This volume offers a fundamentally different way of conceptualizing time and reality. Today, we see time predominantly as the linear-sequential order of events, and reality accordingly as consisting of facts that can be ordered along sequential time. But what if this conceptualization has us mistaking the "exhausts" for the "real thing", i.e. if we miss the best, the actual taking place of reality as it occurs in a very differently structured, primordial form of time, the time-space of the present? In this new conceptual framework, both the sequential aspect of time and the factual aspect of reality are emergent phenomena that come into being only after reality has actually taken place. In the new view, facts are just the "traces" that the actual taking place of reality leaves behind on the co-emergent "canvas'' of local spacetime. Local spacetime itself emerges only as facts come into being - and only facts can be adequately localized in it. But, how does reality then actually occur? It is conceived as a "constellatory self-unfolding", characterized by strong self-referentiality, and taking place in the primordial form of time, the not yet sequentially structured "time-space of the present". Time is seen here as an ontophainetic platform, i.e. as the stage on which reality can first occur. This view of time (and, thus, also space) seems to be very much in accordance with what we encounter in quantum physics before the so-called collapse of the wave function. In parallel, classical and relativistic physics largely operate within the factual portrait of reality, and the sequential aspect of time, respectively. Only singularities constitute an important exemption: here the canvas of local spacetime - that emerged together with factization - melts down again. In the novel framework quantum reduction and singularities can be seen and addressed as inverse transitions: In quantum physical state reduction reality "gains" the chrono-ontological format of facticity, and the sequential aspect of time becomes applicable. In singularities, by contrast, the inverse happens: Reality loses its local spacetime formation and reverts back into its primordial, pre-local shape - making in this way the use of causality relations, Boolean logic and the dichotomization of subject and object obsolete. For our understanding of the relation between quantum and relativistic physics this new view opens up fundamentally new perspectives: Both are legitimate views of time and reality, they just address very different chrono-ontological portraits, and thus should not lead us to erroneously subjugating one view under the other. The task of the book is to provide a formal framework in which this radically different view of time and reality can be addressed properly. The mathematical approach is based on the logical and topological features of the Borromean Rings. It draws upon concepts and methods of algebraic and geometric topology - especially the theory of sheaves and links, group theory, logic and information theory, in relation to the standard constructions employed in quantum mechanics and general relativity, shedding new light on the pestilential problems of their compatibility. The intended audience includes physicists, mathematicians and philosophers with an interest in the conceptual and mathematical foundations of modern physics.

This unique book provides a self-contained conceptual and technical introduction to the theory of differential sheaves. This serves both the newcomer and the experienced researcher in undertaking a background-independent, natural and relational approach to 'physical geometry'. In this manner, this book is situated at the crossroads between the foundations of mathematical analysis with a view toward differential geometry and the foundations of theoretical physics with a view toward quantum mechanics and quantum gravity. The unifying thread is provided by the theory of adjoint functors in category theory and the elucidation of the concepts of sheaf theory and homological algebra in relation to the description and analysis of dynamically constituted physical geometric spectrums.

If there is a central conceptual framework that has reliably borne the weight of modern physics as it ascends into the twenty-first century, it is the framework of quantum mechanics. Because of its enduring stability in experimental application, physics has today reached heights that not only inspire wonder, but arguably exceed the limits of intuitive vision, if not intuitive comprehension. For many physicists and philosophers, however, the currently fashionable tendency toward exotic interpretation of the theoretical formalism is recognized not as a mark of ascent for the tower of physics, but rather an indicator of sway-one that must be dampened rather than encouraged if practical progress is to continue. In this unique two-part volume, designed to be comprehensible to both specialists and non-specialists, the authors chart out a pathway forward by identifying the central deficiency in most interpretations of quantum mechanics: That in its conventional, metrical depiction of extension, inherited from the Enlightenment, objects are characterized as fundamental to relations-i.e., such that relations presuppose objects but objects do not presuppose relations. The authors, by contrast, argue that quantum mechanics exemplifies the fact that physical extensiveness is fundamentally topological rather than metrical, with its proper logico-mathematical framework being category theoretic rather than set theoretic. By this thesis, extensiveness fundamentally entails not only relations of objects, but also relations of relations. Thus, the fundamental quanta of quantum physics are properly defined as units of logico-physical relation rather than merely units of physical relata as is the current convention. Objects are always understood as relata, and likewise relations are always understood objectively. In this way, objects and relations are coherently defined as mutually implicative. The conventional notion of a history as "a story about fundamental objects" is thereby reversed, such that the classical "objects" become the story by which we understand physical systems that are fundamentally histories of quantum events. These are just a few of the novel critical claims explored in this volume-claims whose exemplification in quantum mechanics will, the authors argue, serve more broadly as foundational principles for the philosophy of nature as it evolves through the twenty-first century and beyond.

