# Mathematical Conflict Theory

Updated on June 5, 2012

## Simple Examples of Conflicts

In this hub, we shall begin to develop a mathematical theory of conflict based on several toy examples of simple conflicts. Because the ideas I wish to present are somewhat incomplete in my own mind, there may be some errors, or oversights, or better ways of thinking about conflicts than are presented at first in this hub. If you have any ideas, or thoughts/critiques on anything written here, please feel free to share as this is most definitely a work in progress.

Perhaps the best place to start are with some simple analogies to illustrate the basic ideas underlying the formation of this particular mathematical theory of conflict. I worry about starting with these examples of conflicts as their simplicity conceals a deep fundamental connection to paradoxes and problems in other areas of study, such as for example, a connection to Russel's Paradox, that might be missed if the ideas are dismissed too quickly based solely on the apparent simplicity of the following examples:

Example 1) Suppose one has two dogs, and a single house large enough only to fit one of the dogs. What will happen if one forces the two dogs into the one dog house that is large enough for only one of the dogs? The dogs will fight. This example encodes the fundamental form of conflict we shall study. The conflict always involves a space large enough to fit only one distinct object, and a conflict arises as a result of attempting to force two distinct objects into a space large enough only for one object.

Example 2) This example is a real example that I saw play out personally: two distinct families do not fit into a single house. I witnessed a case where two families moved in together for a period of about two years, the families began to fight and fued, and in the end, the conflict between the families was only resolved when one of the two families moved out into their own home. This conflict has the same form as the conflict in the first example. The space is the single home. The two distinct entities are the two families. In the particular case I observed, the home was simply not large enough to hold the two families.

I have told example 2 to several people, and one of them told me of a similar experience they had with a family that had moved in with them.

Importantly, in both examples 1 and 2, there are two canonical (by canonical, I mean simple runt-of-the-mill) solutions to the conflict. The conflict arises because we have a space large enough only to fit one item, and we attempt to force two items into the one space. The two obvious solutions: build a separate space, and place one distinct item in each of the two distinct spaces, or remove the distinction between the two items so as to have only one item that may then fit into the one space.

In example 2, the family moved into their own house, which was like a distinct space separate from the original home: this is the "build a separate space" solution. The other solution would have been to remove the distinction between the two families, so as to have "one big happy family" so to speak: a single house fits a single family quite nicely. As it turns out, the second solution was quite obviously not feasible based on the context the particular situation provided, but the first solution, to have one of the families move into a separate house, was quite feasible, and quite the obvious solution in the particular situation I observed.

In example 1, the same solutions hold. Either build a second dog house, or remove the distinction between the two dogs (which could mean simply to choose to have only one dog). It is quite obvious that one cannot make two dogs into a single dog, but building another dog house, is quite possible.

For conflicts of the form above, is it always the case that building a separate space is the only feasible solution? The next example shows that the answer to this question is no.

Example 3 (Russel's Paradox, or a version of it): Let X be the set of all objects that are not equal to themselves:

X={O|O ≠ O}, then ask the question does X = X? Suppose X=X, then X fails to satisfy the defining property of X, which is X ≠ X, and so X must not be contained in X. But if X is not contained in itself, then certainly X cannot equal itself by the definition of equivalence of sets. Thus X ≠ X. But if X ≠ X, then X does satisfy the defining property of X and so X is an element of X, and so X is contained in itself, and hence X equals itself, and so X fails again to satisfy its own defining property: what we have proved (please correct me if I am wrong, and I may be wrong) is that the set X ≠ X if, and only if, X=X. In this example, the space is the set X itself, and the distinction is between X and itself. So X plays both the role of the dog house, and the two distinct dogs. An object is only large enough to contain itself, being a single distinct object, and so is not large enough to contain a distinction that creates a dichotomy within the space, or object, itself.

