STRUCTURES ON SETS OF THOUGHTS: TOWARD A THEORY OF THOUGHT SPACES

Abstract

This paper proposes an abstract approach to the study of thinking through the concepts of a set of thoughts and the structures that can be defined on it. The point of departure is the understanding of a thought as a reflection of reality or as an image constructed from other images. It is argued that the character and completeness of reflection are determined not only by the properties of the object itself but also by the capacities of the subject of perception, which depend on experience, language, education, culture, and worldview. Consequently, the set of thoughts available to an individual is considered to be limited by the structures that the individual has acquired.
A thought space is defined as a set of thoughts endowed with a particular structure. Three principal types of structures are distinguished: ordering structures, metamorphosis structures, and composition structures. On this basis, spaces of truth, beauty, expediency, emotions, and other possible thought spaces are introduced. The paper also considers the possibility of comparing different structures within the same thought space and analyzing their relative strength.
To describe processes of thinking, a number of mathematical concepts are employed, including closures, boundaries, open and closed sets, sequences, directed sets, and metric spaces. The search for truth, beauty, or expediency is interpreted as the operation of closing a set of arbitrary thoughts with respect to a corresponding structure. It is argued that truth is dependent on the thought space and its axiomatic basis. A distinction is made between natural closures, which arise from the intrinsic properties of a system, and conventional closures, which are grounded in social agreements and conventions.
True thoughts are proposed to be understood as those standing in relations of isomorphism, and in certain cases homomorphism or homeomorphism, to the objects they reflect. The notion of directedness of thought is introduced as an ordered movement toward a certain result, making it possible to describe processes of proof, inquiry, and creative search. The paper concludes that the development of thinking, science, and culture is associated with the acquisition of new structures and the expansion of corresponding thought spaces within which cognition, evaluation, and the creation of new representations of reality become possible.