**Authors:** Borislav G. Dimitrov

Preface Even a cursory reading of Hegel’s works is sufficient to convince one that the categories of quality and quantity, the concepts of space, time, place are basic to the system. In contrast with Aristotle and Kant, Hegel’s speculative logic exhibits a presuppositionless derivation of the categories. Hegel regards his categories, concepts and notions not as completed and frozen. For him, the categories must be derived, not from presupposed forms of judgement, or from our presuppositions, but from what he calls the sheer ‘simplicity of thinking’, from the indeterminate being of thought and from the indeterminate thought of being. Hegel’s categories of quality and quantity, in particular ‘qualitative quantity’ in relation with his concepts of space, time, place are subject to which much attention has not been paid by Hegel’s expositors and commentators and yet the importance of it cannot be denied. (Haldar,1932). Topological reading of ‘qualitative quantity’ within Hegel’s fourfold (manifold) of multiplicity is subject that lack attention at all from Hegel’s commentators. This conceptual gap, ontological, epistemological, phenomenological is widening within the ‘linguistic turn’, ‘spatial turn’ or ‘topological turn’ and emerging topological approaches to various fields of social science and their need for deep conceptualization based on the philosophical categories, notions and concepts. As Richard Ek establishes in Theorizing the Earth (2010), “The spatial turn is actually a ‘philosophical turn’ repackaged and promoted as a fundamental concern with question of space, place and polity in the social sciences and humanities.” (Ek, Richard, Mekonnen, T. 2010:49-66) John WP Phillips, who critically examines the recent arguments asserting a ‘topological turn’ in culture, the range of topologically informed interventions in social and cultural theory, remarkes in his paper On Topology (2013) , that such contemporary fashionable notions of ‘topological approaches’ and ‘becoming topological of culture’ “demands a greater critical reflection than the notion of a ‘topological turn’ suggests.” I believe that such demand of critical reflection shall be based on Hegel’s categories of logic, notions and concept, and following my assertion that ‘topological’ in intuitively presented by Hegel in his logic, dialectic and method, with the present thesis I argue how Hegel’s categories of qualitative and quantitative, the concept of space, time and place can provide contemporary researcher with the power and methodology of such a greater critical reflection, and ability to readdress in the new and enhanced mode the variety of these toplogical approaches. The methodology of an applied philosophical topology shall be based on the topological re-reading of Hegel and topological hermeneutics. The main objective of the present thesis is to demonstrate how Hegel’s categories, concepts, language, syntax and semantics, his use of rethorical power exibit topological notions and thus the reading of topological (in) Hegel is open for conceptualization. Perhaps it would not be an exaggeration to say that there is no system of thought more intimately bound up with one fundamental principle than is the system of Hegel. My assertion with the present thesis is that topological reading of Hegel reviels true topological system, thus there are reasonable grounds for us to see the doctrine of Hegel, in particular his Science of Logic and Philosophy of Nature as Hegel’s Analysis Situs. A correct interpretation of Hegel’s system depends upon a thorough comprehension of the categories of quality and quantity, the concepts of space, time, and place. If these categories and concepts are neglected, the system must remain a sealed book. The aim of the present thesis is to unseal the topological character of Hegel’s though, that influenced the formation and presence of topological thinking within the tradition of later continental philosophy and directs toward the horizon of new philosophical topology. The concern of the present thesis is to set forth topological reading and interpretation of such categories as quality and quantiny, in particular the qualitative quantity within Hegel’s forfold of multiplicity and such concepts as space, time, place, to emphasize on the importance of topological (in) Hegel for a theory of knowledge, and, in the light of it, and to give some insight toward the current philosophical topology. The aim of the thesis is to critically examine whether it is methodologically possible to combine mathematical rigor – topology with a systematic dialectical methodology in Hegel, and if so, to provide as result of my interpretation the outline of Hegel’s Analysis Situs, also with the proposed models (build on the topological manifold, cobordism, topological data analysis, persistent homology, simplicial complexes and graph theory, to provide an indication of how the merger of Hegel’s dialectical logic and topology may be instrumental to a systematic logician and of how a systematic dialectical logic perspective may help mathematical model builders. Abstract The present study is interdisciplinary, involving the interrelations between philosophy and topology, where topology is understood in both meanings as mathematical discipline and rhetorical notions, and culminating to what I regard as topological philosophy or