*5.2. Definition of the Definition*

"Linda the Bank Teller" seems harmless, but actually, similar fallacies are surprisingly common in philosophical and scientific discourse, becoming a large obstacle in mutual understanding. This is a part of a larger problem in the context of much more serious misunderstanding of the concept of a definition and definability.

Once again, we can see the danger of equivocation, which can be identified as a main source of the Idols of the Common Sense. There are many di fferent meanings of the word "definition" when it is qualified by some adjectives. Definitions always serve the identification of something. For instance, the dictionary definition serves the purpose of the identification of the standard use of words by finding their synonyms or synonymic expressions possibly with many words, typical paraphrases, or by providing contrast to similar words used in a very di fferent way. Dictionary definitions are circular out of necessity, but also intentionally, as it is assumed that someone may know some words but not the others. Another example is an ostensive definition identifying objects by directly pointing at them.

There are multiple other "definitions" serving di fferent objectives. However, in the context of philosophical or scientific inquiries, there is only one concept of a definition within formal logical methodology called genus-species definition. The tradition of this type of definition goes back to Socrates, but the works of Aristotle made it the central tool of philosophical methodology. Formal definitions of universals, i.e., terms with general meaning addressing multiple individuals, became necessary for the development of syllogistics as a methodology of reasoning. After Aristotle, the unqualified term "definition" always refers to genus-species definition. All other definitions, which are diverse forms of identification of a variety of objects or relations, require qualification.

Aristotelian concept of the *genus-species* definition referred to the partial order of universals according to their level of generality. The pair of universals was considered in the genus-species relation if every instance of the latter was an instance of the former. Thus, whenever we have that every A is B, A is species and B is genus. Of course, in the much later adopted biological taxonomy, the meaning of the terms "genus" and "species" changed as names of the specific consecutive levels of such order. In this partially ordered structure of universals, going in the directions of *species* was going in the direction of increasingly smaller classes of individuals, while going upward in the *genus* direction led to increasingly larger classes. Aristotle did not consider individuals being universals, but we could modernize the description of the structure of universals by considering individuals as atoms of the partially ordered set of universals.

To define a universal (*definient*) we have to identify its *genus* (one of them, but preferably *genus proximus*, i.e., the nearest of all genera). Then we have to provide *di*ff*erentia*, i.e., we have to provide the di fference between the universal which we want to define and all other species of this genus. The classical example of the definition was: A human (*definient*) is an animal (*genus*) which is rational. Being rational was the di fferentia which made the distinction between humans and other animals. The genus and di fferentia formed the *definiens*. Of course, we have to be able to identify a genus before we can use it in the definition. Thus, we had to have its definition ready, or we have to give it the status of a category, i.e., primary, undefinable, universal identifiable only with the use of intuition. Aristotle selected his own categories and in the millennia to come, philosophers formed their own selections. The border between the rational and intuitive forms of inquiry is exactly at this point. The selection of categories is beyond our rational capacities and it is left to our intuitive capacities.

The only di fference in the modern formation of a conceptual system is that we do not feel obliged to start from the selection of all categories, but for a particular theory, we choose its primitive concepts (which we do not define) and we formulate a set of axioms as a priori true sentences characterizing the primitive concepts. From the axioms, we derive the truth of all theorems of the theory using valid logical inferences. Of course, the truth of theorems is conditioned by the assumed truth of axioms. To reduce the complexity of the statements, which we want to prove, and give them the status of a theorem of the theory, we can (and actually we do) define derivative concepts using the process of the definition starting with primitive concepts as *genera*. Later we can use, in *definiendum*, already defined concepts as *genera* for consecutive definitions.

It is important that, in principle, we do not have to define additional concepts, and by the cost of extreme complexity of the statements, we could develop an entire theory using only primitive concepts. Here, we can recall another contribution of Eugene Wigner, this time to the Idols of the Common Sense, when, in his paper, "The Unreasonable E ffectiveness of Mathematics in the Natural Sciences" mentioned in the context of the Idols of the Number, he wrote, "Mathematics would soon run out of interesting theorems if these had to be formulated in terms of the concepts which already appear in the axioms" [38]. Of course, in this case there is not much damage as a disproval of this false statement can be found in any introductory textbook to logic. This just shows the dangers of misconceptions which can make even laureates of the Nobel Prize victims of the Idol of Common Sense. This also provides the justification for the inclusion above regarding the elementary explanation of the concept of definition.

This does not mean that definitions do not have practical importance. Not only do they direct the attention of the philosophical or scientific community to a particular direction of research but also, they simplify both reasoning of the author and its reception at the other end of communication. There is a good analogy in the use of the higher level programming languages. Of course, in principle, every program can be written in the machine language, but in such form, it would be practically incomprehensible to other human programmers. Higher level programs use defined subroutines which have a short name easily comprehensible to human programmers, and when they use these names in programming, the reversal of the names to machine language is performed by the compiler.

