Introduction to Research K P Mohanan and Tara Mohanan

18 February 2021

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CHAPTER 3

CONCEPTUAL CLARIFICATION

3.1 Looking Back and Looking Forward

3.2 Examples of Conceptual Inquiry

3.2.1 Defining 3.2.2 Transdisciplinarising 3.2.3 The Structure of Concepts 3.2.4 Constructing Conceptual Theories

3.3 Wrapping up

The Research Gym

3.1 Looking Back and Looking Forward

In Chapter 1, we identified the components of research as in Fig. 3.1 (same as Fig. 1.1):

Figure 3.1

We expanded it subsequently as Fig. 3.2 (same as Fig. 2.3):

Figure 3.2

We then explored the methodological strategies to look for answers to questions.

In exploring different methodological strategies, we discovered that in order to look for an answer, we need to first be clear about what the question means.

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A question is formulated as a sentence, in some language or other. And sentences are made up of words. To understand the meaning of a question that a sentence expresses, we need to understand the concepts that the words in the sentence denote:

Figure 3.3

In examples of research questions that we discussed in Chapters 1 and 2, we had to clarify the concepts in the questions. Thus, in Chapter 1, we pointed to the concepts of ‘scholastic’, ‘academic’, ‘aptitude’, ‘intelligence’, ‘knowledge’, ‘science’, ‘social science’, and so on. In Chapter 2, we encountered the concepts of ‘curriculum’, ‘thinking’, ‘theory’, ‘biological evolution’, ‘ancestry’, and ‘triangle’.

Given that such conceptual clarification is central to research, we need to expand Fig. 3.2 as Fig. 3.4, to include conceptual clarification as a step between the formulation of the question and the process of looking for answers:

Figure 3.4

Conceptual clarification is best illustrated in the branch of philosophy called analytic philosophy. But conceptual inquiry — the investigation of concepts — is also an essential component of mathematics, science, and the humanities (beyond analytic philosophy). In this chapter, we explore some of the strategies of conceptual inquiry.

3.2 Examples of Conceptual Inquiry

3.2.1 Defining Conceptual inquiry begins with a question of the form:

What is x?

where x is a concept. An answer to such questions hinges on a definition of the concept.

We have already used the methodological strategy of defining to answer some of the questions we raised in Chapters 1 and 2. To quote:

“RQb: In discrete geometry, can every line segment be bisected?

“Try to answer this question. To do that, you would first have to define the concept of bisecting.” (Page 4, Chapter 1)

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“Do Curricula in Bachelor’s Programs improve the thinking abilities in students?” (Section 2.2.1, Page 4, Chapter 2)

“Do straight-angled triangles exist?” (Section 2.2.3, Page 9, Chapter 2)

We have also discussed the importance of defining the concepts we use within a definition. (Section 1.2.1, Page 7, Chapter 1)

You will find learning tasks that require you to construct and critically evaluate definitions throughout this course. So it might be useful to say something at this point about what a definition is, and how to go about constructing a definition.

Consider a concept C. A definition of C is Definition of C (def): a statement that allows us to deduce what comes under C and

what doesn’t

For instance, if we define the concept denoted by the word ‘bird’ as:

Bird (def): a creature with wings it follows that parrots and eagles are birds. But it also follows that butterflies and wasps are birds. If we do not wish to judge butterflies and wasps as birds, we would need to revise the definition.

If we define the concept of ‘liquid’ as Liquid (DEF): a concrete entity that has volume, but whose shape is that of its

container it follows that a grain of sand is not liquid. But it also follows that a handful of sand and a long piece of sewing thread are liquid. This is a logical consequence of the definition that we wish to avoid.

On the basis of these examples, we may define a good definition as follows:

A good definition of a concept C is a statement whose logical consequences match our judgements on whether an example E comes under or doesn’t come under C.

There are many strategies of looking for a definition. Suppose we are stuck because we are unable to answer the question, “What is x?” For example, take the question, “What is justice?”