philosophical topology, based in particular on Hegel’s notion of multiplicity, implemented in his logic unfolding the true topological fourfold of infinities, where qualitative and quantitative, spatial and temporal, as well as rhetorical notions such as the four basic tropes of rhetoric: ‘metaphor’, ‘metonymy’, ‘synecdoche’, ‘ironi’ are presented in Hegel’s manifold (Mannigfaltigkeit) of infinities, quality and quantity, time and space. Not only these four basic tropes of rhetoric are equally presented in Hegel’s philosophical narratives, but also the emphasis is on the metonymy seen as metonymy of metonymy or ‘metalepsis’, is strongly presented in Hegel’s logic, in particular in his ‘topological’ notion of Qualitative quantity. The study investigate in particular the categories of ‘quality’, ‘quantity’ and ‘measures’ within the notions of ‘multiplicity in Hegel’s dialectical logic, with an emphasis on the topological notion of qualitative quantity - the category that remained ‘inapparent’ within the well know dialectics of transformation of quality to quality, where the new quality appears as leap (nodal line) exhibiting abrupt changes and discontinuous transformation leading to the new measure. The exhibit form of Hegel’s qulitative quantity is related with continuous changes and smooth, gradual, topological transformations. The gradualness of such transformations demonstrates topological homeomorphism as exhibit form of the category of qualitative quantity, which can be successfully implemented in mathematical, indeed topological models and methods, such as topological manifold, cobordism, topological data analysis, persistent homology, simplicial complexes and graph theory. The topological notions of multiplicity in Hegel’s fourfold of infinities is discussed: (1) the bad qualitative infinity; (2) the good qualitative infinity; (3) the bad quantitative infinity; (4) the good quantitative infinity, related with the fourfold interplay of the two pair of Ancient Greek categories of ‘time’ and ‘space’ as fourfold constructed of ‘Chronos’ and ‘Kairos’, ‘Chora’ and ‘Topos’ as follow: (1) chronochora, chronotopos, kairochora and kairotopos - where the four categorical models implements the following: ‘quantitative quantity’ – ‘quantitative quality’ - ‘qualitative quantity’ – ‘qualitative quality’. The claim supported is that ‘quality’ is in the core of ‘quantitative infinity’, ‘infinity’ is a quality of quantity – ‘Qualitative quantity’. There is homology between Hegel’s fourfold of infinities (multiplicity) and the fourfold of Hegel’s judgments (Judgment of Existence; Judgment of Reflection; Judgment of Necessity; Judgment of the Notion). The true topological character of Hegel’s logic is revealed as coherence between Hegel’s fourfold of infinities (multiplicity) and the fourfold of Hegel’s judgments, between method and subject matter, and this topological character is presented in Hegel through the double negation, where the Understanding and its negation, Dialectical Reason, and the Negation of the Negation – Speculative Reason, can be seen in the fourfold of Hegel’s judgments. In the very last chapter of Hegel’s Science of Logic, method and subject matter supposedly conjoin. (Carlson 2005). Following William Lawvere’s assertion (Lawvere, 1996) that a significant fraction of Hegel’s Logic can be modeled mathematically through the use of “cylinders” (diagrams of shape Δ) in a category, wherein the two identical subobjects (united by the third map in the diagram) are “opposite”.about the use of “cylinders” (diagrams of shape Δ), my interpretation of Hegel’s Objective Logic offer methodological modeling of the categories, based on cylinders - the ‘cobordism’, and on the shape Δ - the ‘simplicial complex’ (simplicial complexes). Hegel’s fourfold of infinities is build on the fourfold of quality and quantity ratio – the notions and ratios of quantitative quantity; quantitative quality; qualitative quantity; qualitative quality. The fourfold of the qualitative and quantitative ratios, relate to the fourfold of the measure. All measures are ratios of two other measures. Measure is twofold, divided into two—the external and the internal measure. This is the dialectical real of the real measure. Measure is the unity of quality and quantity, yet in the center of the series of measures is a master signifier that organizes everything, even while escaping measurement. This ‘master signifier’, the empty center between quality and quantity is like the hole of the torus, the nothing, the void, and Hegel names it ‘substrate.’ The substrate is discontinuous within the series of measure and continuous at the same time. The substrate is what Hegel calls a true infinity. Substrate can be organized in a series of measures. Substrate is abstract measureless. What is important to see at this point in relation to measure is entailed in a duality between the nodal relation of quantity and quality, on the one side, and substrate, on the other side. The first side is measure as such—quantity and quality. The second side—the substrate—is something deeper than quantity and quality. The model of Cobordism (Rene Thom) is proposed as topological model of Hegel’s fourfold of infinities within his concept of multiplicity. For Hegel multiplicity means the quantification of quality and the qualification of quantity, multiplicity of the double-entendre implemented