Axiomatic systems Mathematics typifies an axiomatic system in that it is based on specified primitives e.g. a point which has location only, and relations between them - in conventional geometry a straight line is the shortest distance between two points. From these basic premises the whole system is then deduced. Uninterpreted axiomatic systems remain simply abstract. Interpreted systems are those in which the entities and relations are considered to describe some aspect of reality. Thus y=a+bx is the general uninterpreted equation of a straight line but v=u+at is the equation describing how velocity develops over time for a given acceleration and is interpreted.
Clusters are types, qualitative sets, which 'emerge' from the application of computation to large multi-variate data sets. The clusters depend on what we are looking for, what measurements we have made to begin with, and which of these measured variates we use as classificatory principles.
Complex causes When we have an effect which not only has more than one cause but in which there is interaction between these multiple causes so that the effect is the result of something other than the sum of the separate effects of the causes taken individually we have complex causation.
Contingent causes Here we see the effect of a cause or multiple causes modified by the context in which they operate. This is a special case of complex causation e.g. exposure to an infection has different outcomes depending on the general health of the person exposed and / or whether they have had previous exposure or been immunized in relation to that infection.
Crisp set - see Fuzzy sets
Fuzzy sets are sets whose elements have degrees of membership as opposed to Crisp sets whose membership is assessed in binary terms according to a bivalent condition - an element either belongs or does not belong to the set (0,1). By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set.
Interaction - the effect of two or more variable acting together. An interaction is not simply the sum of their effects taken separately. Instead we find that there are complex emergent properties.
Isomorphism - If a particular mathematical system or algebra is isomorphic with an aspect of reality then the relations within that system correspond exactly to the relations in reality. Thus on a plane surface the shortest distance between two points is a straight line and assuming this works well for travel over short distances so we can map using Euclidean geometry which applies this principle. However, in flying to the west coast of the US and moving over the curved surface of the Earth we need to use another geometry and navigate using a great circle method based on a different geometrical system which is isomorphic with spatial relations on the surface of a sphere.
Methodological pluralism or mixed methods research is a research approach that combines the collection and analysis of quantitative and qualitative data.
Nested systems Complex systems can be contained or nested wholly or partially within other complex systems. For example, a city region is contained within a nation state which itself is contained within the whole world system. All three are complex systems. Complex systems can have fuzzy boundaries.
Triangulation is the use of two or more methods in a research study with a view to double (or triple) checking results. The idea is that confidence in the results is increased if the same results are obtained by different methods.