Abstract

The first paper of a series about the rigorous and ontomathematical definition of postmodernity reinterprets Hegel's dialectics and dialectic logic. His "synthesis", "change", "development", "time" is represented as the third dimension of Hilbert space where the two others correspond to the initial and final state of whether developing "idea" or changing "object" (after Marx's "dialectical/ historical materialism") generating a complete description of reality therefore excluding any hidden variables in principle: after the Kochen-Specker theorem (1967) or forcing a unique probability measure after the Gleason theorem (1957). That approach is obviously geometric rather than logical, thus rather ontomathematical than ontological. Speaking loosely, dialectic logic is not logic in a formal sense: it means only the following negative result after the transition from two-to three-dimensional Hilbert space. The orthomodular lattice of subspaces (projective operators) allowing for a local (i.e., depending on a locally defined context of measurement) Boolean lattice being omnipresently homomorphic (i.e., independent of any locally defined context of measurement) in the two-dimensional case for the Hilbert meta-space of human experience, however, does not allow the same (i.e., both local and universal) Boolean lattice after complementing it by the third dimension of "time", "synthesis", "development", etc. So, dialectic logic is only the "antithesis" of classical logic, but their "synthesis" is not (and fundamentally cannot be) logical, but only geometrical (furthermore originating from Hegel's own understanding of "synthesis": to generate a new quality). Dialectical logic is neither "ontology of change", nor "ontology of time", nor "ontology of contradiction": it is ontomathematics implied by the third dimension, thus beyond logic, but within geometry. It is rather "geometry of reality" than "logic of reality" and quantum mechanics rather than Marx's materialism is what it needs. Speaking loosely, reality starts from the third dimension after the Kochen-Specker and Gleason theorems. Then, Postmodernity versus Modernity can be rigorously and mathematically defined to replace the fundamental metapostulate about the two-dimensionality (e.g., "mindbody"; "objectsubject"; "Self environment"; etc.) of human experience with its three-dimensionality. As that was in fact realized for the first time by Hegel (in a proper philosophical insight since quantum mechanics did not exist in his age), he can be reasonably called the "first postmodern philosopher". The USSR and other socialistic countries' history and their "Marxism-Leninism" provide numerous examples (or counterexamples) for that ontomathematical interpretation of dialectics.