The answer to some of the longstanding issues in the 20th century theoretical physics, such as those of the incompatibility between general relativity and quantum mechanics, the broken symmetries of the electroweak force acting at the subatomic scale and the missing mass of Higgs particle, and also those of the cosmic singularity and the black matter and energy, appear to be closely related to the problem of the quantum texture of space-time and the fluctuations of its underlying geometry. Each region (...) of space landscape seem to be filled with spacetime weaved and knotted networks, for example, spacetime has immaterial curvature and structures, such as topological singularities, and obeys the laws of quantum physics. Thus, it is filled with potentialparticles, pairs of virtual matter and anti-matter units, and potential properties at the quantum scale. For example, quantum entities (like fields and particles) have both wave (i.e., continuous) and particle (i.e., discrete) properties and behaviors. At the quantum level (precisely, the Planck scale) of space-time such properties and behaviors could emerge from some underlying (dynamic) phase space related to some field theory. Accordingly, these properties and behaviors leave their signature on objects and phenomena in the real Universe. In this paper we consider some conceptual issues of this question. (shrink)
Let us start by some general definitions of the concept of complexity. We take a complex system to be one composed by a large number of parts, and whose properties are not fully explained by an understanding of its components parts. Studies of complex systems recognized the importance of “wholeness”, defined as problems of organization (and of regulation), phenomena non resolvable into local events, dynamics interactions in the difference of behaviour of parts when isolated or in higher configuration, etc., in (...) short, systems of various orders (or levels) not understandable by investigation of their respective parts in isolation. In a complex system it is essential to distinguish between ‘global’ and ‘local’ properties. Theoretical physicists in the last two decades have discovered that the collective behaviour of a macro-system, i.e. a system composed of many objects, does not change qualitatively when the behaviour of single components are modified slightly. Conversely, it has been also found that the behaviour of single components does change when the overall behaviour of the system is modified. There are many universal classes which describe the collective behaviour of the system, and each class has its own characteristics; the universal classes do not change when we perturb the system. The most interesting and rewarding work consists in finding these universal classes and in spelling out their properties. This conception has been followed in studies done in the last twenty years on second order phase transitions. The objective, which has been mostly achieved, was to classify all possible types of phase transitions in different universality classes and to compute the parameters that control the behaviour of the system near the transition (or critical or bifurcation) point as a function of the universality class. This point of view is not very different from the one expressed by Thom in the introduction of Structural Stability and Morphogenesis (1975). It differs from Thom’s program because there is no a priori idea of the mathematical framework which should be used. Indeed Thom considers only a restricted class of models (ordinary differential equations in low dimensional spaces) while we do not have any prejudice regarding which models should be accepted. One of the most interesting and surprising results obtained by studying complex systems is the possibility of classifying the configurations of the system taxonomically. It is well-known that a well founded taxonomy is possible only if the objects we want to classify have some unique properties, i.e. species may be introduced in an objective way only if it is impossible to go continuously from one specie to another; in a more mathematical language, we say that objects must have the property of ultrametricity. More precisely, it was discovered that there are conditions under which a class of complex systems may only exist in configurations that have the ultrametricity property and consequently they can be classified in a hierarchical way. Indeed, it has been found that only this ultrametricity property is shared by the near-optimal solutions of many optimization problems of complex functions, i.e. corrugated landscapes in Kauffman’s language. These results are derived from the study of spin glass model, but they have wider implications. It is possible that the kind of structures that arise in these cases is present in many other apparently unrelated problems. Before to go on with our considerations, we have to pick in mind two main complementary ideas about complexity. (i) According to the prevalent and usual point of view, the essence of complex systems lies in the emergence of complex structures from the non-linear interaction of many simple elements that obey simple rules. Typically, these rules consist of 0–1 alternatives selected in response to the input received, as in many prototypes like cellular automata, Boolean networks, spin systems, etc. Quite intricate patterns and structures can occur in such systems. However, what can be also said is that these are toy systems, and the systems occurring in reality rather consist of elements that individually are quite complex themselves. (ii) So, this bring a new aspect that seems essential and indispensable to the emergence and functioning of complex systems, namely the coordination of individual agents or elements that themselves are complex at their own scale of operation. This coordination dramatically reduces the degree of freedom of those participating agents. Even the constituents of molecules, i.e. the atoms, are rather complicated conglomerations of subatomic particles, perhaps ultimately excitations of patterns of superstrings. Genes, the elementary biochemical coding units, are very complex macromolecular strings, as are the metabolic units, the proteins. Neurons, the basic elements of cognitive networks, themselves are cells. In those mentioned and in other complex systems, it is an important feature that the potential complexity of the behaviour of the individual agents gets dramatically simplified through the global interactions within the system. The individual degrees of freedom are drastically reduced, or, in a more formal terminology, the factual space of the system is much smaller than the product of the state space of the individual elements. That is one key aspect. The other one is that on this basis, that is utilizing the coordination between the activities of its members, the system then becomes able to develop and express a coherent structure at a higher level, that is, an emergent behaviour (and emergent properties) that transcends what each element is individually capable of. (shrink)
The physicist's conception of space-time underwent two major upheavals thanks to the general theory of relativity and quantum mechanics. Both theories play a fundamental role in describing the same natural world, although at different scales. However, the inconsistency between them emerged clearly as the limitation of twentieth-century physics, so a more complete description of nature must encompass general relativity and quantum mechanics as well. The problem is a theorists' problem par excellence. Experiment provide little guide, and the inconsistency mentioned above (...) is an important problem which clearly illustrates the intermingling of philosophical, mathematical, and physical thought. In fact, in order to unify general relativity with quantum field theory, it seems necessary to invent a new mathematical framework which will generalise Riemannian geometry and therefore our present conception of space and space-time. Contemporary developments in theoretical physics suggest that another revolution may be in progress, through which a new kind of geometry may enter physics, and space-time itself can be reinterpreted as an approximate, derived concept. The main purpose of this article is to show the great significance of space-time geometry in predetermining the laws which are supposed to govern the behaviour of matter, and further to support the thesis that matter itself can be built from geometry, in the sense that particles of matter as well as the other forces of nature emerges in the same way that gravity emerges from geometry. Scientific research is not a process of steady accumulation of absolute truths, which has culminated in present theories, but rather a much more dynamic kind of process in which there are no final theoretical concepts valid in unlimited domains. (David Bohm). (shrink)
This book proposes a new phenomenological analysis of the questions of perception and cognition which are of paramount importance for a better understanding of those processes which underlies the formation of knowledge and consciousness. It presents many clear arguments showing how a phenomenological perspective helps to deeply interpret most fundamental findings of current research in neurosciences and also in mathematical and physical sciences.