Definitions are not true or false. They are conventional tools reducing complexity of the language, but they are still conventional. In arithmetic, we write 5, not S(S(S(S(S(0), but without the convention of writing digits in some particular way, we cannot understand the meaning of 5 using only arithmetical theory, which describes the primitive concept S(n) in terms of a recursive scheme.

The typical problems arise when the process of defining concepts, which is a syntactic procedure, is confused with semantics. The fact that we provide a definition of a concept does not tell us anything about the relevance of this concept, even if it is formulated in a perfectly correct way. We did not create anything new. We just eliminate a concept by reducing it to other concepts. This is actually an expression of the two main conditions for proper definition called "eliminability*"* and "noncreativity*"* [44]. Herbert Simon writes about them: "These criteria stem from the notion, often repeated in works on logic, that definitions are ('ought to be'?) mere notational abbreviations, allowing a theory to be stated in more compact form without changing its content in any way" [45].

There is extensive classic literature on the modern logical theory of definitions and definability with the particularly highly respected and renowned contributions of Alfred Tarski and Patrick Suppes [44,46]. The form of a logically correct definition is very well established and does not require much more study. The actual subject of the theory of definitions and definability is the transition between deferent theories developed in not necessarily the same conceptual framework of primitive concepts and axioms. This subject is beyond the scope of the present paper. After all, the most important lesson from logic about the concept of a definition is that Humpty Dumpty was right in his teaching Alice about the meaning of words: "When I use a word, 'Humpty Dumpty said, in rather a scornful tone,' it means just what I choose it to mean—neither more nor less" [47].

The idol which Linda the Bank Teller manifests is a quite frequent form of unintended and undesirable restriction of the scope of the concept by adding either additional di fferences or by adding additional axioms for the axiomatic theories based on the primitive concepts. Very often, authors who are not satisfied with the too narrow scope of the existing definition add to it additional conditions or comments, not realizing that this will never make the concept more general, but usually the e ffect is exactly opposite.

The logical definitions may not be su fficient for the purpose of theories describing a part of reality in terms of active engagemen<sup>t</sup> of observers. In this case, very often, operational definitions are used. They describe, for instance, how to construct the object of study through practical manipulations of the environment. This, of course, is very di fferent from the presented before theoretical definitions. However, the di fference can be eliminated if we include a theory of these operations into the more comprehensive theory of the studied fragment of reality. Once we have a theory of operations (for instance, empirical procedures) the operational definition can be formulated in the purely logical form.

Francis Bacon wanted to eliminate the intervention of theoretical reasoning in the form of theoretical description of experimental system, but in the perspective of modern science, his dream is impossible. Even if we could avoid the use of any experimental equipment (we know that we cannot) and restrict all inquiries to direct human observation based on sensory experience, our body is an experimental system and the functioning of our senses cannot be ignored.

The importance of the process of the formulation of the definitions for the concepts forming the conceptual reference frame can be seen in the eternal disputes on subjects, formulated as a question "What is...?" For instance, the concept of culture has been discussed since the 19th century by anthropologists, linguists, scholars of intercultural communication, etc., without ever reaching consensus. Already in 1952, Alfred L. Kroeber and Clyde K.M. Kluckhohn summarized, in a critical review, 164 earlier definitions, adding their own [48]. Arthur Lovejoy, in 1927, studied 66 ways in which the word "nature" has been understood in the context of aesthetics [49,50]. Raymond Williams based on the variety of definitions for nature, calledit "perhaps the most complex word in the language" but he was not aware that no philosophically non-trivial concept has commonly accepted unique meaning [51,52]. Even the concept of meaning has diverse meanings. The classical book of Charles Kay Ogden and Ivor Armstrong Richards, *The Meaning of Meaning* published in 1923, distinguished 16 di fferent ways in which meaning is understood [53].

There is nothing wrong with the diversity of definitions. Actually, this diversity is just an evidence for the relevance. The problem is that the vast majority of so called "definitions" are not definitions at all from the point of view of logic. Quite a typical fallacy is that establishing of a quantitative magnitude is sometimes considered a definition of a concept.

