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

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.

A Meta-Category “People” and Limits of Cognition in the Science of Constitutional Law

Based on approaches existing in domestic science to the interpretation of the term “a people”, the author concludes that in constitutional law the term under consideration is used in at least two different meanings: 1.as a source of power; and 2. as a subject of constitutional rights. The author highlights the irreducibility of this duality and substantiates the thesis that the concept of the people as a source of power is one of the limiting concepts of constitutional law and, therefore, cannot be defined within its framework. Awareness of this problem leads some Russian lawyers to denying the principle of popular sovereignty, and other scholars try to equate the concept of “a people as a source of power” with the concept of electoral corps. Arguing both these approaches, the author highlights that according to Gödel’s theorem on incompleteness, constitutional law as a system of formally defined norms cannot be complete and uncontroversial at the same time. Attempts to construct a “pure” constitutional-legal theory free of any extra-legal elements where definitions of all concepts the theory operates would be derived from the theory itself lead to intractable contradictions. If the system of constitutional law is built as non-contradictory, it is inevitably incomplete, in particular it uses concepts that cannot be defined within its framework. The author investigates such concepts and categories belonging to a higher level of abstraction as meta-categories, or marginal concepts, of the science of constitutional law. The category “the people as a source of power” is one of them. Concerning the dilemma of completeness and consistency of constitutional law as a system of formally defined norms, the author prioritizes consistency.

Gödel's incompleteness theorem in a nutshell

We present a microversion of Gödel's theorem.

Computability and information

The standard definition of randomness as considered in probability theory and used, for example, in quantum mechanics, allows one to speak of a process (such as tossing a coin, or measuring the diagonal polarization of a horizontally polarized photon) as being random. It does not allow one to call a particular outcome (or string of outcomes, or sequence of outcomes) ‘random’, except in an intuitive, heuristic sense. Information-theoretic complexity makes this possible. An algorithmically random string is one which cannot be produced by a description significantly shorter than itself; an algorithmically random sequence is one whose initial finite segments are almost random strings. Gödel’s incompleteness theorem states that every axiomatizable theory which is sufficiently rich and sound is incomplete. Chaitin’s information-theoretic version of Gödel’s theorem goes a step further, revealing the reason for incompleteness: a set of axioms of complexity N cannot yield a theorem that asserts that a specific object is of complexity substantially greater than N. This suggests that incompleteness is not only natural, but pervasive; it can no longer be ignored by everyday mathematics. It also provides a theoretical support for a quasi-empirical and pragmatic attitude to the foundations of mathematics. Information-theoretic complexity is also relevant to physics and biology. For physics it is convenient to reformulate it as the size of the shortest message specifying a microstate, uniquely up to the assumed resolution. In this way we get a rigorous, entropy-like measure of disorder of an individual, microscopic, definite state of a physical system. The regulatory genes of a developing embryo can be ultimately conceived as a program for constructing an organism. The information contained by this biochemical computer program can be measured by information-theoretic complexity.