This volume offers a fundamentally different way of conceptualizing time and reality. Today, we see time predominantly as the linear-sequential order of events, and reality accordingly as consisting of facts that can be ordered along sequential time. But what if this conceptualization has us mistaking the "exhausts" for the "real thing", i.e. if we miss the best, the actual taking place of reality as it occurs in a very differently structured, primordial form of time, the time-space of the present? In this new conceptual framework, both the sequential aspect of time and the factual aspect of reality are emergent phenomena that come into being only after reality has actually taken place. In the new view, facts are just the "traces" that the actual taking place of reality leaves behind on the co-emergent "canvas'' of local spacetime. Local spacetime itself emerges only as facts come into being - and only facts can be adequately localized in it. But, how does reality then actually occur? It is conceived as a "constellatory self-unfolding", characterized by strong self-referentiality, and taking place in the primordial form of time, the not yet sequentially structured "time-space of the present". Time is seen here as an ontophainetic platform, i.e. as the stage on which reality can first occur. This view of time (and, thus, also space) seems to be very much in accordance with what we encounter in quantum physics before the so-called collapse of the wave function. In parallel, classical and relativistic physics largely operate within the factual portrait of reality, and the sequential aspect of time, respectively. Only singularities constitute an important exemption: here the canvas of local spacetime - that emerged together with factization - melts down again. In the novel framework quantum reduction and singularities can be seen and addressed as inverse transitions: In quantum physical state reduction reality "gains" the chrono-ontological format of facticity, and the sequential aspect of time becomes applicable. In singularities, by contrast, the inverse happens: Reality loses its local spacetime formation and reverts back into its primordial, pre-local shape - making in this way the use of causality relations, Boolean logic and the dichotomization of subject and object obsolete. For our understanding of the relation between quantum and relativistic physics this new view opens up fundamentally new perspectives: Both are legitimate views of time and reality, they just address very different chrono-ontological portraits, and thus should not lead us to erroneously subjugating one view under the other. The task of the book is to provide a formal framework in which this radically different view of time and reality can be addressed properly. The mathematical approach is based on the logical and topological features of the Borromean Rings. It draws upon concepts and methods of algebraic and geometric topology - especially the theory of sheaves and links, group theory, logic and information theory, in relation to the standard constructions employed in quantum mechanics and general relativity, shedding new light on the pestilential problems of their compatibility. The intended audience includes physicists, mathematicians and philosophers with an interest in the conceptual and mathematical foundations of modern physics.

In Natural Communication, the author criticizes the current paradigm of specific goal orientation in the complexity sciences and proposes an alternative that envisions a fundamental architectonics of communication. His model of "natural communication" encapsulates modern theoretical concepts from mathematics and physics, in particular category theory and quantum theory. From these fields it abstracts precise concepts such as to constitute a terminological basis for this theory which offers the opportunity to open up novel ways of thinking about complexity. The author is convinced that it is only possible to establish a continuity and coherence with contemporary thinking, especially with respect to complexity, through looking into the past.

If there is a central conceptual framework that has reliably borne the weight of modern physics as it ascends into the twenty-first century, it is the framework of quantum mechanics. Because of its enduring stability in experimental application, physics has today reached heights that not only inspire wonder, but arguably exceed the limits of intuitive vision, if not intuitive comprehension. For many physicists and philosophers, however, the currently fashionable tendency toward exotic interpretation of the theoretical formalism is recognized not as a mark of ascent for the tower of physics, but rather an indicator of sway-one that must be dampened rather than encouraged if practical progress is to continue. In this unique two-part volume, designed to be comprehensible to both specialists and non-specialists, the authors chart out a pathway forward by identifying the central deficiency in most interpretations of quantum mechanics: That in its conventional, metrical depiction of extension, inherited from the Enlightenment, objects are characterized as fundamental to relations-i.e., such that relations presuppose objects but objects do not presuppose relations. The authors, by contrast, argue that quantum mechanics exemplifies the fact that physical extensiveness is fundamentally topological rather than metrical, with its proper logico-mathematical framework being category theoretic rather than set theoretic. By this thesis, extensiveness fundamentally entails not only relations of objects, but also relations of relations. Thus, the fundamental quanta of quantum physics are properly defined as units of logico-physical relation rather than merely units of physical relata as is the current convention. Objects are always understood as relata, and likewise relations are always understood objectively. In this way, objects and relations are coherently defined as mutually implicative. The conventional notion of a history as "a story about fundamental objects" is thereby reversed, such that the classical "objects" become the story by which we understand physical systems that are fundamentally histories of quantum events. These are just a few of the novel critical claims explored in this volume-claims whose exemplification in quantum mechanics will, the authors argue, serve more broadly as foundational principles for the philosophy of nature as it evolves through the twenty-first century and beyond.