Topological Ontology and Logic of Qualitative quantity

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Buffalo. 8 Barry Smith and Pierre Grenon initiated the Basic Formal Ontology (BFO) 9 Project in 2002 with a series of publications. 10 The inspirations of Basic Formal Ontology are Aristotle, Husserl, Roman Ingarden, Ingvar Johansson, Wittgenstein’s Tractatus (picture theory of language). In his paper “Mereotopology: A Theory of Parts and Boundaries” 11 , remarkable contribution to formal ontology, Barry Smith use topological means in order to derive ontological laws pertaining to the boundaries and interiors of wholes, to relations of contact and connectedness, to the concepts of surface, point, neighbourhood. and so on. The basis of the theory is mereology, the formal theory of part and whole, a theory which is shown to have a number of advantages for ontological purposes over standard treatments of topology. The roots of topology in Husserl can be found in the forerunners of the school of formal ontology (founded by Husserl 12 ) - Bernard Bolzano and Franz Brentano 13 , within the origins of the main traditions of contemporary ontology: the Phenomenological, the Analytical, and the Austro-Polish (with the main exponents as Adolf Reinach, Roman Ingarden and Nicolai Hartmann). During the period from 1884 through 1887, Husserl attended the lectures of Franz Brentano. Under the mentorship of Brentano, Husserl came to view philosophy as complementary to science. Husserl became concerned with linking mathematics and philosophy. Husserl’s first text, The Philosophy of Arithmetic, was published in 1891 with a dedication to Brentano. After Kant's rejection of the possibility of a general ontology 14 , the first philosopher who contributed to the new ontological turn was Bernard Bolzano. In his 1837 “Theory of 7 Barry Smith and Kevin Mulligan, "Husserl's Logical Investigations," Grazer Philosophische Studien 27: 199- 206 (1986). 8 Barry Smith is also one of the principal scientists of the National Center for Biomedical Ontology (NCBO), an NIH Roadmap National Center for Biomedical Computing, a member oft he Scientific Advisory Board of the Gene Ontology Consortium, and a PI of the Protein Ontology and Infectious Disease Ontology projects. He is a Coordinating Editor of the OBO Foundry initiative, and plays a guiding role in Basic Formal Ontology and the Ontology for General Medical Science, the Environment Ontology, and the Plant Ontology initiatives. 9 See: http://www.ifomis.org/bfo 10 See: (http://www.ifomis.org/bfo/publications). 11 Barry Smith, “Mereotopology: A Theory of Parts and Boundaries”, from: “Mereotopology: A Theory of Parts and Boundaries”, Data and Knowledge Engineering, 20 (1996), 287–303: http://ontology.buffalo.edu/smith/courses03/tb/SmithMereotopology1.pdf 12 E. Husserl, Formal and Transcendental Logic (1929), English translation: The Hague: Martinus Nijhoff, 1969, 27, p. 86.): “The idea of a formal ontology makes its first literary appearance in Volume I of my Logische Untersuchungen (1900), [Chapter 11, The Idea of Pure Logic] in connection with the attempt to explicate systematically the idea of a pure logic -- but not yet does it appear there under the name of formal ontology, which was introduced by me only later. The Logische Untersuchungen as a whole and, above all, the investigations in Volume II ventured to take up in a new form the old idea of an apriori ontology -- so strongly interdicted by Kantianism and empiricism -- and attempted to establish it, in respect of concretely executed portions, as an idea necessary to philosophy." 13 Brentano, Franz 1988 Philosophical Investigations on Space, Time and the Continuum, translated by Barry Smith, London/New York/Sydney: Croom Helm. 14 I. Kant, Critique of Pure Reason (A247/B304), Cambridge: Cambridge University Press, 1998, pp. 358- 359 : "The Transcendental Analytic accordingly has this important result: That the understanding can never accomplish a priori anything more than to anticipate the form of a possible experience in general, and, since that which is not appearance cannot be an object of experience, it can never overstep the limits of sensibility, within which alone objects are given to us. Its principles are merely principles e of the exposition of appearances, and the proud name of an ontology, which presumes to offer synthetic a priori cognition of things in general in a systematic doctrine (e.g., the principle of causality), must give way to the modest one of a mere analytic of the pure understanding." 4