The classical example of this fallacy of taking the definition of a mathematical formula for some magnitude as a definition of the concept is "Shannon's definition of information" which supposedly was written by Claude Shannon in his famous 1948 paper, "A Mathematical Theory of Communication" later published together with Warren Weaver in book format [54]. Shannon claimed to be interested in the fundamental communication problem of reproducing, at one point, either exactly or approximately a message selected at another point. In his paper, he formulated a mathematical concept of entropy

characterizing probability distributions and wrote in Section 6, with the title, Choice, Uncertainty and Entropy: "Quantities of the form H = − - pi log pi (the constant K (omitted in the formula, m.j.s) merely amounts to a choice of a unit of measure) play a central role in information theory as measures of information, choice and uncertainty" [54] (p. 20). There is not much more directly about information in this historical paper, ye<sup>t</sup> it is considered that Shannon defined here "information". It is clear that the two idols, of the Number and of the Common Sense, are responsible for this opinion. The former prompts people to believe that something expressed as a number giving value to some magnitude must be an entity. The latter idol just obscures the meaning of the definition as a concept.

Even when all definitions of some diverse attempts to define a concept are correct, the Idol of the Common Sense may prevent their e ffective use. The disputes on the definitions are often performed as if it was a matter of truth or falsity or of correctness. The definitions of concepts (if correctly formulated) can be evaluated exclusively on the adequacy of the theory which they serve, not by the form or content of the definition. If the theory (i.e., its syntactically true sentences or claims) describes objects of reality in the way which can be empirically confirmed, then we can consider the definition useful, but, of course, not true.

Another possible criterion of the evaluation of a definition can be formulated through the analysis of its conceptual framework (concepts involved in the *definiendum*). If a definition gives a wide range of relations with other relevant concepts, then this gives the evidence of its potential value, but this, too, can be assessed only by the analysis of the theory and its consequences. We have to remember that a definition of the concept is basically a selection of already defined or primitive concepts, something which metaphorically we could describe as a "conceptual system of coordinates". The same way as coordinate systems may be convenient or not is less important than finding the rules which govern phenomena framed by the coordinate system.

### **6. The Idols of the Elephant**

The Idols of the Elephant can be easily recognized because of a well-known parable of "The Blind Men and an Elephant" accompanying many threads of Indian philosophical tradition, and going back before its first historical appearance in the Buddhist texts more than two millennia ago. The parable is now well known all over the world. A group of blind men tries to learn what a large object is, in their way. They use their tactile sense, but without having ability to see, they cannot compare and synthesize their individual experiences derived from touching small portions of the object. This may look like a too simplistic metaphor of the fragmented vision of reality provided by science. Certainly, the parable is of high relevance for Natural Philosophy as an integrated system of knowledge of entire reality, as it suggests that we should look for some form of sense of sight (or insight) to achieve integration.

Actually, not all the Idols of the Elephant are as obvious as that represented by the ancient parable. The other idols which prevent us in achieving our goal of integrated vision may not be like that in the parable, where the men are aware of their handicap. As in the cases of other idols, we may not be aware of our handicaps.

The division into the types of idols is not exclusive and not straightforward. In the description of the Idols of the Common Sense, the central fallacy was related to the relationship between the rational and intuitive forms of inquiry which have consequences for the relationship between objective and subjective forms of inquiry. However, this distinction can be associated with the Idols of the Elephant, too. Objectivity can be viewed as intersubjectivity, i.e., invariance with respect to the transition between the di fferent human knowing subjects rather than the di fferent, more general and not necessarily human observers or reference frames.

Let us change the parable and let all blind men touch at the same time, the same part of the elephant. Why should we expect that their reports should be the same? Each of the men has a di fferent experience in touching the object of their environment. They may have di fferent levels of sensitivity in tactile perception. Finally, they may have di fferent skills in expressing their perceptions. Thus, we have to avoid oversimplified one-dimensional conclusions about the value of the collective inquiry.

The danger here is in the habits perpetuated by the language, promoting the view of the complementary objective and subjective forms of experience and inquiry falsely considered as exclusive, contradictory, competing, and requiring the dominance of one form over the other. The priority is usually given to the former, objective form. This can be seen in the normative character of the terms, "objective" (good), "intersubjective" (neutral) and "subjective" (bad) in everyday language.

Usual studies of objectivity are focused on preventing bias coming from the conflict of interest present in social life or from psychological determinants such as the trait ascription bias exhibited in a tendency to describe own behavior as flexible, adapting to the situational factors while the behavior of others is by ascribing fixed dispositions to their personality. Objectivity of science is expected to be achieved by the requirement of the judgment of many disinterested and independent reviewers. Sometimes, objectivity is considered in more abstract terms of independence of the evidence from that or whom it serves. Peter Kosso considers more general description"Objective evidence is evidence that is verified independently of what it is evidence for" [55].

We could see in the discussion of the Common Sense that misunderstanding of the concept of definition may generate di fficulties in coordination of individual inquiries and formulation of a consistent vision of reality. However, this looks like a matter of communication between the blind men in the parable—but is the deficiency of communication the main problem? The problem is rather in the lost sight, i.e., missing tool of integration at the level of the acquisition of knowledge.

