Environment and Planning B: Urban Analytics and City Science

In this paper, aspects of four areas of controversy which have recently surrounded the development and application of Q-analysis are examined. It is suggested that the difficulties over which the controversy has been generated are underpinned by important substantive problems, which those wishing to utilise the approach should be aware of, but that further development and application of the style of analysis afforded by Q-analysis and related methods should continue.

Since each of the modern sciences—physical, biological, and human—broke their traditional ties with philosophy they have reflected less upon foundational problems of their methodologies. The essay constitutes an invitation to reflect upon Q-analysis in an historical perspective, raising the question of the meaning of mathematization in the human sciences. Functional description implies a mechanical framework for the human world. By enlarging the descriptive possibilities to the mapping and relation, Q-analysis raises the question of allowing and forbidding, but not requiring, geometries. It also opens up difficult questions of language in the human sciences, the constrained nature of current methodologies, the effects of aggregation, and a concern for proper definitions—including the notion of structure itself.

The purpose of this paper is to examine the following questions: are the assumptions and approach of Q-analysis sufficiently novel for it to be accepted as offering a fundamentally new and important perspective for social scientific inquiry? And, if this proposition is at least partially sustainable, are those assumptions and the perspective they define appropriate to the current practical and epistemological context of applied social science? Specifically, the author attempts to distil, from a sometimes obscure and highly technical literature, the presuppositions which are crucial with respect to the issue of defining an approach to the subject matter in terms of concepts such as structure, dynamics, and human agency. He then attempts to assess the potential of the technique for assisting in the generation of interpretations and explanations which are at once both refreshingly different and consistent with a framework of rules which reflect a theory of knowledge that is neither purely mechanical nor naively individualist in its orientation. The eventual goal of the paper, however, is to bring the discussion back down from the levels of both mathematical and epistemological abstraction to the practical issues of planning and policy guidance.

Q-analysis involves modelling a relation Λ between two sets, A and B, by a pair of ‘conjugate’ simplicial complexes Λ, Λ′. To ‘understand’ Λ, we study the geometry of Λ and Λ′, hoping that the features we recognise can be carried back to the data that gives rise to Λ. There is a ‘Galois connection’ between Λ and Λ′ which yields a decreasing sequence of subcomplexes Λ ^r of Λ: Λ = Λ⁰ ⊇ Λ¹ ⊇ … ⊇ Λ^r ⊇ Λ^r+1 ⊇ … (and similarly for Λ′), Q-analysis conventionally calculates the numbers, Q_r, the number of components of Λ ^r , but this is merely zero-dimensional information. By using homology theory, we can calculate ‘Betti numbers’ R₀(Λ ^r ), R₁(Λ ^r ), …, R_q(Λ ^r ), … (where R₀ = Q_r) to obtain much more information about the original relation Λ. The purpose of this paper is to draw attention to the existence of these methods, and to give geometrical illustrations. Remarks will be made about the conventional exposition of Q-analysis, to cover the word ‘simplify’ in the title.

At the basis of Q-analysis are the notions of well-defined sets, and of hierarchies constructed out of the power sets of such sets. It is here argued that such set-theoretic hierarchies are in general inadequate in themselves for representing the structure of empirical phenomena, because sets defined as collections of individual objects or of other sets cannot, on the whole, be made to correspond directly to specific empirical entities. This is because empirical concepts correspond to more or less complex structures, rather than collections, of other concepts, whereas a set is, by definition, an unstructured collection of elements. This point is illustrated through several examples of sets of ‘things’ that purport to represent certain higher-order ‘things’, but in fact do not. There follows an analysis of the problem that draws from mathematical philosophy and logic on the one hand, and from the theory of Boolean algebras on the other. Both these perspectives point to the same conclusion, namely, that Q-analysis is a specialized methodology applicable to a rather restricted, though potentially very interesting class of problems. The paper ends with an evaluation of the significance of such a shift in our understanding of Q-analysis, and some tentative ideas about possible areas of fruitful application.

Fundamental both to Q-analysis and to social network analysis is the assumption that relational structure influences the behavior of the underlying population. This assumption is expressed in Q-analysis by saying that the backcloth structure permits or forbids traffic on the backcloth. Backcloth structure and traffic are represented rather differently in the two disciplines. In this paper, the two representations are contrasted, with the goal of obtaining a useful cross-fertilization.

The simplicial complexes of a relation have a star structure which is richer than the usual q-connectivity structure of Q-analysis. A mathematical theory of stars is developed in which the star and hub mappings are analogous to a Galois pair. Parameters can be associated with the new structures which have q-nearness and q-connectivity as a special case. These ideas clarify the notion of t-force and resolve various problems in shomotopy analysis. Stars and hubs form isomorphic lattices analogous to Ho's Galois lattice construction. It is argued that the theory of stars will enrich the hierarchical backcloth-traffic theory of Q-analysis, and improve its value as an applied methodology.

An alternative representation of the binary relationship used in Q-analysis is presented. A unique decomposition is constructed without loss of information, and the elements of the decomposition are ordered in a Galois lattice. This lattice is an algebra identified with the intuitionistic logic of Brouwer. Comparisons are made with Boolean systems. Throughout references are made to the work of Charles Sanders Peirce who pioneered relational logic and diagrammatic methods in the nineteenth century.