HOW ABSTRACTION WORKS
Leon Horsten, Hannes Leitgeb · 2009
In this paper we describe and interpret the formal machinery of abstraction processes in which the domain of abstracta is a subset of the domain of objects from which is abstracted. ABSTRACTION IN THE HISTORY OF PHILOSOPHYPlato distinguished the world of ideas or forms from the world of experience.His distinction has been with us ever since.Plato's theory of the orderly world of forms was a first description of what we would now call the abstract realm.The question about the ontological relation between the abstract realm and the realm of the given proved to be very difficult.One famous bone of contention was the issue of ontological dependence.Plato held that the existence of abstract entities is in some sense independent of the sensory realm.Indeed, he thought that the existence of the world of experience is dependent on the abstract realm: the sensory world consists of "shadows" of ideas.Aristotle riposted that it is rather the other way round.The existence of the forms depends on the sensory objects in which they are realized; the forms only exist in the sensory objects.Aristotle added that on the cognitive side there is a constructive mental process associated with the relation between forms and objects.The human mind actively abstracts or extracts forms from the objects in which they are realized.This is done by "forgetting" some aspects of objects and focussing on others.Note that Aristotle does not say much about how relations (rather than properties) are abstracted: his was a monadic instead of a polyadic point of view.A standpoint intermediate between Plato and Aristotle can be adopted.One can hold with Aristotle and against Plato that the existence of an abstraction somehow depends on the given, and at the same time hold with Plato and against Aristotle that an abstraction does not exist in the objects from which it is abstracted.In other words, abstract objects are not multi-