This contribution tentatively outlines the presumed conceptual duality between the issues of incompleteness and incomprehensibility—The first being more formal in nature and able to be declined in various ways until specified in the literature as theoretical incompleteness. This is theoretical and not temporary, which is admissible and the completion prosecutable. As considered in the literature, theoretical incompleteness refers to uncertainty principles in physics, incompleteness in mathematics, oracles for the Turing Machine, logical openness as the multiplicity of models focusing on coherence more than the optimum selections, fuzziness, quasiness, e.g., quasi-crystals, quasi-systems, and quasi-periodicity, which are intended as the space of equivalences that allow for coherent processes of emergence. The issue of incomprehensibility cannot be considered without reference to an agent endowed with cognitive abilities. In this article, we consider incomprehensibility as understood here as not generally scientifically explicable, i.e., with the available knowledge, as such incomprehensibility may be temporary, pending theoretical and technological advances, or deemed to be absolute as coincident with eventual definitive, theoretical non-explicability, and incomprehensibility. We considered the theoretically incomprehensibility mostly in three main ways: as the inexhaustibility of the multiplicity of constructivist reality as given by the theoretically incomprehensible endless loop of incomprehensible–comprehensible, and by existential questions. Moreover, theoretical incomprehensibility is intended as evidence of the logical openness of both the world and of understanding itself. The role of theoretical incomprehensibility is intended as a source of theoretical research issues such as paradoxes and paradigm shifts, where it is a matter of having cognitive strategies and approaches to look for, cohabit, combine, and use comprehensibility and (theoretical) incomprehensibility. The usefulness of imaginary numbers comes to mind. Can we support such research for local, temporary, and theoretical incomprehensibility with suitable approaches such as software tools, for instance, that simulate the logical frameworks of incomprehensibility? Is this a step toward a kind of artificial creativity leading to paradigm shifts? The most significant novelty of the article lies in the focus on the concept of theoretical incomprehensibility and distinguishing it from incomprehensibility and considering different forms of understanding. It is a matter of identifying strategies to act and coexist with the theoretically incomprehensible, to represent and use it, for example when dealing with imaginary numbers and quantum contexts where classical comprehensibility is theoretically impossible. Can we think of forms of non-classical understanding? In this article, these topics are developed in conceptual and philosophical ways.
Computational Logical Frameworks and Generic Program Analysis Technologies
