Some Conceptual Principles with Mathematical Background for Interdisciplinary Developments in the Sciences and Beyond

Abstract

This chapter discusses some of the principles of interdisciplinarity and emphasizes those principles that may provide some relevant components to a potentially unified background to many areas of natural sciences. This extension even reaches into fields that are typically regarded as humanities. The discussion focuses on the three concepts of “paradox,” “analogy,” and “fractals,” which are prevalent in the basis of our understanding of nature and related principles. Although these concepts are well defined in mathematical terms, the current description relies on limited hints towards the mathematical background and avoids detailed mathematical treatment. Emphasis is on the conceptual aspects, while some parts of the mathematical basis, relying on fuzzy sets, fuzzy logic, and a treatment of analogies using the functor model of category theory, are presented in the Appendices 1 and 2.

Appendices

Appendix 1

1.1 Main Concepts and Definitions of Fuzzy Set Theory

Following Zadeh’s original exposition (Zadeh, 1965), the fundamentals of fuzzy sets can be described as shown below:

Take X as a set of elements denoted by x, thus

$${\text{X}} = \left\{ {\text{x}} \right\}.$$

(10.4)

A fuzzy set A in X is characterized by a membership function f_A(x) assigning to each element x of X a real number from the interval [0, 1] that represents the “grade of membership” of x in A.

For a fuzzy set A, if this grade of membership f_A(x) of element x is close to 1, then x is a “strong member” of fuzzy set A. However, if this grade of membership f_A(x) of element x is close to 0, then x is a “weak member” of fuzzy set A. This fuzzy set approach also includes the traditional set approach as well, because if for a formally fuzzy set A the membership function f_A(x) can take only one of the values of 0 or 1, then we find that set A is actually a traditional ordinary set and is not fuzzy. In this case, the element x of X is either a “full member” of set A, or not a member at all of set A.

For the complement A′ of a fuzzy set A, the membership function f_A′(x) is defined as

$${\text{f}}_{{{\text{A}}^{\prime } }} \left( {\text{x}} \right) = {1} - {\text{f}}_{{\text{A}}} \left( {\text{x}} \right).$$

(10.5)

Unions and intersections of fuzzy sets are also defined in terms of the respective membership functions:

For the union C of two fuzzy sets A and B,

$${\text{F}}_{{\text{C}}} \left( {\text{x}} \right) = {\text{Max}}\left\{ {{\text{f}}_{{\text{A}}} \left( {\text{x}} \right),{\text{f}}_{{\text{B}}} \left( {\text{x}} \right)} \right\},$$

(10.6)

whereas for the intersection D of two fuzzy sets A and B,

$${\text{f}}_{{\text{D}}} \left( {\text{x}} \right) = {\text{Min}}\left\{ {{\text{f}}_{{\text{A}}} \left( {\text{x}} \right),{\text{f}}_{{\text{B}}} \left( {\text{x}} \right)} \right\}.$$

(10.7)

Appendix 2

2.1 A Functorial Approach to Analogies

The concept of analogy is commonly perceived as being more than similarity. If one finds several types of similarities between two phenomena, then a comparison between the two often suggests a deeper level of similarity: an analogy. In a formal sense, analogies can be considered as systems of similarities. If such systems of similarities, taken as families of “interconnected” similarities, can be characterized and described by mathematical means, then one should be able to take advantage of the formalism so created.

For a systematic study of similarities, one may use simple, common sense approaches without sophisticated mathematical tools (Mezey, 2019a, 2019b, 2019c). However, some of the more formal mathematical treatments also offer advantages, and in some recent developments the application of the functor model of category theory has been suggested for the analysis of analogies (Mezey, 2019a, 2019b, 2019c). Although functorial models are powerful and adaptable tools, and connect to some deeper chapters of category theory, functors and their use in describing analogies can be presented in a rather simple way, without the need for excessive detail on the category theory background.

