Topological Ontology and Logic of Qualitative quantity

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According to Paterson, “in Hegelian logic, the Concept is self-identical in its three moments: roughlyspeaking, the thinking through of any one of them brings to light the others.” Thus the thinking through “quality” brings to light the “quantity” and in reverse, the thinking of “quantity” brings us to “quality”. Paterson concludes that “since the intuitions are the concept of mathematics as Particularity, the Concept (in the Hegelian approach) will produce these “intuitions” underlying formal systems as its own and explicitly show how it is self-identical in them. In the process, the relations between these intuitions will be made explicit. In formal logic, by contrast, the intuitions just lie there unconnected.” Paterson progresses with the discussion on Hegel’s infinities, pointing out that “in this selfidentity of the Concept in its particularization (and its other moments) we have the full ``genuine infinite'' (das wahrhafte Unendliche) of the Concept contrasting with the ``bad infinite'' (die schlechte Unendlichkeit), the latter being the finitized form of the former in which is endlessly posited the same repeating process in the impossible task of `àttaining'' the genuine infinite. The correct picture of the genuine infinite is, for Hegel, that of a circle that closes back on itself while that of the bad infinite is that of an infinite line which goes on ``for ever'', or of an infinite sequence (in Hegelian terms, an infinite progress (Progreß)) that never stops.” The notion of “genuine infinity” and the “bad infinity” is discussed in detail by Hegel in the chapter on Dasein in the Wissenschaft der Logik. Paterson states that “in the externalizing of formal systems, we are ultimately trapped in the bad infinite almost by definition - the genuine infinite cannot be expressed in terms of sequences of meaningless wff's linking onto one another, the allowable linkings being checked mechanically. The conceptual `ìn-itself'' of the system, driving the system, which is a genuine infinite must eventually make explicit the bad infinite of the formal system, the latter's incongruity with its concept.” Later on Paterson links the “bad infinity” with the the methods of mathematical logic itself – “The explicit exhibiting of this bad infinite - coming out of the methods of mathematical logic itself - is to be found in the fourth inadequacy discussed in § 5 (from Paterson’s paper), that concerning Gödel's second incompleteness theorem.” Paterson explains how “a formal system tries to investigate itself in its own terms. This is nothing other than the reflexivity of the genuine infinite of Concept”. The failure of the system to prove its own consistency is nothing but the illustration of the thesis that “the bad infinite of the formal system is not true (= not genuine in the Hegelian sense) - it cannot reach back on itself to establish that it does not contain a contradiction. The bad infinite in the sense of Hegel is evident in the never stoping attempts and sequence of formal systems to prove its own consistency. According to Paterson, Hegelian logic provides an explanation for the five inadequacies of formal systems, which inadequacies Paterson himself discussed in § 5 of his paper. With the help of Hegelian logic, Paterson establishes that “these inadequacies are not to be treated as difficulties to be lamented but rather understood positively, as asserting the reflexivity and infinitude of mathematics. As such, the formalism of mathematical logic is restored, not adopting a pose external to mathematics and putting it to rights, as it were, but as part of the activity of mathematics itself which it presupposes and from whose meaningfulness it is derived.” 50
For Paterson, the “true mathematical logic is the self-determining of the concept of mathematics, in which one starts at the simplest, most abstract stage of intuitive mathematics and teases out what it implicitly contains in a logical development ( Entwicklung). In Hegelian logic, the starting point is, of course, abstract, indeterminate Being. A major difference from the formal approach to logic is that such a development is conceptual in character, the bringing out explicitly of the riches implicit in mathematical concepts and their interconnection.” Thinking of the centre of the debate between Gadamer’s hermeneutics and Alain Badiou’a attack on the core of dialectical hermeneutics – the human finitude”, we may follow Alan Paterson’s claim about “the conceptual finitude of the formal system.” This conceptual finitude, as determined in mathematical form, for Paterson is a mathematical question whose resolution required the genius of Gödel and, of course, a great deal of formal proof and calculation.” This conceptual finitude is not only mathematical question or hermeneutical question, but dialectical question in Hegelian sense. The answer to this question of the conceptual finitude will be probably given by the same dialectics, for which Gadamer expected to retrieve itself in hermeneutics and also by the hermeneutics that may retrieve itself in the language and logic of mathematic and topology in particular. For Paterson “such a development requires much detailed argument, and the classical Hegelian logic on which it is based will need to be substantially adapted and supplemented.” In the conclusion of his paper, Paterson provides brief indication how Hegel's logic contains the key to three of the difficult problems of the philosophy of mathematics: “these are how to deal with 1) the semantic paradoxes (such as the classical liar paradox), 2) the logical paradoxes (such as the Russell-Zermelo paradox) and 3) the applicability of mathematics. These all center round the infinite in one form or another, whether in the form 1) of selfreflexive statements,47 or 2) of ``large'' infinite sets or 3) of the axiom of infinity (as in Russell's brand of logicism).” For Paterson, “these have their resolution in the three major transitions of the infinite in Hegel's treatment of Being (Sein). The first is the infinite qualitative progress (der qualitative unendliche Progreß) in the Section on Quality, the second is the infinite quantitative progress (der quantitative unendliche Progreß) in the Section on Quantity and the third is the infinite of the specification of measure (die Unendlichkeit der Spezifikation des Maßes) in the Section on Measure. In conclusion, Paterson claim is that “the reflexivity, effectively asserted in the failure of the formal approach to mathematics, poses the need for a framework which only the logic of Hegel has the resources to cope with. Hegel's method of inquiry in terms of a conceptual development is the correct method for investigating the human activity that is called mathematics simply because it is intrinsic and lets the subject develop itself (as a genuine infinite, a circle closed on itself) rather than assuming an external `òbjective'' pose in which it is put in front of the investigator and 51
Page 1 and 2: Topological Ontology and Logic of Q
Page 3 and 4: agreements about the representation
Page 5 and 6: Science” (Wissenschaftslehre), Bo
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
Page 49: Hegel's great work on Logic, the
Page 53 and 54: The term “social topology” is u
Page 55 and 56: of Leonard Euler and later with the
Page 57: To Meet the Other is to have the id

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