When that happens, it is a good idea to ask, “What is NOT-X?” That is, what is the opposite of X? In this case, we ask, “What is injustice?” The second question is easier to answer than the first. And once it is answered, that answer would give us a clue to answer the first question.

We will keep returning to defining as a methodological strategy of research throughout this course.

3.2.2 Transdisciplinarising

Another useful strategy for defining a concept is to transdisciplinarise it.

What does that mean? If we are stuck with the question, “What is X?”, and X, Y and Z are sub-concepts of M, it is often useful to ask, “What is M such that X, Y and Z are instances of M?”

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Transdisciplinarising a concept is the process of generalising it across disciplines. This is a strategy that allows us to make connections in knowledge across domains. So when we are confronted with the question of the form, “What is X?” it is often a good idea to

~ generalise the concept across disciplines — to transdisciplinarise it;

~ define that transdisciplinary concept; and

~ add further specifications to that definition for the concept in the particular context.

We have already seen the beginnings of that strategy in Chapter 1 when we asked the question: What do we mean by ‘species’ in ‘biological species’, ‘chemical species’, and

‘cultural species’?

Using the question form, “What is x?” we may reformulate this question as:

RQ3.1 What is a ‘species’ such that biological species, chemical species, and cultural species are all instances of species?

Likewise, our question in Chapter 2 on the meaning of the term ‘evolution’ would be:

RQ3.2 What is ‘evolution’ such that evolution of the physical universe we live in (including that of the solar system, and of the earth), evolution of physics, and evolution of regional cuisines are all instances of evolution?

Before we talk about evolution, let us tentatively define ‘evolutionary change’ as follows:

Evolutionary change (DEF): the process of the emergence of a novel type of something (an entity, property, relation, function) that did not exist before.

Let us look at this definition in the context of the evolution of the physical universe. There was a time in the history of the universe when there were no atoms. The universe then evolved into a state in which atoms had come to exist. At this stage, there were no molecules. But the universe evolved, and molecules began to appear.

Does this mean that all change in the universe is evolutionary change? No, there are non-evolutionary changes as well. For instance, neither the cooling nor the warming of the earth is an evolutionary change, because on its own, it does not involve the emergence of anything novel.

Take another example, this time in the domain of the physical world. We know that the water in the seas is salty, but the water in rivers is not. Imagine a time when salt crystals never existed. Suppose someone filled a water tank with sea water. In summer, part of the water in the tank evaporated, and in winter, it cooled. As it cooled, it became a supersaturated solution. A child threw a stone into the tank. The disturbance resulted in the formation of salt crystals in the tank. Before the stone disturbed the water, there were no salt crystals on earth. After the event, salt crystals emerged. Neither the evaporation of the water nor its cooling are evolutionary changes. But the emergence of salt crystals is an evolutionary change.

When we come up with a definition of x, where x is any kind of entity or process, it is useful to give examples of both what we include as x, and what we don’t include as x, as in the above example of evaporation, cooling, and crystal formation. This will form the basis for critically evaluating definitions, whether our own, or those of others.

The concepts of ‘evolution’ and ‘evolutionary change’ are related. So, given the definition of ‘evolutionary change’, we can define ‘evolution’ as:

Evolution (DEF): a series of evolutionary changes (the history of those changes)

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Why did Darwin use the title The Origin of Species for his book on evolution? We now begin to understand. He must have viewed speciation — the emergence (origin) of novel species — as an evolutionary change.

Using the above definition of the transdisciplinary concept of evolution, we may define biological evolution as: Biological Evolution (DEF): the process of the emergence of novel types of species

that did not exist before.

We can now extend this to other instances of evolution as well: Evolution of organs (DEF): the process of the emergence of novel types of organs

that did not exist before. Evolution of biomolecules (DEF): the process of the emergence of novel types of

biomolecules that did not exist before.

Given these definitions, all the following are instances of biological evolution: the emergence of organs like eye, limb, vertebra, and the neocortex; the emergence of proteins; and the emergence of traits like eye colour and hair colour.

