Topological Ontology and Logic of Qualitative quantity

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homology theory. Eilenberg and Mac Lane later wrote that their goal was to understand natural transformations; in order to do that, functors had to be defined, which required categories. Categorical logic is a branch of category theory within mathematics, adjacent to mathematical logic but more notable for its connections to theoretical computer science. In broad terms, categorical logic represents both syntax and semantics by a category, and an interpretation by a functor. The categorical framework provides a rich conceptual background for logical and type-theoretic constructions. The subject has been recognisable in these terms since around 1970. Categorical logic originated with Bill Lawvere’s Functorial Semantics of Algebraic Theories (1963), and Elementary Theory of the Category of Sets (1964). Lawvere recognised Alexander Grothendieck’s topos theory, introduced in algebraic topology as a generalised space, as a generalisation of the category of sets (Quantifiers and Sheaves (1970). With Myles Tierney, Lawvere then developed the notion of elementary topos, thus establishing the fruitful field of topos theory, which provides a unified categorical treatment of the syntax and semantics of higher-order predicate logic. The resulting logic is formally intuitionistic. Andre Joyal is credited, in the term Kripke– Joyal semantics, with the observation that the sheaf models for predicate logic, provided by topos theory, generalise Kripke semantics. Historically, the major irony here is that in large-scale terms, inuiyionistic logic had reappeared in mathematics, in a central place in the Bourbaki-Grothendieck program, a generation after the messy Brouwer-Hilbert controversy had ended, with Hilbert the apparent winner. Bourbaki, or more accurately Jean Dieudonne, having laid claim to the legacy of Hilbert and the Gottingen school including Emmy Noether, had revived intuitionistic logic's credibility (although Dieudonné himself found Intuitionistic Logic ludicrous), as the logic of an arbitrary topos, where classical logic was that of 'the' topos of sets. This was one consequence, certainly unanticipated, of Grothendieck’s relative point of view; and not lost on Pierre Cartier, one of the broadest of the core group of French mathematicians around Bourbaki and IHES. Cartier was to give a Seminaire Bourbaki exposition of intuitionistic logic. The grounds towards the merger between the categories of quality and quantity in the Hegelian Concepts, and the category theory in mathematics and categorical logic, could be found in the series of works of Alan L. T. Paterson, Adjunct Professor of Mathematics, University of Colorado, Boulder, the so called mathematical/Hegelian philosophy papers 113 43 and the works by Dirk Damsma 114 . 113 G. W. F. Hegel: Geometrical Studies - translated with Introduction and Notes, Bulletin of the Hegel Society of Great Britain 57/58, 2008, 118-153.; The Hegelian Concept and set theory, 15 pages, 2007.; A modern Hegelian Philosophy of Special Relativity, 32 pages, 2006.; Hegel's Early Geometry, Hegel Studien 39/40, 2004/2005, 61-124.; Does Hegel have anything to say to modern mathematical philosophy, Idealistic Studies 32:2, 2002, 143- 158.; The Successor Function and Induction Principle in a Hegelian Philosophy of Number, Idealistic Studies 30 (1) 2000, 25-61.; Frege and Hegel on concepts and number, 22 pages.; Self-reference and the natural numbers as the logic of Dasein, Hegel Studien 32(1997), 93- 121.; Alan Paterson, Towards a Hegelian philosophy of mathematics, Idealistic Studies, 27(1997), 1-10. On Paterson’s website (http://sites.google.com/site/apat1erson/) we will find five papers on the merits of a Hegelian philosophy of mathematics. Three of these (Paterson 1997b; 1999; 2000) are about the philosophy of Number, one is about the concept of the propositional calculus (Paterson 1994) and one is about the Hegelian philosophy of mathematics in general (Paterson 2002). In each of these, Hegelian philosophy is proposed as a solution to the problems ‘which arise out of the existence in mathematics of self-referential, nonconstructive concepts (such as class)’ (Paterson 2002: 143). 114 Dirk Damsma: Set Theory and Geometry in Hegel (2011), In Hegel Gesellschaft, Hegel-Jahrbuch. Berlin: 48
Hegel's great work on Logic, the “Science of Logic” (Wissenschaft der Logik), appeared in three volumes in 1812, 1813 and 1816, contains long discussions of mathematics. In particular in “Science of Logic” Hegel develops philosophies of the natural numbers, the negative numbers and also of the rational numbers. In Hegel’s logic we will find an extensive and erudite account (with the real numbers presupposed for the sake of the argument) of how this philosophical understanding of mathematics ``makes sense'' of the (informal) infinitesmimal manipulations of the differential and integral calculus. Alan Paterson argues that Hegel has much to say to modern mathematical philosophy, although the Hegelian perspective needs to be substantially developed to incorporate within it the extensive advances in post-Hegelian mathematics and its logic. According to Paterson, the Hegelian approach provides a framework for answering the philosophical problems, discussed by Kurt Gödel in his paper on Bertrand Russell, which arise out of the existence in mathematics of self-referential, non-constructive concepts (such as class). In his paper “Does Hegel have anything to say to modern mathematical philosophy”, Paterson insists that “it is time to grapple with what Hegel was trying to do in his treatment of mathematics. Of course a lot of the details of Hegel's view of mathematics belong to his time and sound quaint(!) (but no more so than Newton's discussions of fluxions), and Hegel did not have the benefit of coming after Weierstrass, Dedekind, Cantor and Frege. But 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 ``objective'' pose in which it is put in front of the investigator and inspected ``from the outside'', only for it to creep in from the back in the very act of investigation.” 115 For the purpose of the question above and in particular for the issue of human infinity as conceptual infinity, I will focus on the discussion given by Alan Paterson in his paper “Towards a Hegelian philosophy of mathematics”. According to Alan Paterson, as he establishes in “Towards a Hegelian philosophy of mathematics” 116 46 - “the well-known difficulties in modern mathematical logic require a Hegelian grounding for their resolution and that, in the terminology of Hegelian logic” (Hegel's “Science of Logic”). In his paper Paterson describes how “the discussion of the problems with the formal approach to mathematics generates a Hegelian approach to mathematics, one in which the great results of mathematical logic are aufgehoben, sublated their integrity preserved, while at the same time, the rigid separation of mechanical logic and mathematical intuition - that is, the obsessive abstractive operation of Verstand, the Understanding – is overcome.” Paterson establishes that “the Hegelian Concept, however, does not have just two moments. It has three, those of Universality, Particularity and Singularity (Allgemeinheit, Besonderheit, Einzelheit).” The same statement is valid in thinking of the categories as “quality” and “quantity”. Akademie Verlag, http://www1.fee.uva.nl/pp/bin/311fulltext.pdf; Qualitative and Quantitative Analysis in Semantic Dialectics: Marx vs. Hegel and Arthur vs. Smith (2010); On the Dialectical Foundations of Mathematics.(2006); 115 Alan Paterson, Does Hegel have anything to say to modern mathematical philosophy, Idealistic Studies 32:2, 2002, 143-158. 116 Alan Paterson, “Towards a Hegelian philosophy of mathematics”, Idealistic Studies, 27(1997) 49
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: David Gray Carlson’s “A Comment
Page 51 and 52: For Paterson, the “true mathemati
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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