Continental (vs.) Analytical Philosophy Research Papers (original) (raw)

The present work is devoted to the exploration of some formal possibilities suggesting, since some years, the possibility to elaborate a new, whole geometry, relative to the concept of “opposition”. The latter concept is very important and vast (as for its possible applications), both for philosophy and science and it admits since more than two thousand years a standard logical theory, Aristotle’s “opposition theory”, whose culminating formal point is the so called “square of opposition”. In some sense, the whole present enterprise consists in discovering and ordering geometrically an infinite amount of “avatars” of this traditional square structure (also called “logical square” or “Aristotle’s square”). The results obtained here go even beyond the most optimistic previous expectations, for it turns out that such a geometry exists indeed and offers to science many new conceptual insights and formal tools. Its main algorithms are the notion of “logical bi-simplex of dimension m” (which allows “opposition” to become “n-opposition”) and, beyond it, the notions of “Aristotelian pq-semantics” and “Aristotelian pq-lattice” (which allow opposition to become p-valued and, more generally, much more fine-grained): the former is a game-theoretical device for generating “opposition kinds”, the latter gives the structure of the “opposition frameworks” containing and ordering the opposition kinds. With these formal means, the notion of opposition reaches a conceptual clarity never possible before. The naturalness of the theory seems to be maximal with respect to the object it deals with, making this geometry the new standard for dealing scientifically with opposition phenomena. One question, however, philosophical and epistemological, may seem embarrassing with it: this new, successful theory exhibits fundamental logical structures which are shown to be intrinsically geometrical: the theory, in fact, relies on notions like those of “simplex”, of “n-dimensional central symmetry” and the like. Now, despite some appearances (that is, the existence, from time to time, of logics using some minor spatial or geometrical features), this fact is rather revolutionary. It joins an ancient and still unresolved debate over the essence of mathematics and rationality, opposing, for instance, Plato’s foundation of philosophy and science through Euclidean geometry and Aristotle’s alternative foundation of philosophy and science through logic. The geometry of opposition shows, shockingly, that the logical square, the heart of Aristotle’s transcendental, anti-Platonic strategy is in fact a Platonic formal jungle, containing geometrical-logical hyper-polyhedra going into infinite. Moreover, this fact of discovering a lot of geometry inside the very heart of logic, is also linked to a contemporary, raging, important debate between the partisans of “logic-inspired philosophy” (for short, the analytic philosophers and the cognitive scientists) and those, mathematics-inspired, who begin to claim more and more that logic is intrinsically unable to formalise, alone, the concept of “concept” (the key ingredient of philosophy), which in fact requires rather geometry, for displaying its natural “conceptual spaces” (Gärdenfors). So, we put forward some philosophical reflections over the aforementioned debate and its deep relations with questions about the nature of concepts. As a general epistemological result, we claim that the geometrical theory of oppositions reveals, by contrast, the danger implicit in equating “formal structures” to “symbolic calculi” (i.e. non-geometrical logic), as does the paradigm of analytic philosophy. We propose instead to take newly in consideration, inspired by the geometry of logic, the alternative paradigm of “structuralism”, for in it the notion of “structure” is much more general (being not reduced to logic alone) and leaves room to formalisations systematically missed by the “pure partisans” of “pure logic”.