Abstract
Analogy belongs to the class of concepts notorious for a variety of definitions generating continuing disputes about their preferred understanding. Analogy is typically defined by or at least associated with similarity, but as long as similarity remains undefined this association does not eliminate ambiguity. In this paper, analogy is considered synonymous with a slightly generalized mathematical concept of similarity which under the name of tolerance relation has been the subject of extensive studies over several decades. In this approach, analogy can be mathematically formalized in terms of the sequence of binary relations of increased generality, from the identity, equivalence, tolerance, to weak tolerance relations. Each of these relations has cryptomorphic presentations relevant to the study of analogy. The formalism requires only two assumptions which are satisfied in all of the earlier attempts to formulate adequate definitions which met expectations of the intuitive use of the word analogy in general contexts. The mathematical formalism presented here permits theoretical analysis of analogy in the contrasting comparison with abstraction, showing its higher level of complexity, providing a precise methodology for its study and informing philosophical reflection. Also, arguments are presented for the legitimate expectation that better understanding of analogy can help mathematics in establishing a unified and universal concept of a structure.
Highlights
The concept of analogy belongs to the class of concepts which can be considered elusive, i.e., concepts which almost everyone claims to understand well, which are used in a majority of the domains of inquiry from philosophy to computer science as well as in the everyday discourse, but which under scrutiny escape any commonly accepted definition
As in cases of other elusive concepts such as identity, structure, information, computation, or mind, the omnipresence of the word “analogy” in everyday discourse where there is no expectation for the semantic clarification makes it very difficult to establish a common foundation for its analysis
While the paper has as its main objective to contribute a mathematical formalism to the methodology for the study of analogy and similarity, it has an additional objective to present the possibility of using analogy for the purpose of establishing a clear meaning for the general concept of a structure
Summary
The concept of analogy belongs to the class of concepts which can be considered elusive, i.e., concepts which almost everyone claims to understand well, which are used in a majority of the domains of inquiry from philosophy to computer science as well as in the everyday discourse, but which under scrutiny escape any commonly accepted definition. An analogical argument is an explicit representation of a form of analogical reasoning that cites accepted similarities between two systems to support the conclusion that some further similarity exists” [1] This definition, as with many other attempts, is guilty of the common sin of addressing the meaning of hard words which consists of sweeping the dirt under the carpet of another hard but undefined word in the definiendum. The main objective of this paper is to search for the common ground and to establish a firm foundation for the study of analogy in the experience of mathematics and in its conceptual framework It is a surprisingly well-kept secret, that there is a quite extensive mathematical theory of general similarity relation initiated by Eric Christopher Zeeman more than fifty years ago, under the misleading term of tolerance relation [2,3]. While the paper has as its main objective to contribute a mathematical formalism to the methodology for the study of analogy and similarity, it has an additional objective to present the possibility of using analogy for the purpose of establishing a clear meaning for the general concept of a structure
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