The history and philosophy of science are destined to play a fundamental role in an epoch marked by a major scientific revolution. This ongoing revolution, principally affecting mathematics and physics, entails a profound upheaval of our conception of space, space–time, and, consequently, of natural laws themselves. Briefly, this revolution can be summarized by the following two trends: by the search for a unified theory of the four fundamental forces of nature, which are known, as of now, as gravity, electromagnetism, and (...) strong and weak nuclear forces; by the search for new mathematical concepts capable of elucidating and therefore explaining such a relationship. In fact, the first search is essentially dependent on the second; that is to say, that in order for a new theory of physics to come to light, the development of a deeper geometric theory capable of explaining the structure of space–time on a quantum scale appears to be necessary. On careful consideration, we notice that both of these developments converge in the direction of a unitary and fundamental tendency of modern science—which is the geometrization of theoretical physics and of natural sciences. This new emergent situation carries within it a profound conceptual change, affecting the way in which relations are conceived of, first and foremost, between mathematics and physics. This new paradigm can be summed up by the intimately interdependent points: the immense variety of physical phenomena and of natural forms follows from the equally infinite variety of geometric and topological objects that can be made out in space and from which space is made up; the second point, which ensues from the former one and which is of great historical and epistemological significance, is that mathematics is involved in rather than applied to phenomena. In other words, phenomena are effects that emerge from the geometrical structure of space–time. There is no doubt that this new conception of the relationship between the universe of mathematical ideas and objects and the world of natural phenomena is the true scientific revolution of our century, of great conceptual importance, and consequently, capable of changing our view of science and of nature at one and the same time. It is all at once of a scientific, philosophical and aesthetic order. (shrink)
According to Kant, the axioms of intuition, i.e. space and time, must provide an organization of the sensory experience. However, this first orderliness of empirical sensations seems to depend on a kind of faculty pertaining to subjectivity, rather than to the encounter of these same intuitions with the real properties of phenomena. Starting from an analysis of some very significant developments in mathematical and theoretical physics in the last decades, in which intuition played an important role, we argue that nevertheless (...) intuition comes into play in a fundamentally different way to that which Kant had foreseen: in the form of a formal or “categorical” yet not sensible intuition. We show further that the statement that our space is mathematically three-dimensional and locally Euclidean by no means follows from a supposed a priori nature of the sensible or subjective space as Kant claimed. In fact, the three-dimensional space can bear many different geometrical and topological structures, as particularly the mathematical results of Milnor, Smale, Thurston and Donaldson demonstrated. On the other hand, it has been stressed that even the phenomenological or perceptual space, and especially the visual system, carries a very rich geometrical organization whose structure is essentially non-Euclidean. Finally, we argue that in order to grasp the meaning of abstract geometric objects, as n-dimensional spaces, connections on a manifold, fiber spaces, module spaces, knotted spaces and so forth, where sensible intuition is essentially lacking and where therefore another type of mathematical idealization intervenes, we need to develop a new form of intuition. (shrink)
Nous analysons le développement mathématique et la signification épistémologique du mouvement de géométrisation de la physique théorique, à partir des travaux fondamentaux d’E. Cartan et de H. Weyl jusqu’aux théories de jauge non-abéliennes récentes. Le principal propos de cet article est d'étudier ces développements qui ont été inspirés par les tentatives de résoudre l'un des problèmes centraux de la physique théorique au siècle dernier, c’est-à-dire comment arriver à concilier la relativité générale et la théorie quantique des champs dans un cadre (...) théorique unitaire du monde physique. Ces développements ont produit un changement conceptuel profond concernant tout particulièrement la façon de concevoir les rapports entre structures mathématiques et phénomènes physiques. Nous mettons l’accent sur les points suivants : plutôt que de simplement s’appliquer aux phénomènes, les mathématiques sont impliquées dans leur constitution, autrement dit, les phénomènes sont autant d’effets qui émergent de la structure géométrique de l’espace-temps ; la structure géométrique et topologique de l’espace-temps est à l’origine de la dynamique de ce dernier; les symétries « internes » dictent les différentes interactions entre forces et entre particules ; l’invariance de jauge est un principe « universel » régissant les forces fondamentales et les interactions entre les champs de matière. (shrink)
On May 11th a round table discussion was held on the subject "The Interactions of Science and Art under the Conditions of the Revolution in Science and Technology ," organized by the editorial boards of the journals Voprosy filosofii and Voprosy literatury.