Similar questions that call for generalising from the discipline-specific level to the transdisciplinary level include:

RQ3.3 What is ‘change’ such that change of location (through motion), change of the states of affairs, historical change, societal change, developmental change, economic change, and temperature change are all instances of change?

RQ3.4 What is ‘structure’ such that structure of the atom, structure of the molecule, structure of the cell, structure of the skeleton, structure of the heart, structure of a sentence, structure of a poem, and structure of an argument are all instances of structure?

We leave it to you to grapple with these questions as we navigate through this course.

3.2.3 The Structure of Concepts As we have seen, when exploring concepts, we ask, “What is x?” And the response is often a definition. But there may be words in the definition itself that trigger questions about the concepts that each of those words denotes. As an example, take the question:

RQ3.5 What is a square?

A reasonable response would be the following definition: Square (DEF): A square is an equilateral rectangle.

The above definition is a combination of two statements: Square (DEF): A square is

i) a subcategory of a rectangle; and ii) equilateral.

Why do we divide a definition into the atomic statements that it is made of?

This will become clear as we proceed.

Notice that while (i) and (ii) jointly form a definition, neither (i) nor (ii) by itself is a definition.

But statements (i) and (ii) in the definition talk about ‘rectangle’ and ‘equilaterality’. How do we define these concepts?

We can define ‘rectangle’ as follows: Rectangle (DEF): A rectangle is an equiangular quadrilateral.

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We then unpack the definition as: Rectangle (DEF): A rectangle is iii) a subcategory of a quadrilateral; and iv) equiangular.

And we can define ‘equilaterality’ as: Equilaterality (DEF): the attribute of all sides being of equal length.

Statements (iii) and (iv) raise two further questions: How do we define ‘quadrilateral’ and ‘equiangularity’?

And the answers would be: Quadrilateral (DEF): A quadrilateral is a four-sided polygon.

We unpack this definition as:

Quadrilateral (DEF): A quadrilateral v) is a subcategory of a polygon; and vi) has four sides.

And we can define ‘equiangularity’ as: Equiangularity (DEF): the attribute of all angles being equal.

And what is a ‘polygon’? Polygon (DEF): A polygon is a figure with three or more vertices such that each

vertex is connected to exactly two other vertices through straight lines.

Unpacking this definition, we get: Polygon (DEF): (vii) A polygon is a figure with three or more vertices. (viii) Every vertex is connected to exactly two other vertices. (ix) The vertices are connected through straight lines.

Let us pause for a moment and reflect on the nature of the statements in (i)-(ix). What is most obvious is that every answer to a question of the form, “What is x?” triggers further questions of the form “What is y?” about the concepts in the answer. This process can in principle be never ending (called ad infinitum in Latin), so at some point, we have to decide to stop digging further, and not continue to ask, “What is x?” (At what stage we make this decision is a question we will consider at a later point.)

Next, notice that the answers in (i), (iii), and (v) are of the form, “x is a subcategory of y.” This is a statement of a relation between two entities.

If we use the notation of a dot to represent an entity, and the notation of a line between two dots to represent a relation, we may diagrammatically represent “x is a subcategory of y” as in Fig. 3.5: Figure 3.5

Using this notation, we may represent the relations expressed by (i), (iii) and (v), as well as their logical connectedness, as in Fig. 3.6, where the numbers to the left of the lines specify the relations:

Figure 3.6

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To this, we may add the specifications of the properties of the entities ((ii), (iv), (vi), and (vii)-(ix)) as in Fig. 3.7:

In the relation ‘x is a subcategory of y’, we will use the terms ‘mother’ to refer to the y, and ‘daughter’ to refer to the x.

Figure 3.7

For a question of the form, “What is x?” what is the advantage of expressing the answer in terms of the subcategory relation?

The answer is this: The subcategory relation is governed by the principle of Logical Inheritance stated below, which allows us to make inferences about a daughter in the structure from the specifications on the mother:

The Principle of Logical Inheritance A daughter inherits the attributes of the mother.

