From a set of parts to an indivisible whole. Part I: Operations in a closed mode Leonid Andreev Equicom, Inc., 10273 E Emily Dr, Tucson, AZ 85730, U.S.A. E-mail: [email protected] February 29, 2008 Abstract This paper provides a description of a new method for information processing based on holistic approach wherein analysis is a direct product of synthesis. The core of the method is iterative averaging of all the elements of a system according to all the parameters describing the elements. It appears that, contrary to common logic, the iterative averaging of a system's elements does not result in homogenization of the system; instead, it causes an obligatory subdivision of the system into two alternative subgroups, leaving no outliers. Within each of the formed subgroups, similarity coefficients between the elements reach the value of 1, whereas similarity coefficients between the elements of different subgroups equal a certain constant value of 0>Ω<1. When subjected to iterative averaging, any system consisting of three or more elements of which at least two elements are not completely identical undergo such a process of bifurcation that occurs non- linearly. Successive iterative averaging of each of the forming subgroups eventually provides a hierarchical system that reflects relationships between the elements of an input system under analysis. We propose and discuss a definition of a natural hierarchy that can exist only in conditions of closeness of a system and can be discovered upon providing such an effect onto a system which allows its elements interact with each other based on the principle of self- organization. We show that self-organization can be achieved through an overall and total cross- averaging of a system's elements. We propose an algorithm for performing such cross-averaging through iterative averaging transformations of a system's similarity matrix, wherein the very first of the iterative transformations turns any system under processing into a closed type system that does not allow an addition of new elements or removal of any of its existing systems as it would result in drastic changes as compared to the original state of the input data system. A system subdivision into groups occurring in the course of iterative averaging performed in an autonomous unsupervised mode displays a highly intelligent analysis of part-whole relations within the system, which proves that the resulting hierarchical structures reflect the system's natural hierarchy. This method for data processing, named by us 'matrix reasoning', can be effectively utilized for analysis of any kind and any combination of data. We demonstrate new methods for construction of hierarchical trees, dendrograms, and iso-hierarchical structures which allow effective visualization of results of a hierarchical analysis in the form of a holistic picture. We demonstrate the application potentials of the proposed technology on a number of examples, including a system of scattered points, randomized datasets, as well as meteorological and demographical datasets. Keywords: Iterative averaging algorithm, Nonlinearity, Holism, Natural hierarchy, Similarity matrix, Metrics, Scattered points, Random systems, Meteorology, Demography 1 about Nature's objects and phenomena still fail to 1. Introduction provide a holistic picture of the world. Extreme Part-whole relations are one of the structuring forms of the concept of holism exist largely due bases of the universe. One may assume that since to unavailability of scientifically grounded the ancient times the problem of 'part-whole' methods – or even ideas that would promise a relations has been most stimulating for potential capability of development of such development of philosophical understanding of methods – for synthesis of a whole which could the nature of human-being and the environment. provide that a resulting whole, rather than being a The principle of approaching a whole from the sum of the component elements, would acquire standpoint of its parts and treating the properties new properties that were not present in the of a whole as the sum of properties of its parts is component elements. known as merism (from the Greek 'meros' , 'part') Analysis of the behavior of parts from the and is the subject of studies in mereology [1-3]. viewpoint of the whole is not characteristic of the Alternatively, a position which emphasizes the classical science. In classical science, analysis is inequality between a whole and the sum its parts based on breaking down a phenomenological because a whole, due to its parts, acquires new whole into its parts and examining the parts, and properties as compared to its parts, is known as this reductionist process is not complemented holism (from the Greek word 'holos' , 'whole') [4- with a reverse process – from parts to a whole – 5]. Holism is based on the idea that all properties and therefore it does not provide an integral of a given system cannot be determined or picture of a system under analysis. The problem explained by the sum of its component parts of relationships between real, identifiable subjects alone. Instead, the system as a whole determines and the appearances whose emergence in an important way how the parts behave. supposedly involves participation of those Holism as a philosophical paradigm is aimed at a subjects seems to be the most critical and "holistic perception of the world", i.e. at resolving complicated issue of the modern scientific the conflict between the subjective and objective, knowledge. As was emphasized by Craig Dilworth between irrational and rational. The holistic [6], "The debate over empiricism and realism approach concerns all the areas of philosophy as concerns the very nature of modern science: what it deals with the general principles of scientific it is or what it ought to be. Empiricism, in its discovery of knowledge. 'Wholeness' as a display extreme form, claims that there is no reality of the properties of a subject under investigation, behind appearances and that it is the task of hence the entire cognizable world, is viewed by science to determine what the appearances are holistic science not as something that directly and and what the formal relation are that obtain obviously follows from interrelations between the among them." A functional whole may contain elements of a system, but as something that is any kind of elements (elements are parts of a manifested in the existence of specific and stable system which do not consist of subsystems, and properties of such wholeness. Holism maintains the notion of 'element' includes also peculiarities that one of the fundamentally important of an element's interactions with other elements properties of a whole is nonreducibility of the of the whole), even something that is unknown to properties of a whole to the properties of its science and lives and evolves on its own, component parts, which is in contrast to analytic independently from us, the humankind. tradition of establishing the properties of a whole Notwithstanding the overwhelming amount of through analysis of cause-and-effect relations and publications on philosophical understanding of relationships between the parts of a whole which the problem of 'whole-part' from the standpoint have a fixed set of properties. From the of holism, science does not know of mathematical standpoint of holism, the requirement of logical ideas and exact methods that would be able to deducibility of properties of a whole from initially demonstrate, on the quantitative level, the set conditions, as it is maintained by traditional relations between parts which underlie the scientific methodologies, is the reason of why all functioning of an integral indivisible whole. This the diverse and extensive domains of knowledge problem is profoundly important and hardly 2 solvable in general from the position of linear artificial and natural intelligence, risk assessment, logic prevailing in science. Kurt Koffka, one the modeling of unpredictable situations, catastrophe classics of Gestalt psychology, wrote, "It has been theory, statistics and economics, medical and said: The whole is more than the sum of its parts. social psychology, including collective behavior, It is more correct to say that the whole is quantum physics, astrophysics, and many others. something else than the sum of its parts, because Solutions for many critical problems of the summing up is a meaningless procedure, whereas nowadays science directly depend on novel the whole-part relationship is meaningful." [7]. approaches to holistic processing of information. Eighty years ago, Jan Smuts, who coined the term There has to be a certain universal methodology 'holism' and made an important contribution to providing a capability to objectively evaluate a the philosophy of holism, outlined the problem given set of elements from the point of view of its that would be faced by the science were it to ability to become – upon certain variations in attempt a description of holism: "This is … the conditions of their interactions – a non-additive, case where cell a unites with cell b to form a new indivisible, specific entity, i.e. a phenomenon entity, in which both a and b disappear finally and whose scientific analysis allowed for distinction of irrecoverably, and whose character and behaviour those elements. Currently, a determination on cannot be traced mathematically or mechanically how relevant is a given set of subjects in the to those of a and b." [8]. It is "…impossible to say emergence of an indivisible whole under analysis where the whole ends and the parts begin, so is made by using specific and subjective intimate is their interaction and so profound their approaches that are based on and result from a mutual influence" [9]. combination of such factors as experience, Due to its universality, the problem of whole- opinions and beliefs of the analyst or a team of part relations is inexhaustible for analysis and analysts, as well as methods in mathematical understanding.