## Reflections on Abstractions: Classification and Generalisation by Conceptualisation

Abstraction Awareness is about deeper understanding of abstraction, a concept so basic to human thinking. Subsequently abstraction is discussed by the means of basic Formal Concept Analysis.

Subsequently, we want to focus on entities with properties, as opposed to entities with relations, which we considered in former posts by means of Graph Theory.  The figure (context) shows an example, where the entities ‘1’ to ‘5’ have assigned properties like ‘even’, ‘odd’, ‘prime’, etc.
In Formal Concept Analysis this is said to be a context, with an intension (entities) and an extension (properties).
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Sole visualisation doesn’t take us much further, for the purpose of examining such structures.  Representations like in figure (visualisation) sometimes can be found in literature. However, they are only a bit nicer to read, since they do not enrich the analysis by any structural concept.
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A Concept in Formal Concept Analysis, is a combination of intensional and extensional elements as in figure (concept): each property is owned by each of the concept’s entities, and each entity owns all the properties of the concept. Here the concept of odd primes (let’s call them ‘opps’) is shown, in the context of the integers 1 to 5. More examples can be found here.

Conceptualisation is classification, seen from the abstractional perspective.  That is, in order to represent a concept, any of its entities on the correct abstraction level can be chosen. For example, dividing the ‘4’ by 2 results in another integer.  This is a property all evens share.  However, while ‘4’ by 2 is an even number again, this is not true for all evens. It would require a deeper abstraction level, with some ‘enhanced evenness’.

Ordering concepts is generalisation.  Structures usually consist of quite a lot of concepts, that are related to each other by the entities or properties they share.

Altogether they make the concept lattice (fundamental theorem of formal concept analysis).  In figure (concept lattice) the highest abstraction level is at the bottom, which is the concept of the integers 1 to 5.  It is broken down by specialisation, into odds, primes, and evens, and on a deeper level we find our concept of ‘opps’ again, as specialisation of odds and primes.

A concept lattice is more than plain visualisation, since it comprises conceptualisation, and thus, adds additional value to the examination of entity x property structures.
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Thus, we have seen how the basic notion of Concept in Formal Concept Analysis covers two of the fundamental notions of Abstraction: Classification and Generalisation.  Hence, the above formalises the practical experience, that conceptualisation, i.e. finding concepts (, naming) and ordering them, is a very helpful approach to gain a deeper understanding of structures.

So Long
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