For instance, a polygon has three attributes, (vii)-(ix), specified in Fig. 3.7. By the above principle, these properties are inherited by its daughter, a quadrilateral. A quadrilateral has an additional property, (vi), so its daughter, a rectangle, inherits (vi), as well as (vii)-(ix). A rectangle has an additional attribute, (iv), so by the principle, a square inherits all of these attributes: (iv) and (vi)–(ix). A square has the additional attribute (ii). So we can infer that a square has the following attributes:

Its vertices are connected through straight lines. (ix) Every vertex is connected to exactly two vertices. (viii) It has four sides (and hence four vertices). (vii), (vi) It is equiangular. (iv) It is equilateral. (ii)

Notice that, except for (ii), these attributes are not stipulated on the square: they are derived through, or predicted by the application of the principle of Logical Inheritance.

What we have outlined above is a set of logically consistent statements that allow us to make predictions about geometrical entities. This set of statements may be thought of as a theory of squares as a sub-theory of a theory of polygons in geometry.

If you now combine this theory with theories of other shapes, including lines and ellipses, you get an initial sketch of the theory of geometry that you learnt in school.

Let us pause to reflect on two points.

1. We began with the question, “What is a square?” and ended up with the rudiments of a theory of geometry.

2. Many abstract concepts have a rich structure, which we can recognise and appreciate only through what is revealed by a careful investigation of questions of the form, “What is x?”

Read this section again for a deeper understanding of these two points.

Exercise: To understand what we are trying to do, it would be useful for you to construct a theory of regular triangles along the same lines as done above for squares.

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3.2.4 Constructing Conceptual Theories

Consider this question:

What should we do to make sure there is true democracy in the world?

We cannot find an answer to this question without first answering another question:

RQ 3.6 What is democracy?

Let us consider the following candidates for a definition: Democracy (DEF A): a political system of voting to elect the rulers of a country.

Democracy (DEF B): a system in which everyone who is affected by a decision has an opportunity to influence that decision.

It may be possible to define democracy in other ways. But for the purpose of learning how to construct and evaluate definitions to clarify abstract concepts, let us use these two competitors, and think of our challenge as choosing between them.

In evaluating scientific theories, we often need to use the methodological strategies of experimental observations to test the predictions of the theory — to check if the logical consequences of the theory match the results of experiments. In evaluating philosophical theories, such as a theory of democracy, we use a similar strategy: we design thought experiments to check if the logical consequences of the theory match the results of those thought experiments.

Let us conduct a few thought experiments.

A thought experiment is an experiment that we conduct in our mind, as distinct from a real experiment.

Scenario 1: Imagine a group of eight friends, who are inseparable. They are always together, whether doing their homework, playing, watching movies, or having their meals. But they do have their individual preferences: two are vegetarian; one hates garlic; three dislike martial arts movies; and so on. Whenever they go to a restaurant, or for a movie, they make sure that everyone’s preferences are accommodated. Is this practice of the group a democratic one?

Scenario 2: Imagine a family in which all decisions affecting the children’s lives are made jointly by the father, the mother, and the children, through rational discussion, negotiation, and consensus. They jointly decide whether the father/mother should accept a job offer, or a promotion with transfer to a different city; what subjects the children should study, what extra-curricular activities they should join, what TV programs they can watch and for how long, and what specialization they choose for their higher studies. Is this practice of the family a democratic one?

Scenario 3: In HW School, all decisions affecting students and their learning are made jointly by the Principal, administrative staff, teachers, and students. If students are interested in learning something that is not currently part of the school curriculum, they discuss it with the Principal and teachers, and if feasible, the school offers the course. Syllabi, textbooks, homework, assignments, and deadlines are negotiated between teachers and students. If there is a discipline problem, a committee of teachers and students figures out a solution, as well as a penalty, if needed. Is this practice of the school a democratic one?

