Hilbert mathematics versus (or rather “without”) Gödel mathematics: V. Ontomathematics!

Abstract

The paper is the final, fifth part of a series of studies introducing the new conceptions of“Hilbert mathematics” and “ontomathematics”. The specific subject of the present investigation is theproper philosophical sense of both, including philosophy of mathematics and philosophy of physics notless than the traditional “first philosophy” (as far as ontomathematics is a conservative generalization ofontology as well as of Heidegger’s “fundamental ontology” though in a sense) and history of philosophy(deepening Heidegger’s destruction of it from the pre-Socratics to the Pythagoreans). Husserl’sphenomenology and Heidegger’s derivative “fundamental ontology” as well as his later doctrine after the“turn” are the starting point of the research as established and well known approaches relative to thenewly introduced conception of ontomathematics, even more so that Husserl himself started criticizing his“Philosophy of arithmetic” as too naturalistic and psychological turning to “Logical investigations” andthe foundations of phenomenology. Heidegger’s “Aletheia” is also interpreted ontomathematically: as arelation of locality and nonlocality, respectively as a motion from nonlocality to locality if both arephysically considered. Aristotle’s ontological revision of Plato’s doctrine is “destructed” further from thepre-Socratics' “Logos” or Heideger’s “Language” (after the “turn”) to the Pythagoreans “Numbers” or“Arithmetics” as an inherent and fundamental philosophical doctrine. Then, a leap to contemporaryphysics elucidates the essence of ontomathematics overcoming the Cartesian abyss inherited from Plato’sopposition of “ideas” versus “things”, and now unifying physics and mathematics, particularly allowingfor the “creation from nothing” instead of the quasi-scientific myth of the “Big Bang”. Furthermore,ontomathematics needs another interpretation of arithmetic, propositional logic and set theory in thefoundations of mathematics, where the latter two ones are both identified with Boolean algebra, and theformer is considered to be a “half of Boolean algebra” in the exact meaning to be equated to it afterdoubling by a dual anti-isometric counterpart of Peano arithmetic. That unified algebraic realization of thefoundations of mathematics is related to Hilbert mathematics in both “narrow and wide senses” where thelatter is isomorphic to the qubit Hilbert space, thus underlying all the physical world by the newlyintroduced substance of quantum information being physically dimensionless and generalizing classicalinformation measured in bits. The substance of information, whether classical or quantum, visualizes theway of the unification of physics and mathematics by merging their foundations in Hilbert arithmetic andHilbert mathematics: thus how ontomathetics is a “first philosophy”. The relation of ontomathematics tothe Socratic “human problematics”, furthermore being fundamental for Western philosophy in Modernity,is discussed. Ontomathematics implies its “substitution” by abstract information (or by “subjectlesschoice” relevant to it), thus “obliterating the human outline on the ocean beach sand” (by MichelFoucault’s metaphor). A reflection back from the viewpoint of mathematic to Western philosophy as thephilosophy of locality ends the study.