realize that we had to pay a price: the new definition gained in precision at the expense of its generality; many symmetry phenomena don’t fit with the new definition.
Order: rigor at the expense of meaning As a matter of fact, a similar thing happens with the idea of order: in its empirical-intuitive version, it is very vague, it displays a lot of variants, some of them were indicated above. As soon as we precise it mathematically, as it happens in the first pages of a mathematical handbook (the order relation is a binary relation with some precise mathematical properties), it becomes a very restricted concept.
Ultimately, the complexity is in question Indeed, the real distinction, behind order-disorder, is that between two kinds of order: perceived and hidden. As a matter of fact, both ‘perceived’ and ‘hidden’ are not binary predicates, they are a matter of degree. An algorithm whose time complexity is exponential is a clear example of disorder within order; disorder refers to ‘exponential’, while order refers to ‘algorithm’. One of the main problems of computer science is to invent algorithms of low complexity for important problems in economics, finance, communication etc.
Limitations of human performance The example we have just presented, of exponential time complexity, concerns a limitation of human performance in keeping under control the parameter time. But this limitation is within the existing human competence, not beyond it. The machine, be it an algorithm, a generative grammar, an automaton or any special type of a Turing machine defines a corresponding type of human competence. But all types of such devices have to face the problem of their complexity (that can be time complexity, space complexity, energy complexity or other types of complexity); if the cost of this complexity is too high, then it follows that the respective device is not able to perform its task effectively; for instance, if the time we need to process the data regarding whether prediction for the next 24 hours is larger than 24 hours, then the respective prediction is useless.
An unavoidable gap between competence and performance There is always a gap between human competence (which is potentially infinite), on the one hand, and human performance (which is always finite), on the other hand. The problem is not to cancel this gap, but to reduce it as much as possible. The set of well-formed strings in English or in any other natural language is infinite, but our linguistic performance captures only a finite part of these strings; a similar situation occurs in any human language, be it natural or artificial, formal or non-formal. Computer programming languages follow the same scenario.
When human competence stops It stops where our macroscopic surrounding stops. This surrounding defines the area of action of our sensorial, psychic and mental capacities; this area is represented by what science, culture in general were able to develop until the end of the 19th century. Around one hundred years ago, we learned that what seemed to represent the whole reality was only its macroscopic part, while other parts, such as the part investigated by relativity theory and that studied by quantum physics were beyond the macroscopic universe, beyond the sensorial, psychic and mental human competence; because human beings are, by their history, macroscopic beings.
Prisoners of a lot of limitations So, we are prisoners of the Euclidean geometry in the way we perceive the space; of the Galileo-Newtonian physics, in the way we perceive time, motion and energy; of the traditional logic of identity, non- contradiction and excluded middle, in the way we use logic and reasoning; of the foundation axiom of set theory, claiming that no set can be an element of itself, in the way we use sets; of the Church–Turing hypothesis, claiming that any human computation is of the type proposed by Turing with his abstract machine; of the human language and more general, of the human semiosis.
How does it work the language prison ? Please compare the hospitality of natural languages in finding ways to express situations in Euclidean geometry, in traditional logic, in Galileo-Newtonian physics etc in contrast with the difficulty to express situations beyond these fields. For instance, it is easy to observe that language is well prepared to give expression to logic relations and operations: logical proposition vs linguistic proposition; logical predicate vs linguistic predicate; logical disjunction vs linguistic use of or; logical conjunction vs linguistic use of and; etc. The whole architecture of traditional logic is similar to that of natural languages; but similar does not mean identical. There are logical propositions which are not linguistic propositions, while a logical predicate may not be a linguistic predicate. On the other hand, cancelation of past-present-future relations and their replacement with a continuous present, as they take place in quantum physics hardly can be represented in natural languages. Human language, human semiosis in general have to face this linguistic crisis. They try to escape from this trap by inventing all kinds of metaphors and allegories, as macroscopic ways to capture non- macroscopic situations. Human semiosis seems to be in difficulty when subject and object are no longer sharply separated and this is just what happens frequently in contemporary science, art and philosophy.
