Levels of abstraction

Dynamics of Symmetry Group Theorizing (Part #6)

The progressive comprehension of higher orders of symmetry is intimately related -- in mathematics at least -- with a particular understanding of increasing levels of abstraction. A proper understanding of abstraction is considered to be a prerequisite for explaining mathematical concepts and for doing mathematics. As a concern of philosophy, however, abstraction has been a focus of continuing debate since Aristotle -- who introduced the notion to the west (cf Andrzej Maryniarczyk, Abstraction, Powszechna Encyklopedia Filozofii). Mehmet Fatih Ozmantar and John Monaghan (A Dialectical Approach to the Formation of Mathematical Abstractions. Mathematics Education Research Journal, 2007) offer a useful summary of current perspectives on the nature of abstraction.

In a helpfully extensive review of the challenges of abstraction from an educational perspective, Agnes Edling (Abstraction and Authority in Textbooks: the textual paths towards specialized language, 2006) makes the point that:

Abstraction can be seen as an important feature of many varieties of specialized language. The function of specialized language is not just to use language in a pretentious way which excludes the uninitiated. It is rather a necessary part of that knowledge. Academic contexts are constituted through abstraction in texts.

Abstraction in general can be viewed as an essential feature of specialized texts in a society where expertise and specialization are increasingly significant. The focus of the study is on the different levels of abstraction and the transitions between them in literary, social science and natural science texts. This requires clarification of the nature of abstraction, presented in this case in terms of the scales concrete-abstract and specific-general. The point is made that:

But the view of abstraction as enabling large meanings in a concise form is not the only understanding of abstraction. In some contexts, abstraction is regarded as a feature that complicates and hinders students' access to texts. In a pedagogical context, where researchers discuss ways of facilitating the encounter between reader and text, abstraction, as well as other features of specialized discourses, is sometimes seen as problematic.

In the case of objected-oriented programming -- so fundamental to modern computing -- Alex Mueller (What is abstraction? 2006) argues:

Abstraction, in my opinion, is one of the more complex concepts to understand in object-oriented programming. First, the understanding of abstraction requires a good cognizance of objects, classes, inheritance, encapsulation, and polymorphism. Second, the definition of abstraction is somewhat abstract, in that it is not specific. Third, the ability to apply abstraction when designing a system demands OO concept recognition, which takes time and practice.

The argument is echoed in a detailed, cross-disciplinary review of the literature relevant to computer programming by Jeff Kramer (Abstraction: the key to computing?), noting the constrained capacity of some students with respect to the handling of complexity, the production of elegant models and designs, and the applicability of various modelling notations and other subtle issues:

They tend to find distributed algorithms very difficult, do not appreciate the utility of modelling, find it difficult to identify what is important in a problem, and produce convoluted solutions that replicate the problem complexities. Why? What is it that makes the good students so able? What is lacking in the weaker ones? Is it some aspect of intelligence? I believe that the key lies in abstraction: the ability to perform abstract thinking and to exhibit abstraction skills.

Whilst this clearly has fundamental implications for the possibility of abstraction in policy-making, with respect to mathematics, Raymond Duval (A Cognitive Analysis of Problems of Comprehension in a Learning of Mathematics. Educational Studies in Mathematics, 2006) notes:

To understand the difficulties that many students have with comprehension of mathematics, we must determine the cognitive functioning underlying the diversity of mathematical processes. What are the cognitive systems that are required to give access to mathematical objects? Are these systems common to all processes of knowledge or, on the contrary, are some of them specific to mathematical activity?

Victor Katz et al (The Role of Historical Analysis in Predicting and Interpreting Students' Difficulties in Mathematics, 2000, p. 153) note:

A final common student difficulty involves the transition to abstraction. As a typical example, many instances of what today are called groups were known in the first decades of the nineteenth century -- and some were known even earlier. Yet it was not until 1882 that the first complete formal definition of this abstract concept was given. Nevertheless, many current textbooks in abstract algebra begin by giving a formal definition of a group before the student has experienced many of these examples. It is not surprising that students have difficulties making the leap to abstraction; too little attention has been paid to the necessary steps that historically preceded this leap.

One of the continuing challenges is the nature of the "objects" that are "discovered" through the development of mathematical insight and the vexed issue of whether they "exist" independently of that discipline or its practitioners -- as presented, for example, by Øystein Linnebo (Epistemological Challenges to Mathematical Platonism, Philosophical Studies, 2006; The Nature of Mathematical Objects, 2008).

Do the skills that enable the discovery of those object, and others beyond ordinary understanding, reflect a view of symmetry by people with a particular mindset, with particular skills, and with a particular understanding of generality? To what extent is their appreciation of higher orders of symmetry an "acquired taste" -- dependent on special training of the "palette"?

To the extent that their beautiful symmetry can be understood by others, why are so many such mathematical objects apparently without relevance to the psycho-social domain -- other than as nice posters and art objects?

Is it to be assumed that the "objects" discovered will be described for all time through the notations favoured for comprehension at this time? Is the mathematical language through which they are described to be understood as a kind of cognitive "exoskeleton" -- with the possibility that artificial intelligence might come to be analogous to a powered human exoskeleton or even to a cyborg -- empowered to appreciate even higher orders of symmetry beyond human ken? Might this be an implication of the "scaffolding" explored by Mehmet Fatih Ozmantar (An investigation of the formation of mathematical abstractions through scaffolding, 2005)?

How many levels of abstraction might there be? To what extent is abstraction self-reflexively defined, possibly in the case of symmetry as:

a sequence of operations, of which the last involves the application of that sequence to itself?
in which the nature of the "operations" is suggested geometrically by the developement: point into a line, line into a polygon, polygon into a polyhedron, etc -- more generically understood as progressively more complex polytopes (hence n-polytope):
- recognizing however that the operational progression may be more generally defined dynamically, rather than in terms of a geometric form?
- raising the question of whether application of such an abstraction operation is constrained in some way by the operational symmetry so implied?

Might there be other ways of comprehending such symmetry? In other words, are the objects discovered more intimately related to particular modes of understanding than is currently suspected? How could one demonstrate otherwise?

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Levels of abstraction