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 invention of such a logically formalized axiomatic epistemology system, in which it is possible to construct 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 hypothetical–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 deductively 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.

Formally Inferring Galileo Galilei Principle of Relativity of Motion in an Axiomatic System “Sigma+V” from a Triple of Nontrivial Assumptions

“…The formal axiomatic theory Σ+V is an outcome of significant mutations in (and additions to) the logically formalized axiomatic epistemology-and-axiology systems Σ [38] [40] [41], Σ+C [39], and [42]. For constructing a sufficiently precise definition of the formal axiomatic system Σ+V, it is indispensable to provide exact definitions of its basic concepts, namely, the concept "alphabet of object-language of Σ+V", the abstract syntactic notion "term of Σ+V", the abstract syntactic concept "formula of Σ+V", and, finally, the fundamental notion "axiom of Σ+V".…”

“…For constructing a sufficiently precise definition of the formal axiomatic system Σ+V, it is indispensable to provide exact definitions of its basic concepts, namely, the concept "alphabet of object-language of Σ+V", the abstract syntactic notion "term of Σ+V", the abstract syntactic concept "formula of Σ+V", and, finally, the fundamental notion "axiom of Σ+V". Exact definitions of the mentioned basic concepts of Σ+V are analogous to the definitions of corresponding concepts of Σ [38] [40] [41] and Σ+C [39].…”

“…Today, some significant aspects (subproblems) of the indicated difficult compound theoretical problem are already solved by means of a formal axiomatic theory Σ (originally formulated and published in [38]) and by its various modifications, for instance, Σ+C [39]. In particular, the formal logical (deductive) derivation of the formula representing the principle of inertia in the formal axiomatic theory Σ (given that an appropriate physical interpretation of Σ is provided) from the nontrivial assumption of knowledge a-priori-ness is already discovered (or invented, i.e.…”