What about randomness and chaos? Roughly speaking, a random string is one that cannot be compressed, i.e., there is no description of the string shorter than the string itself. Obviously, this informal definition cannot replace in investigation the formal one. Again roughly speaking, a dynamical system is chaotic if to small differences in the initial conditions may correspond important differences in behavior. Mathematics is able to bring a great surprise, pointing out that things such as randomness and chaos are not from our world. I have conceived this presentation avoiding any mathematical technicality, but now, due to the importance of facts, I will mention two theorems to be considered only by what they suggest to your intuition; however, for those readers that want to see the details, I will provide the bibliographic references: Most finite strings over a finite alphabet are random (se M. Li, P. Vitanyi, An Introduction to Kolmogorov complexity ad its applications. Berlin: Springer, 1993; Cristian Calude: Information and Randomness; an Algorithmic Perspective. Berlin, New York: Springer, 1994); T.I. Lin (“Rough patterns on data”. Foundations of Computing and Decision Sciences 18, 18, 1993, ¾, 225-239) shows that most finite strings have no pattern. The other surprising theorem belongs to J. Piorek (“On generic chaos” Annales Polonici Mathematici 52, 1990, 139-146): Most dynamical systems are chaotic. (In all these cases, ‘most’ has a mathematical meaning.
Evaluate the surprise So, against expectations, both randomness and chaos are not exceptional situations, they are just the typical ones. But the next fact deceives the first one: These theorems are not able to capture the respective typical situations. The theorems are not constructive, they cannot provide means to effectively identify the numerous random strings and the numerous chaotic dynamical systems. To a large extent, the universe of random strings and the universe of chaotic dynamical systems remain far away from what is effectively available.
Another fairy tale: the fractals According to Benoit Mandelbrot, Euclid geometry is the world of manmade objects, while fractals are the creation of nature: clouds, snowflakes, ocean coasts, Brownian motion etc. Why are fractals placed in the world of disorder? Because fractals are invisible, they are obtained by asymptotic processes, i.e. processes consisting of infinitely many steps. You start with most simple Euclidean objects and applying them iteratively the same operation you reach some invisible, but perfectly intelligible objects: the fractals. So, fractals, as objects far away from sensorial, empirical- intuitive perception, are characterized by high complexity, as opposed to the simplicity, low complexity of the most objects in the Euclidean universe. Moreover, their strong connection with the chaotic dynamical systems, as it results from the fact that, in most significant situations, the attractors of a chaotic dynamical system are fractals. By their fractional Hausdorff dimension, fractals are obviously shocking us as a special type of complexity, i.e., disorder whose hidden order is most remarkable: their self-similarity. It is interesting also the way the move from order to disorder, from low to high complexity is not abrupt, it is gradual. This fact suggests that at least some times the move from macroscopic to non-macroscopic is not abrupt, but gradual. But this move involves infinity; the distinction macroscopic-non-macroscopic is essentially related to the distinction finite-infinite.
Zero and its company Let us consider entities such that zero, emptiness, nothingness, absence, silence, stillness, quietness, vacuum, void, naught, non-being, zero-sign (in linguistics), empty set (in set theory).
Zero’s plurality of faces If we take zero as their prototype, we observe its rich semantics: origin of a coordinate system; neutral element in some operations; term of reference in defining the infinitely small; center of symmetry between negative and positive numbers; number that is simultaneously natural, integer, rational, real and complex number, but has no referent in the empirical world.; logical, purely qualitative symbol (in computer science, in logic); symbol of start in various actions; metaphor expressing nothingness, emptiness.
Zero: both simplicity and complexity The semiotic status of “zero” is fundamentally different from the semiotic status of any other natural number 1, 2, 3,… and this fact is one of the reasons it deserves so much attention; here are two examples of authors of deep investigations of zero: Brian Rotman: Signifying nothing. The semiotics of zero. New York: St Martin’s Press, 1987; Robert Kaplan: The nothing that is. A natural history of zero. New York: Oxford University Press, 1999. Zero offers the spectacle of highest simplicity, in some respect: you have nothing to pay; but at the same time of high complexity, in some other respects, for instance the infinite hierarchy of infinitely small functions of successive order: the successive powers of x when x is approaching zero.
