A Logically Formalized Axiomatic Epistemology System Σ + c and Philosophical Grounding Mathematics as a Self-Sufficing System

Abstract

The subject matter of this research is Kant’s apriorism underlying Hilbert’s formalism in the philosophical grounding of mathematics as a self-sufficing system. The research aim is the in-vention of such a logically formalized axiomatic epistemology system, in which it is possible to con-struct formal deductive inferences of formulae—modeling the formalism ideal of Hilbert—from the assumption of Kant’s apriorism in relation to mathematical knowledge. The research method is hy-pothetical–deductive (axiomatic). The research results and their scientific novelty are based on a logically formalized axiomatic system of epistemology called Σ + C, constructed here for the first time. In comparison with the already published formal epistemology systems X and Σ, some of the axiom schemes here are generalized in Σ + C, and a new symbol is included in the object-language alphabet of Σ + C, namely, the symbol representing the perfection modality, C: “it is consistent that…”. The meaning of this modality is defined by the system of axiom schemes of Σ + C. A deductive proof of the consistency of Σ + C is submitted. For the first time, by means of Σ + C, it is deduc-tively demonstrated that, from the conjunction of Σ + C and either the first or second version of Gödel’s theorem of incompleteness of a formal arithmetic system, the formal arithmetic investigated by Gödel is a representation of an empirical knowledge system. Thus, Kant’s view of mathematics as a self-sufficient, pure, a priori knowledge system is falsified. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.