Here, we present a somewhat simplified, but conceptually sufficient description of a special but relevant functor model. We may regard a functor as a transformation that operates on two levels, transforming two types of mathematical entities. First, a functor establishes a connection between two families of sets: a “sets to sets” connection. Second, the functor also establishes a connection between some mappings and transformations present among the first family of sets, connecting them to the mappings and transformations present among the second family of sets: a “mappings to mappings” connection. That is, a functor transforms a family of sets to another family of sets, and at the same time, it transforms the relations among members of the first family of sets to the relations among members of the second family of sets.

Given that our primary interest is analogies used in interdisciplinary areas, one may consider two scientific fields, with sets of concepts, problems, and approaches in both, as well as connections, relations, and comparisons in both. An analogy between these two fields of science may be modelled by a functor, where the concepts, problems, and approaches of one field are related to those of the other scientific field, and beyond all these, the individual mappings among the sets within one family of problems in the first scientific field are also connected to those of the second scientific field. Whereas individual mappings can be regarded as expressions of similarities, the complete functor itself can then be taken as an expression of a higher level of similarity: an analogy. In other words, this entire comparison can then be taken as a description of the similarity of similarities; that is, an expression of an analogy.

A simplified example may provide some insight of the power of a functorial approach. In this case, we use the example of only three sets in the first family of sets, X, Y, and Z, and another three sets, U, V, and W, in the second family of sets. We assume that among the sets of each family of sets, there are several mappings, such as

$${\text{f}}_{{{\text{iXY}}}} \left( {{\text{i}} = {1}, \ldots {\text{ n}}_{{{\text{XY}}}} } \right),$$

(10.8)

$${\text{f}}_{{{\text{iXZ}}}} \left( {{\text{i}} = {1}, \ldots {\text{ n}}_{{{\text{XZ}}}} } \right),$$

(10.9)

$${\text{f}}_{{{\text{iYZ}}}} \left( {{\text{i}} = {1}, \ldots {\text{ n}}_{{{\text{YZ}}}} } \right),$$

(10.10)

and

$${\text{g}}_{{{\text{iUV}}}} \left( {{\text{i}} = {1}, \ldots {\text{ n}}_{{{\text{UV}}}} } \right),$$

(10.11)

$${\text{g}}_{{{\text{iUW}}}} \left( {{\text{i}} = {1}, \ldots {\text{ n}}_{{{\text{UW}}}} } \right),$$

(10.12)

$${\text{g}}_{{{\text{iVW}}}} \left( {{\text{i}} = {1}, \ldots {\text{ n}}_{{{\text{VW}}}} } \right).$$

(10.13)

Here we also assume that for each index i, and for all choices of sets A and B, the mapping f_iAB maps set A to set B.

A functor F can then be defined for a functorial relation between the two families of sets and for the corresponding mappings, as follows:

Functor F maps the first three sets, X, Y, and Z to sets U, V, and W, respectively. It also converts for each index i the mapping f_iXY into g_iUV, the mapping f_iXZ into g_iUW, and the mapping f_iYZ into g_iVW.

Indeed, the above construction illustrates the most fundamental properties of functors, especially in the context of modelling analogies. In the above example, one may also regard the mappings f_iST as expressions of similarities. Consequently, the above functor can be regarded as an expression of a higher level of similarity between the properties of the family of the first three sets, X, Y, and Z, and their interrelations, on the one hand, and the family of the other three sets, U, V, and W, and their interrelations, on the other hand, where the actual similarities within each family of sets are described by the individual mappings f_iST, as presented in Eqs. (10.8)–(10.13).

Such functor models can be used as mathematical tools to model various complex problems of comparisons and analogies, which may occur on several levels of interdisciplinary studies. Analogies often may appear to be exploitable in a new field if various levels of comparisons, and possible re-uses of earlier approaches and methodologies, which were originally introduced for a different set of problems, are possible. Functors are useful to formalize such analogies, and can be beneficial, especially if interdisciplinary considerations apply.