# Complexity: Quantity or Quality

Russell K. Standish (Department of Mathematics and Statistics, University of New South Wales (UNSW), Kensington, NSW, Australia)
DOI: 10.4018/ijsss.2014010103
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## Abstract

The term complexity is used informally both as a quality and as a quantity. As a quality, complexity has something to do with our ability to understand a system or object—people understand simple systems, but not complex ones. On another level, complexity is used as a quantity, when people talk about something being more complicated than another. In this article, the author explores the formalisation of both meanings of complexity, which happened during the latter half of the twentieth century.
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## Complexity As A Quantity

We have an intuitive notion of complexity as a quantity; we often speak of something being more or less complex than something else. However, capturing what we mean by complexity in a formal way has proved far more difficult, than other more familiar quantities we use, such as length, area and mass.

In these more conventional cases, the quantities in question prove to be decomposable in a linear way, ie a 5cm length can be broken into 5 equal parts 1 cm long; and they can also be directly compared a mass can be compared with a standard mass by comparing the weights of the two objects on a balance.

However, complexity is not like that. Cutting an object in half does not leave you with two objects having half the complexity overall. Nor can you easily compare the complexity of two objects, say an apple and an orange, in the same way you can compare their masses.

The fact that complexity includes a component due to the interactions between subsystems rapidly leads to a combinatorial explosion in the computational difficulty of using complexity measures that take this into account. Therefore, the earliest attempts at deriving a measure simply added up the complexities of the subsystems, ignoring the component due to interactions between the subsystems.

The simplest such measure is the number of parts definition. A car is more complex than a bicycle, because it contains more parts. However, a pile of sand contains an enormous number of parts (each grain of sand), yet it is not so complex since each grain of sand is conceptually the same, and the order of the grains in the pile is not important. Another definition used is the number of distinct parts, which partially circumvents this problem. The problem with this idea is that a shopping list and a Shakespearian play will end up having the same complexity, since it is constructed from the same set of parts (the 26 letters of the alphabet—assuming the shopping list includes items like zucchini, wax and quince, of course). An even bigger problem is to define precisely what one means by “part”. This is an example of the context dependence of complexity, which we'll explore further later.

Bonner and McShea have used these (organism size, number of cell types) and other proxy complexity measures to analyse complexity trends in evolution (Bonner, 1988; McShea, 1996), and in a later paper, hierarchical maximum (number of hierarchy levels, similar to the graph radius measure introduced in the next section) (Marcot & McShea, 2007). They argue that all these measures trend in the same way when figures are available for the same organism, hence are indicative of an underlying organism complexity value. This approach is of most value when analysing trends within a single phylogenetic line, such as the diversification of trilobytes.

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