5. Dimensions

Dimensions are an extension of the concept of dichotomy.

A dichotomy is the simplest form of dimension, which is just a two-way division. More generally, a dimension can have any number of divisions. In less mathematical contexts, dimensions are also known as scales . Scales are typically divided into four types: nominal, ordinal, and interval, and ratio (here, the ratio scale is treated as merely a type of interval scale). These types might also be called unsorted, sorted, and measured dimensions.

Nominal dimensions have unordered parts.

A nominal dimension is unordered in the sense that there is no basis to assign relative positions to things. In the figure below, we depict a nominal dimension by showing three things: everything, a named part ( “something” ), and the complement of that part ( “not something” ). As this is a nominal dimension, the relative left-right position of the children is not an essential characteristic.[23] For example, if “something” were to the right of “not something” in the diagram below, it would not make a significant difference:


Again, nominal dimensions may determine any number of parts. For a nominal dimension with N parts (or children), we refer to the corresponding division as an N-way division. If the parts do not overlap one another, which is the case for the diagrams in this book, this is also an N-way partition.

Ordinal dimensions are nominal dimensions that have an associated order.

In an ordinal dimension, the relative positions of the divisions have significance. In other words, if a dimension is ordinal, then it imposes an order (or at least a partial order) on the parts that it defines. As an example, finishing first, second, or third in a marathon constitutes an ordinal dimension: knowing the position does not tell you exactly what the winner's time was, it only conveys that one time was greater or less than another.

A diagram depicting an ordinal relationship is shown below, which shows a whole and two parts. We know that child things (parts) must be smaller than their parents, so we are able to determine a partial order between the nodes labeled “Whole” , “Part” and “Part of a Part” in the figure below. However, although we know that each part is smaller than its parent part, we don't necessarily know any of the sizes:


Part hierarchies, such as the one depicted above, represent ordinal dimensions because the parthood relation (the vertical dimension) imposes a partial order which the horizontal dimension does not. To reiterate, whether one sibling is to the left or right of another is not (structurally) meaningful, whereas it is meaningful to ask if one part is the parent of another.

Interval dimensions are ordinal dimensions that have an associated measure.

An interval dimension introduces an additional relation between parts that results in a measurable distance (metric) between the designated parts. As a numerical example, 1 is the same distance from 2 as 2 is from 3. One way of creating an interval dimension is to use the same condition for division at each level of the tree. For example, the figure below depicts a structure which is composed of exactly two types of things, a unit element and a sum:


Trees which have a fixed metric structure can be described with very few parameters. For example, because every numerical node in the tree above is identical, one can compute the number corresponding to any node in the tree. In the general case, interval dimensions are more flexible than this example illustrates: for example, they need not be linear (e.g. a logarithmic scale could be established by using multiplication instead of addition).



[23] The fact that the order of the children is not meaningful is also a characteristic of mathematical sets.