in the inevitable double-meaning of Qualitative quantity. For Hegel, the inability to read history as two-sided, as both continuous and discrete, to co-board between, into, within this twofold, to see beings as double, as both qualitative and quantitative, is the road to the destruction of spirit. Trough the ‘Topological’, Hegel’s logic and dialectics are seen in the current. Today, it is Hegel who give us the true, current, topological reply on the question - What does it mean to think multiplicity as both quality and quantity? Topological is subject to philosophical systematization, as well as the philosophical is subject to topological (mathematical) formalization. In the development of philosophy and mathematics (mathematical and philosophical Mind) there is mathesis universalis and some "problematic situations" that have a common "categorial" structure to study them and the "method" of the study can not be other than "mathematical (topological) modeling" or "explication" of philosophical "categories" and together with it – categorial (philosophical) "interpretation" of the mathematical (topological) "structures". Philosophical-topological (mathematical) Mind and mathematical (topological)-philosophical Mind have one and the same "subject", to whom study apply with the two "polar-opposite” of its "form" methods, or studying the same "subject" in the two polar-opposite "forms" through "method", which is inherently "the same". Novel ‘topological’ reading of Hegel’s notions of spatial and temporal, qualitative and quantitative, is proposed as fundamental in re-thinking of the evolution of hierarchical systems. The evolution of hierarchical systems is approached topologically as specific ‘problem situation’ for mathematical and philosophical mind, in the term of Philosophical Analisis Situs, where philosophical aspect is present through the categories of Hegel’s multiplicity – the dialectics and logic of ‘qualitative’ and ‘quantitative’, spatial and temporal, and mathematical aspect is presented through the models, such as Cantor Set, logistic map, bifurcation diagrams and topological notion of Cobordism. In addition to the Hierarchical (vertical evolution) relations, an emphasis on the topological notion of qualitative quantity in Hegel, reveals the role of Heterarchy and heteronomy in Evolution (horizontal evolution), since the exhibit form of Qualitative quantity is related with continuous changes and smooth, gradual, topological transformations. The gradualness of such transformations demonstrates topological homeomorphism as exhibit form of the category of Qualitative quantity, which could be successfully implemented in mathematical, indeed topological models and methods. Hegel’s notions of manifold presented in the model of Rene Thom’s cobordism is discussed and implemented through the logistic map of Feigenbaum bifurcation Diagram accepted as universal scenario of development, change and evolution. The proposed outcome demonstrate that within the interval of chaos, marked in the Feigenbaum Diagram, where the parameter ‘a’ increase over the value of 3.0 in the higher octaves of 4.33 , the ‘voices’ of Hegel and Cantor are present within the region of chaos known as Cantor dust. In the zone of chaos and Cantor dust, Hegel’s multiplicity and four measures works in progress and logic breaths the thin air of being retrieving itself in metaphysic. Beyond the octaves of the values 4.33 in Feigenbaum bifurcation Diagram, within the chaos, there are the heads of Canto comets or the divergence diagrams, where can be found the seeds of the new orders and multiplicity (manifolds) of new bifurcations and Feigenbaum diagrams. The visible ‘white’ corridors of homeostasis are windows open for new hierarchies of order and possibilities of development. Based on the said proposition, the focus of the present paper is on the evolutionary scenario, where in addition to the currently accepted paradigm of hierarchy of evolutionary systems where the core of representation is through the phylogenetic tree structure (vertical evolution), the thesis of reticulate (horizontal evolution) is asserted and discussed as exhibiting the heterarchy and heteronomy of evolutionary systems. The standard evolutionary representation, the phylogenetic tree, and the notion of hierarchy of evolutionary systems, faithfully represents the vertical evolution, but cannot capture horizontal, or reticulate, events, which occur when distinct clades merge together to form a new hybrid lineage. Both hierarchy and heterarchy of the tree structure and the structure of reticulate events could be mathematically investigated, modeled and represented through the field of algebraic topology known as Topological Data Analysis, where the primary mathematical tool considered is a homology theory for point-cloud data sets—persistent homology—and a novel representation of this algebraic characterization— simplicial complex and barcodes. The persistent homology in evolution, which characterizes global properties of a geometric object that are invariant to continuous deformation, such as stretching or bending without tearing or gluing any single part of it, and the properties that includes such notions as connectedness, is the current method of implementation of Hegel’ topological logic of multiplicity and Qualitative quantity.

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