Science” (Wissenschaftslehre), Bolzano attempted to provide logical foundations for all sciences, building on abstractions like part-relation, abstract objects, attributes, sentenceshapes, ideas and propositions in themselves, sums and sets…The logical theory of Bolzano developed in his work “Theory of Science” has come to be acknowledged as groundbreaking. Bolzano was mainly concerned with three realms: (1) The realm of language, consisting in words and sentences; (2) The realm of thought, consisting in subjective ideas and judgements; (3) The realm of logic, consisting in objective ideas (or ideas in themselves) and propositions in themselves. The distinction between parts and wholes play a prominent role in Bolzano’s system. Bolzano's work “The Paradoxes of the Infinite” was greatly admired by Charles Sanders Peirce, Georg Cantor and Richard Dedekind. Bolzano’s works was rediscovered by Edmund Husserl 15 and the Polish philosopher and logician Kazimierz Twardowski 16 , both students of Franz Brentano. Through Husserl and Twardowski , Bolzano became a formative influence on both phenomenology and analytic philosophy. The influence of Bolzano on Heidegger is witnessed by Heidegger himself. 17 The first work of Brentano On the Several Senses of Being in Aristotle (1862) and the Logical Investigations (1900) by Husserl were at the origin of the interest in philosophy of the most authoritative exponent of Continental ontology, Martin Heidegger. If the place of ontology in modern philosophy is naturally topological, same is valid for the critical categories of ontology – quality and quantity. Ontology is the theory of objects and their ties. Topology, as a branch of mathematics, can be formally defined as "the study of qualitative properties of certain objects (called topological spaces) that are invariant under a certain kind of transformation (called a continuous map), especially those properties that are invariant under a certain kind of equivalence (called homeomorphism)." To put it more simply, topology is the study of continuity and connectivity. Topology is concerned only with the geometric relationship between the objects (figures) or elements of a network, not with the kind of objects or elements themselves. Ontology provides criteria for distinguishing different types of objects (concrete and abstract, existent and nonexistent, real and ideal, independent and dependent) and their ties (relations, dependencies and predication). We can distinguish: a) formal, b) descriptive and c) formalized ontologies. 15 Hermes Scholz, Concise History of Logic (1931), English translation: New York: Philosophical Library, 1961, p. 47. : "With such illogicality did things happen in the history of logic which we are pursuing here that this great, born logician fell prey to a fate which beats the fate of Joachim Jungius. For the latter at least was read, and read by a Leibniz; but that cannot even be said of Bolzano. Hence we cannot even maintain in his case that he was forgotten. All the greater is the merit of Edmund Husserl who discovered Bolzano." 16 the founder of the Lvov-Warsaw School of logic, together with Alfred Tarski and Jan Lukasiewicz, Kazimierz Twardowski formed the troika which made the University of Warsaw, during that period, the most important research center in the world for formal logic. 17 Martin Heidegger, Preface to: William Richardson, Heidegger. Through Phenomenology to Thought, The Hague: Martinus Nijhoff, 1963. p. X.: "The first philosophical text through which I worked my way, again and again from 1907 on, was Franz Brentano's dissertation: On the Manifold Sense of Being in Aristotle." 5
Page 1 and 2: Topological Ontology and Logic of Q
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Page 7 and 8: Leśniewski's student Alfred Tarski
Page 9 and 10: Mereotopology is a first-order theo
Page 11 and 12: only that between process and event
Page 13 and 14: hierarchies to represent their mutu
Page 15 and 16: meta attribute value counting_valu
Page 17 and 18: Fig. 2 Detail version of top-level
Page 19 and 20: differentiation enables us to captu
Page 21 and 22: 3.Quality: It is a property associa
Page 23 and 24: 3. The Topological Notion of Qualit
Page 25 and 26: series of Measure Relations” /b/,
Page 27 and 28: The true resurgence of Hegel’s Qu
Page 29 and 30: Kairos, Chora and Topos as follow:
Page 31 and 32: In Philosophical or Dialectical Her
Page 33 and 34: The roots of Hegel’s Qualitative
Page 35 and 36: Richter mentions these developments
Page 37 and 38: Name: Firstness. Secondness. Thirdn
Page 39 and 40: of something in its other. In his e
Page 41 and 42: This is not to say, however, that t
Page 43 and 44: One of the main concerns of hermene
Page 45 and 46: concept of spatiality. But they did
Page 47 and 48: David Gray Carlson’s “A Comment
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Page 51 and 52: For Paterson, the “true mathemati
Page 53 and 54: The term “social topology” is u
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of Leonard Euler and later with the
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To Meet the Other is to have the id
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