Zero and infinite need each other There are reasons to consider zero as forming an organic couple with infinite. The most elementary example relevant in this respect is the sequence 1, 1/2, 1/3, …, 1/n, …; when n is increasing and tends to infinite, 1/n is decreasing and becomes as near as we want to zero. Fundamental notions of limit and convergence of a sequence or of a series; limit, continuity, differentiability and integrability of a function involve the interplay of two quantities, traditionally denoted by the Greek letters epsilon and delta whose relevance follows from their freedom to be as near to zero as we want, in a scenario with infinitely many steps; this scenario is usually called the epsilon-delta approach. Richard Courant and Herbert Robbins (What is Mathematics. Oxford, England: Oxford University Press, 1941) arranged this scenario in a theatrical play with two characters, one o them manipulating the values of epsilon, while the other manipulates the values of delta.
Zero is not from this world Indeed, in the scenario above the quantities epsilon and delta can be as near as we want to zero, but never equal to zero, which remains a Fata Morgana perfectly symmetric with the similar status of infinite. Zero and infinite have no correspondent in the tangible world of our empirical sensorial existence. Henry Wedsworth Longfellow, in his poem “Fata Morgana” (1873) describes the situation: I approach and you vanish away I grasp you, and you are gone But ever by night and by day The melody soundeth on. […] So I wander and wander along And forever before me gleems The shining city of song In the beautiful land of dreams. But when I would enter the gate Of that golden atmosphere It is gone, and I wonder and wait For the vision to reappear.
Zero’s exotic neighbours: Leibniz’s infinitesimals The so-called Archimedes’ axiom states that, given two quantities, each of them, enough multiplied, overcomes the other. It is in agreement with our empirical-sensorial perception. Why should we formulate such apparently trivial things? During almost two thousand years, science spected Archimedes’ axiom. Leibniz was the first to challenge it and obtained, by its negation, the existence of infinitesimals, as quantities that are inferior to any number 1/n (n=1, 2, 3,…), but different from zero. But only in the 20th century infinitesimals obtained a rigorous status within the framework of the so-called non-standard analysis invented by Abraham Robinson. It became clear that infinitesimals are not usual numbers, they are objects of a fictional universe, the non-standard universe, in contrast with the infinitely small and the infinitely large, which belong to the macroscopic universe.
Infinitely many negligible disorders may lead to a new order This happened with Robinson’s non- standard infinitesimals. They were able to help scholars in the field of economics to understand what happens with the exchange economy when the number of participants to this economy is increasing. The idea was to include this economy into a larger one, with infinitely many participants and so, it became a non- standard exchange economy, where the impact of each participant is negligible, while their total impact is very significant. The accumulation of infinitely many local negligible disorders may lead to a new order.
A typology of silence Now let us consider other situations in the family of zero. One of the most interesting is silence, to which a rich literature was dedicated. I remember in this respect the work of Lisa Block de Behar A Rhetoric of Silence and other Selected Writings (Approaches to Semiotics 122, Mouton de Gruyter, 1995)concerned manly with silence in literature, a book by Sergiu Celibidache (Textes et entretiens pour une phenomenologie de la musique. Arles, Actes Sud, 2012) and another one by Elizabeth Sombart (Pour que les sons deviennent musique-to be published), related to silence in music. A lot of questions claim for an answer. What is the relation, the difference between silence and quietness? Between natural silence, in a forest for example, and the human silence? Why, when, for what reasons a human being remains quiet. We remain quiet between two speeches; because we got tired; because we have nothing to say; because we have no partner for a dialogue; because we feel it would be risky to speak (for instance, if we live in a political dictatorship); because we cannot face the cultural, intellectual level of our partners; because we have something to hide; etc.
Wittgenstein’s silence But there is also another reason for which we would like to keep silence. There is a slogan according to which we understand more than we can say; recognizing a human face would not be possible only by its description in words. There is also the Wittgenstein’s famous slogan: Where one cannot speak, thereof must one be silent. But it seems that he ignored the main argument in this respect, due to Niels Bohr (see David Favrholdt’s “Niels Bohr’s views concerning language” Semiotica 94, 1993, 1/2): our macroscopic status stops the competence of human language beyond the macroscopic universe. Obviously, this is not the only reason of failure of human semiosis; high complexity of various kinds may stop our capacity of understanding even within the macroscopic universe. Essentially, what we said about zero remains valid for silence too. Its ambiguity is infinite.