4. Dichotomy

Dichotomy both collectivizes and dichotomizes, without being intrusive on the dichotomized domain.

Although the universe may be divided into things, the dividing line itself does not have any concrete existence. Neither do any number of dividing lines: the dividing line itself does not occupy the same space that the objects occupy. However, this does not entail that the dividing line is insignificant: it is essential for the formation of sets. Conceptually, there is a difference between a set of apples and a set of pairs of apples, even if these sets ultimately refer to the same apple material (i.e. if they have the same spatial extent). In this section, we explore the nature of the boundary that is created by dichotomy.

Sets are discrete: they may be divided into their members in only one way. Wholes are continuous: they may be divided into further parts in arbitrary ways.

If something is created out of everything by a process of dichotomy, then it is a part of everything, but it is not a subset of everything. Hence, there is a very distinct difference between parthood and subsethood. With respect to parts, if my hand is a part of my arm, and my arm is a part of my body, then my hand is a part of my body. With respect to subsets, however, if my hand is a subset of my arm, and my arm is a subset of my body, then it is not true that my hand is a subset of my body: my hand is a subset of a subset of my body. Expressed mathematically, the transitive property does not hold for subsets, but it does hold for (spatiotemporal) things.

The table above compares some of the terminology typically associated with set theory and mereology. In general, both sets and wholes are things , and both subsets and parts are somethings . All of the differences between set theory and mereology are ultimately due to the fact that in set theory, the curly braces have (ontological) significance. In other words, the curly braces cannot be taken away without consequences; they establish a significant boundary, such that the set of a thing is not equivalent to that thing. The set, therefore, is more than the sum of its parts.

Despite the ontological significance of boundaries, however, they are not of the same nature as the things that they collect; they are not parts themselves. To reiterate, sets and subsets have boundaries that are in some sense real, while wholes and parts do not. In both cases, however, the boundaries are not intrusive on the things that they contain (or divide): boundaries, even when they have some reality, are not of the same nature as things. The difference between set boundaries and mereological boundaries is also apparent when we consider collections (either unions or fusions) of subsets and parts: set boundaries are preserved, but mereological boundaries collapse (the parts fuse together, which is why a mereological union is known as a fusion). Similarly, intersection (as defined for sets) is not a valid operation on a continuum: hence, a mereological division is referred to as a dichotomy or a partition.

A universe has no boundaries

Universes do not have boundaries; they are by definition unbounded. If a universe did have a boundary, then there would be something in it which it did not contain (and therefore it would not be a universe). In this section, we briefly examine the subjective/objective boundary from the point of view of both the subjective universe and the objective universe.

From the inside of a subjective universe, there are no boundaries: you do not see what you do not see (we cannot experience the objective world in a manner other than that in which it comes to us through our subjective experience). Although you experience a limited subjective world, it is impossible to experience the edge of the subjective world. To be more precise, you can know that the subjective world has a boundary or edge, but you cannot perceive it: to perceive an edge as such entails perceiving both of its sides. For example, from the outside, you may view yourself as coextensive with your body. But from the inside, your senses extend right through this boundary; they sense as far as they can, and vision perceives a good deal further than the exterior of the body. So when viewed from the inside, there is no inherent boundary at the edge of your body: in fact, there is no boundary at all.[20]

Similarly, from the outside looking in, there are no boundaries. Psychology has been looking inward (into the brain) for a long time, expecting to find the seat of the soul, but it cannot find the boundary point at which we cross from the objective world into the subjective world.[21] This boundary seems to retreat endlessly, no matter how far into the neural pathways you look. If you look from the outside-in, it looks like sensation continues all the way through (and before you know it, you wind up in action). It may turn out to be somewhat of a doomed endeavor to try to localize a subjective experiencer in the first place, if the boundary between the experiencer and what is experienced does not exist in the way that we think it does.

To summarize a few oddities about this elusive subject: universes themselves have no boundaries, parts of universes are created by boundaries that are not really there, and sets of things are demarcated by boundaries that are there in some sense (although we have not been explicit about their nature).

True and false are the essence of categorization.

Many statements may be either true or false: these statements are traditionally called propositions. These statements may not be anything other than true or false: hence, if a statement is not true, then we may infer that the statement is false. If it is not false, then we may infer that the statement is true. In slightly more technical terms, these statements are propositional functions which yield either a true or false result. For example, either an object has the property Px, or it has the property not-Px (which we abbreviate by writing ¬Px). The fact that there is no third alternative is known in the field of logic as the Law of the Excluded Middle. This law adds power to our reasoning: it allows us to infer statements on the basis of other statements, which might not otherwise be possible. This law is central to everyday reasoning, and it is closely related to dichotomy.

The law of the excluded middle is not just about binary (true/false) logic. For example, in the field of fuzzy logic, which is an extension of binary logic, the predicates take on true and false values, as well as values in between: for example, statements may be eighty percent true. For example, we may feel that an Asian pear is only somewhat of an apple. If we feel that it is seventy-five percent apple, then the equivalent of the law of the excluded middle in the fuzzy logic context allows us to infer that the Asian pear is twenty-five percent not-apple.[22]

Despite the power of the Law of the Excluded Middle, it is not always applicable. In particular, predicates have a range of valid application, or a set of things to which they can be applied. It is only in the case that a predicate can be validly applied that it divides a set of objects into those objects which have the property and those objects which do not. For example, assume that the predicate “green” can operate effectively only on things which are capable of having a color (i.e. things that emit radiation within the visible spectrum). Particles which are invisible in this sense may not be any color, so it would be meaningless to apply the distinction “green/notgreen” to such particles. If we are determined to apply this predicate, then we are forced to say that:

  1. The particle isn't green

  2. The particle isn't not-green

However, the combination of both of these statements is problematic under standard logical analysis, where the first statement, “The particle isn't green” , may be transformed into “The particle is not-green” , which contradicts the second statement.

Therefore, when we invoke the law of the excluded middle, we must be sure to take into account the domain on which the predicate operates. If we wish to be able to conclude that a thing is not-green, we require the following two preconditions:

  1. The thing isn't green

  2. The predicate green can be applied to the thing (i.e. the thing is in the domain of the function green)

[20] Note that we are considering boundaries to be things which divide one thing from another: the notion of an edge with only one side is paradoxical.

[21] Many psychologists expected to find something they called a homunculus, which literally means “little person” : it represents some smaller agent at the controls of the body, perhaps located in the brain. This type of thinking is paradoxical if it leads to the expectation that homunculi must themselves have homunculi.

[22] Another context in which negation finds a home is in set theory, where negation is represented as the set complement operation. Complementation plays the same role (in terms of forming a Boolean logic) as negation, although it generalizes to more entities than just true or false.