Scenario 4: In SVS School, all teachers and students have equal voting rights. The

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system works like this. There is one teacher for every ten students. So the children mostly get what they want by voting for them — holidays, food, movies, picnics. If a teacher thinks that learning something is valuable for the children, but the children are not interested, they can outvote the teacher’s proposal. Is this practice of the school a democratic one?

Scenario 5: In AVA School too, all teachers and students have equal voting rights. The system works as follows. Every year, they vote to elect the School Administrator (SA) and Assistant School Administrator (ASA). The Principal and Deputy Principal (DP) nominate a set of four candidates. The candidate with the highest number of votes becomes the SA, and the one with the next highest votes becomes the ASA. The Principal and the DP give advice on various matters, but all decisions in the school are taken by the SA and the ASA. The Principal and the DP have the power to fire the SA and the ASA. They also have the power to re-nominate the same people for the positions of SA and ASA each year. Is this practice of the school a democratic one?

Scenario 6: Imagine country A where elections are held every five years. There is a large extended extremely wealthy family in this country. Only members of that family are allowed to stand for election. In every election, members of the family are nominated, and the people vote to elect ten of them as ministers. The ministers elect one of them as the Prime Minister. Does Country A have democracy?

Scenario 7: Country B also holds elections every five years. There are four extended wealthy families in the country. For every election, each of the four families nominates ten candidates. The people vote to elect ten of the forty as ministers. The ministers elect one of them as the Prime Minister. Does country B have democracy?

Scenario 8: Country C too holds elections every five years. There are four criminal mafias in the country. For every election, each mafia nominates ten candidates. The people vote to elect ten of the forty as ministers. The ministers elect one of them as the Prime Minister. Does country C have democracy?

Did you judge the practices of the groups in Scenarios 2 and 3 to be democratic? These groups do not resort to voting. So if your answer is ‘yes’, then clearly, voting is not a necessary requirement for a system to be considered democratic.

How about country C in Scenario 8? If you judged country C to not have democracy, then voting is not a sufficient requirement either. In other words, voting is neither sufficient nor necessary for a system to be democratic. Voting then is only one of the mechanisms designed to implement democracy. If it doesn’t achieve democracy, the mechanism fails.

What is democracy then? We leave that question for you to chew on.

Incidentally, the school in Scenario 3 is not an unrealistic fantasy. It is possible to ensure collective decision-making in a family or in a school where all decision-makers can come together in a room or a large hall to make decisions. The video on Summerhill (2008) at https://www.youtube.com/watch?v=TxngqMavda0&t=1387s is precisely about such a democratic school.

[Note: For those who might be interested, here is an excellent article on democracy, by Justice P.B. Sawant, a former judge of the Supreme Court of India and former chairman of the Press Council of India, published by The Wire. https://thewire.in/rights/how-much-of-a-democracy-is-india-really ]

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3.4 Wrapping up

In this chapter, we expanded our model of research in Fig. 3.4 to include conceptual clarification, the component of research that raises questions on abstract concepts. Even though conceptual clarification is the hallmark of Analytic Philosophy, this strand of inquiry is fundamental to all forms of academic knowledge: mathematics, the physical-biological-human sciences, and the humanities.

Conceptual inquiry begins with a question of the form, “What is x?” where x is an abstract concept. The response to it is a definition of the concept.

In Section 3.2.1, we used the concepts of species and evolution to illustrate a powerful strategy of clarifying discipline-specific concepts, by:

(i) transdisciplinarising it — abstracting it in a way that it is generalised to hold across and beyond disciplinary boundaries;

(ii) defining that transdisciplinary concept; and

(iii) adding further specifications to narrow it down to the discipline under consideration.

In Section 3.2.2, we illustrated the complex structure of concepts — by asking the question, “What is x?” recursively, and defining concepts that appear as part of the definition of another concept. We used the concept denoted by the word square to demonstrate how this strategy may lead us towards constructing a theory of polygons, with multiple statements interlinked through logical connections, showing the logical structure of academic concepts.

