DOI: 10.25136/2409-8728.2023.12.69100

Abstract: The aim of the research is to study the relationship between logic and mathematics in the structure of axiomatized and formalized scientific theories. The object of the study is the explication of this connection and its explanation. The subject of the study is syntactic and semantic views on the structure of scientific theories, the relationship between logic and mathematics has not been studied in detail in them. In the syntactic view, the structure of the theory is understood as a linguistic construct build from various logical propositions of the theoretical level, correspondence propositions and observation propositions. The structure of the theory does not take into account the variety of model representations of the theory that generate a variety of language constructs. The semantic view overcomes this disadvantage, and in it the structure of the theory is presented as a hierarchy of models: from axioms to theoretical-level models, experimental models and data models. The structure of the theory, the connection of logic and mathematics were studied using comparative analysis, methods of interpretive analysis and reconstruction of scientific theories. The methods made it possible to explicate mathematical concepts in the structure of the theory and correlate them with logic and natural language. Comparative analysis has shown that in the syntactic view, the connection between logic and mathematics lies in the fact that mathematical concepts of physics are interpreted in the language of logic of first-order predicates with equality. The connection between mathematical concepts is provided by the axiomatic method, which serves as a means of formalizing concepts. Mathematics comes down to logic. In the semantic approach, in order to identify the connection between mathematics and logic, it was necessary to reconstruct the structure of non-relativistic quantum mechanics. With the help of the set-theoretic predicate of Suppes, its axioms were determined, the connection between mathematical structures, postulates of the theory, axioms, and observable quantities was established. Logic and mathematics are related to each other in such a way that metamathematics or linguistics is a part of mathematics. Mathematics includes set theory and model theory, i.e. mathematical logic. The connection of mathematical formalisms with phenomena and with natural language remains problematic, and there is this drawback in the syntactic approach. The novelty lies in the fact that the research contributes to the methodology and logic of science, to the explanation of the connection between logic and mathematics in scientific theory, which was illustrated by various examples from various fields of physics.

DOI: 10.25136/2409-8728.2021.12.36840

Abstract: The subject of this research is the method of interpretation in theoretical chemistry as a combination of cognitive procedures and approaches on the example of interaction of the classical theory of structure and quantum chemistry within the framework of their history and logic of development. It is demonstrated that the process of interpretation encompasses several historical stages of the development of quantum chemistry, marking the transition from meaningful symbolic concepts of the theory of structure towards formal-logical quantum-chemical terms, and the reverse interaction of these theories – the implementation of the latter into the theory of structure. The interpretational method in quantum chemistry contributes to the construction of more complex mathematical schemes underlying the natural scientific content. Such schemes include various approximations and assumptions, as well as the elements of arbitrariness in selection of the mathematical schemes by the theoretician, which reduces the accuracy of explanations and predictions of quantum chemistry. The object of this research is the methodology of theoretical chemistry, in terms of which takes place the interaction between quantum chemistry and classical theory of structure, their cognitive abilities, structure and dynamics of theoretical knowledge. The novelty lies in the fact that the interpretation in natural sciences is yet to be fully research; the study of interpretation in the context of constructivist approach in the philosophy of science allows revealing the logical-methodological and gnoseological aspects of interpretation. The acquired results contribute to the methodology of chemistry, epistemology, and philosophy of science. It is concluded that the process of interpretation is the construction of more complex mathematical schemes, which leads to the gap between mathematical and natural scientific content of the concepts; between mathematical description, natural-scientific theoretical representations, and experiment. The gap is accompanied by origination of the new concepts of quantum chemistry as a result of integration of the various fields of knowledge and extinction of concepts of the classical theory of structure, as well as determination of the limits of mathematical method in chemistry.