A New Account of Abstraction?

Software engineering, as a discipline, could benefit from a more rigorous grounding in epistemology, e.g., for the basic account of the pervasive concept of abstraction. So, let’s see what we can learn from Raymond Turner (2018) Computational Artifacts (Ch. 21: Data Abstraction) [1]:

Traditional Account of Abstraction

To give a common understanding of the traditional account of abstraction, Turner refers to Lewis’ three ways of distinguishing abstracta from concreta:

Way of Abstraction:
Abstract entities are abstractions from concrete entities. They result somehow from subtracting specificity, so that an incomplete description of the original entity would be a complete description of the abstraction.

Way of Conflation:
The distinction between concrete and abstract entities is just the distinction between individuals and collections, or between particulars and universals, or perhaps between particular individuals and everything else.

Way of Negation:
Abstract entities have no spatio-temporal location; they do not enter into causal interaction.

See Burgess and Rosen [2]. Further examples given, can be categorized in one or more of these ways. Notice that unlike Negation, the ways of Abstraction and Conflation do not depend on actual physical laws; they are essentially purely structural (up to isomorphism).

New Account of Abstraction

As the essence of traditional accounts of abstraction, Turner extracts two aspects to give a new account of abstraction:

A process of similarity recognition;

The formation of a new idea or concept on the basis of these similarities.

Following Hale and Wright [3], the former can be formalized, with a function f and an equivalence relation R, by

f(a) = f(b) if and only if R(a, b)

e.g., the parents of a = the parents of b if and only if a is sibling to b. Upon this, the latter aspect can be seen as a corresponding kind Kf such that x is a Kf if and only if, for some y, x = f (y), like the concept of siblings.
Subsequently, the author employs this account to analyse the abstraction process in abstract data types, and thus is able to depict “… how the type-theoretic way of abstraction provides a way of implementing the new abstract notions in terms of the old ones”.

Takeaway

Fig.1: Formal Concept

As a professional abstractor (in the software business), a well-founded understanding of abstraction is definitely a good thing to have. For this purpose, I’ve been happy with Formal Concept Analysis. It provides the concepts of extension – objects like a and b above – and intension – properties like f above – and joins them in the concept of the formal concept, as sketched in Figure 1.
Moreover, it comes with a fundamental theorem – that the set of all concept constitutes a lattice – and the so called reading rule to conveniently formulate higher abstract concepts on the top of lower ones. It also provides a solid grounding for discussing further epistemic aspects, like if a formal concept presupposes an intension etc.

Thus, I prefer to stick with Formal Concept Analysis as my basic account of abstraction, for practical and epistemic purposes.

Opinions welcome,
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Raymond Turner (2018) “Computational Artifacts (Theory and Applications of Computability)” ch. 21 Data Abstraction, Springer Berlin Heidelberg.
DOI: 10.1007/978-3-662-55565-1
J. Burgess, G. Rosen (1997) “A subject With No Object” Ch. What is Nominalism?, Oxford University Press
ISBN: 9780198236153
B. Hale, C. Wright (2009) “The Metaontology of Abstraction” in Chalmers, Manley, Wasserman (Eds.), Metametaphysics: New Essays on the Foundations of Ontology; Oxford University Press.
ISBN 9780199546046
B. Ganter, G. Stumme (2003) “Formal Concept Analysis: Methods and Applications in Computer Science”
lecture notes