To do this, we abstracted away atomic statements about subcategory relations, showing how the principle of logical inheritance allows for predictions, needed for constructing theories, and as we will see in later chapters, for evaluating theories as well. This statement requires two further remarks:

Remark 1: You would have noticed that the subcategory relation we saw when constructing a theory of triangles — and by extension, a theory of polygons, and looking more broadly, a theory of geometry — points to the transdisciplinary concept of categories. To take an example from chemistry, ‘electron’ and ‘muon’ denote categories of fundamental particles; ‘oxygen atom’ and ‘hydrogen atom’ are categories of atoms; and ‘water molecule’ and ‘carbon dioxide molecule’ are categories of molecules.

The compositionality relation is different from the subcategory relation. When we say that X is composed of Y, we may be talking either about substances or about units. We say that a compound is composed of elements, where both the compound and the element are substances. When we say that a molecule is composed of atoms, and atoms are composed of fundamental particles, we are talking about a relation between units and subunits.

Remark 2: You must have also noticed that the notation we used for the diagrammatic representation of subcategory relations is that of a tree diagram, where a node represents an entity and a line between two nodes represents the relation of subcategorization. This notation is used for classificatory (subcategorisation) trees in biology, where statements like, “Humans are a subcategory of primates, primates are a subcategory of mammals, mammals are a subcategory of vertebrates, and vertebrates are a subcategory of animals,” are represented in terms of the notation of tree diagrams.

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It is good to keep in mind that such classificatory trees are distinct from the trees that represent ancestry relations (“x is an ancestor of y” = “y is a descendant of x”) in phylogenetic (ancestry) trees.

We will return to the themes of subcategories, subunits, compositionality, and ancestry at a later point in the course.

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THE RESEARCH GYM (GYM 3)

QUESTION 1

Using the strategies we used in building a theory of squares in Section 3.2.2 as an example, build a theory of the human species by asking, “What are human beings?” To do this, you could begin with the following statements:

1. Humans are a subcategory of primates. 2. Primates are a subcategory of mammals. 3. Mammals are a subcategory of vertebrates. 4. Vertebrates are a subcategory of animals.

Your task: a) Draw a tree diagram for (1)-(4). b) In the tree, include the categories bird, fish, lizard, insect, butterfly, ant, and worm. c) On the basis of the information you can gather from Wikipedia entries, specify at

least two attributes of each category, as in Fig. 3.7. d) For the category node of humans, specify which attributes are inherited on the

basis of the principle of logical inheritance.

QUESTION 2 [This is an advanced question, and hence optional. ☺]

Consider the following theoretical principle: Principle: Organisms and species that are ill-adapted to their environment are less

likely to survive than their well-adapted counterparts; when the ill-adaptedness of organisms and species increases beyond a threshold, they die/become extinct.

Your task:

1) Try to clarify the concepts of adaptation, adaptedness, and ill-adaptedness.

2) Using the concept of ill-adaptedness you have formulated, examine the following scenarios to evaluate the credibility of the theoretical principle given above. a) Place a fish, a frog, a mouse, and a butterfly immersed in water for an hour.

What do you think would happen? b) Place a fish, a frog, a mouse, and a butterfly on a land surface for an hour. What

do you think would happen? c) For a month, leave a few butterflies and cows enclosed in a grassland, where

there are no flowers. What do you think would happen? d) For two months, leave a few koala bears and polar bears in an enclosed forest

where the only vegetation is eucalyptus trees. What do you think would happen? e) A dozen highly educated academics are stranded for months on a deserted island

with dense tropical forests; a dozen aboriginals are stranded for months on another deserted island of the same kind. What do you think would happen?

Use the results of the thought experiments in (a)-(e) to argue for or against the proposed theoretical principle.

A QUICK FUN READ: “Topology 101: How-Mathematicians-Study-Holes,” by David S. Richeson at https://www.quantamagazine.org/topology-101-how-mathematicians-study-holes-20210126/