As with the examples above, we could choose to create two distinct objects, say X and Y, and then it would be perfectly non-paradoxical to assert X ≠ Y as two distinct objects are somehow large enough to encode a single distinction (think, as an analogy, a single classical state encodes exactly one bit of information, that is, one distinction). The other solution is to remove the distinction between an object and itself, that is to assume as an axiom that for any object X, an equivalence relation on X must satisfy the axiom of self in-distinction, or self-equivelance (called the axiom of reflexivity for equivalence relations [1] ) X=X. This shows that it is not always the case that building a separate space is the only feasible solution to a conflict of the form listed in examples 1 and 2. Also note, that example 3 is a conflict that abstractly has the same form as the conflicts given in examples 1 and 2. This shows that there is a connection between Russel's Paradox, and conflicts, including human conflicts, of the form given in examples 1 and 2.

The fourth example shows that there are non-trivial examples of conflicts having the same form given in examples 1 and 2.

Example 4 (Conflict between a Scientific Claim and a Religious Claim): Consider the predicate

____ originated the species Homo sapiens according to _______. This predicate is our single space.

Compare the statements "God originated the species Homo sapiens according to Spiritual Text A" and "Evolution originated the species Homo sapiens according to the Theory of Evolution." These two statements conflict in two ways: the concept of God is ontologically distinct from the concept of Evolution, and Spiritual Text A and the Theory of Evolution are distinct authorities. In this case, the clause "____ originated the species Homo sapiens" attempts to contain both the word "God" and the word "Evolution", which represent ontologically distinct concepts, and the interpretation of the statement itself, like the dog-house, is only large enough to hold one distinct concept. If the two statements had read

"God originated the species Homo sapiens according to Spiritual Text A" and "Evolution originated the species Homo sapiens according to the Spiritual Text A", then the two statements together would have merely formed a contradiction. This suggests two logically contradictary claims are an example of a conflict of the form given in examples 1 and 2; and in fact, a conflict is a kind of generalization of the notion of a logical contradiction.

Like in example 3, we may attempt to resolve the conflict by removing the distinction between the scientific method and spiritual text, or we could try to create two distinct spaces, one in which to discuss religion, the other in which to discuss science. It turns out, both have been attempted. Scientology, and creationists' arguments, are examples of attempts to remove the distinction between science and religion in order to remove one or more conflicts between science and religion. The other attempt, which is to create two distinct "spaces," one in which to discuss religion, the other, to discuss science are as follows: the field of study known as Philosophy of Religion (from what I remember of my philosophy of science class years ago) is the space created to discuss religions in general. From my studies, or rather my take on what I learned, in my philosophy of science class, the field of Philosophy of Science is the space created in which to discuss science. Both the spaces "philosophy of religion" and "philosophy of science" are somewhat distinct fields of study, and so placing questions of religion squarely in the field of philosophy of religion, and placing questions of science squarely in the field philosophy of science is analogous to placing the one dog in the one dog house, and the other dog in the other dog house, both dog houses being distinct, so to speak. Note however, that placing questions of religion squarely in the field of philosophy of religion, and placing questions of science squarely in the field philosophy of science would resolve the conflict between science and religion, per cannonical solution 2 above, only if these two fields of philosophy were to be completely independent, i.e. were to be completely distinct, which means they would need to not overlap in any way significant enough to induce conflict of the form given in examples 1-3 above.

The four examples above motivate the development of a mathematical theory of conflict. The utility of a theory which generalizes, and studies, conflicts of the form above are that a better understanding of the two canonical solutions to conflicts of the form above can yield strategies to resolve those conflicts. Example 4 gives a real-world example of a kind of conflict which is important to many people, and so provides further evidence to support the importance of studying conflicts of the form given in examples 1-3 above.

The following are some open questions that are worth researching:

1) Can all human conflicts be reduced to ones of the form given in examples 1-3 above?

2) Does there exist an example C of a conflict of the form given in examples 1-3 in which the two canonical solutions provided are completely impractical, or impossible, for the example C?

In the posts that follow, I, and any who wish to work with me below, will attempt to bore out an abstract mathematical theory of conflict using the examples above as a guide to develop the theory.

Any thoughts, critiques, or suggestions related to the content mentioned above are most welcome!

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