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.