Certainly, it is very important to establish social mechanisms eliminating the influence of external factors and interests on the inquiries and their outcomes. Equally important is to foster good communication coordinating and integrating collective forms of inquiry. However, the most dangerous Idols of the Elephant are highly non-trivial and di fficult to identify and control. They may not be related to the problems of coordination of inquiries performed by di fferent individuals. They may put the obstacles on the path of inquiry of an individual inquirer.

In this paper, only one example of such non-trivial form of the Idols of the Elephant will be considered. This is a tendency to avoid the recognition of the hierarchic character of reality or in the attempts to giving, without any justification, the privileged status of reality to one particular level of this hierarchy. The blind people in the parable experience separate parts of reality (the elephant) in this parts' geometric separation. Each of the blind men is touching di fferent parts of the surface of the elephant. It is still quite easy to reassemble the picture of the animal by gluing together fragmented images. We can consider ye<sup>t</sup> another version of the parable of the (rather science fiction) ability to penetrate the body of the elephant to di fferent depths. Once again, their reports will be di fferent.

Reality can be analyzed from another perspective of having multiple levels of the collections of its components. In the mathematical language of the set theory, these levels can be constructed with the concept of a power set. We start from some set S which forms the first level of the hierarchy. Then we consider the set of all its subsets, which is called the power set of set S. Of course, we can form the power set of power set and the constructions can continue forever. On the other hand, all (usual) set theories do not have any separation of sets and elements. All objects of these theories are sets (no matter how strange it may seem to non-mathematicians). The concept of an element is relative. One set, let us say, set x, can be an element of another set y, which for the purpose of simplicity is expressed as "x is an element of y" (x ∈ y), but it does not give x and y di fferent status. It is just an expression of the relationship between sets. This gives us the possibility of the infinite downward hierarchy.

We already know from physics that moving from one level of this hierarchy to another requires some change of the conceptual tools for the study of the collective phenomena which do not have any meaning at the lower level. We have examples of emergen<sup>t</sup> phenomena whose description or prediction cannot be derived from the lower level. These are well-known ideas. Little less known or recognized is the fact that for the description of symmetry, the most important concept in many disciplines of science and humanities, we have to consider three levels of this hierarchy. If we want to consider higher order symmetries, we have to consider more levels [23,56,57]. Thus, there is no reason to think or to believe that these levels are just a creation of the human mind. At least, we should consider this hierarchic

structure of reality a central subject of study and we should assess its ontological status based on the results of this study.

Now, the Idols of the Elephant appear here because there is a consistent tendency in many domains of inquiry to flatten the vision of reality. Traditionally, this tendency was expressed in reductionist forms of physicalism. In this position, we can only consider as real, one level of the hierarchy. Originally, this distinguished level was a stage of the set of points of space-time in which atoms or point masses were actors. Later atoms were replaced by elementary particles and the empty stage of points was equipped with the assigned to them vectors of the fields. All collectives of the higher level were just abstract creations of our inquiry without any right for independent existence.

Both the physicalist and reductionist position have lost attractiveness and are currently retreating, mainly under the influence of the reflection on the forms and mechanisms of life. However, the tendency of flattening reality remains, for instance, in the form of a variety of doctrines of structural realism initiated by John Worrall in 1989 [58]. The change in this direction consists in giving exclusive or, at least, primary existence to the second level instead of the first. Whichever level we choose, it may be the perspective of an individual blind man. To avoid this type of the Idols of the Elephant, we should wait for giving the priority to any particular level. If (for some unlikely reason) there is a reason to prioritize some levels over the other, this has to be justified by an empirically testable explanation and justification. In the absence of this justification, the hierarchic architecture should be retained until demonstrated otherwise.

The "flattening" tendency can be identified not only in natural sciences, physics, or philosophy. In some sense, it can be identified in mathematics, too. The shift in mathematics with some analogy to the shift in philosophy towards structural realism can be identified, for instance, in the category theory. In the admittedly oversimplified summary of the category theory, it can be described as an attempt to eliminate the first level. We give the priority to morphisms acting on objects. In the traditional perspective of set theory, both objects are sets of elements from the first level and therefore, morphisms are elements of the second level. This can be considered as a highly efficient way to deal with complexity by lifting the level of abstraction. Such an argumen<sup>t</sup> would have been convincing, if we had only two or, at most, a few levels of the hierarchy. However, the hierarchy is infinite, so lifting or lowering by one level is irrelevant. This does not preclude the fact that the category theory and its easy diagrammatic representation may have multiple good contributions to mathematics and its applications.

**Funding:** This research received no external funding.

**Acknowledgments:** The author would like to express his gratitude for the encouragement, criticism, and helpful comments from the reviewers and the editor.

**Conflicts of Interest:** The author declares no conflict of interest.
