The previous posts established that services involve objects with properties, events that change those properties, and systems in which those events unfold. The next question is representational: how do we formalise any of this? And specifically, how do we bridge the gap between how people understand services and how computational systems represent them?
Multiple frameworks in cognitive science, social psychology, and organisational theory address how people represent knowledge - and why different people represent the "same" domain differently. Not all of them provide the mathematical apparatus needed for the bridge this series is building. This post surveys the main contenders and argues that one - geometric vector spaces - is particularly relevant to service design for agentic and generative systems.
Distributed Cognition
Edwin Hutchins's distributed cognition framework (1995) reframes cognition as something that happens not inside individual heads but across systems of people, artefacts, and environments. A cockpit doesn't "think" through its pilots alone; it thinks through the coordinated interaction of pilots, instruments, checklists, and procedures. Cognitive processes are distributed across the components of the system.
This was the foundational perspective in the Cognitive Ergonomics department at Brunel where I completed my MPhil. It explains how coordination works in practice - how knowledge lives in forms and protocols and shared routines, not just in people's minds. The series returns to this in the post on boundary objects, where Star's framework addresses exactly how artefacts mediate between different communities. Blomkvist (2014) draws on Hutchins directly in his work on service prototyping, arguing that prototypes function as distributed cognitive artefacts.
But distributed cognition doesn't provide a formal mathematical structure for representation itself. It tells you cognition is distributed; it doesn't give you a formalism for what's distributed. There are no dimensions, no distance metrics, no way to formally compare the structure of one person's representation with another's. You can say "these two groups understand the system differently" but you can't specify the structural difference precisely enough to compute with it.
Cultural-Historical Activity Theory
Vygotsky's cultural-historical framework, developed by Leont'ev and most influentially by Engeström (1987, 2002), treats all human activity as mediated by tools, embedded in social rules, and oriented toward objects within a community. The activity system triangle - subject, object, tools, rules, community, division of labour - captures the structural relations within any purposeful activity. Engeström's third-generation activity theory extends this to multiple interacting activity systems, identifying contradictions between them as the engine of development.
Activity theory is powerful for analysing work systems and understanding why interventions fail. Spinuzzi (2019) uses it to study how organisations try to predict the future through planning processes. Clemmensen and Kaptelinin (2016) demonstrate its value for HCI research through the concepts of tool mediation and context. Cornet and Voida (2017) apply it to self-care in chronic illness.
The limitation is the same: the formalism is descriptive rather than mathematical. You can draw activity triangles and identify contradictions, but you can't express the representational structure in terms that a computational system could work with. The triangle captures relationships between components of an activity system; it doesn't capture the internal structure of how those components represent meaning.
Social Representations Theory
Moscovici's social representations theory (1961, 2001) explains how groups construct shared understandings through two processes: anchoring (linking the unfamiliar to existing knowledge) and objectification (making abstract ideas concrete through images or metaphors). Professional communities develop different social representations of "the same" phenomenon - a doctor and a social worker may understand "rehabilitation" through entirely different anchoring points and objectifications.
Strindlund (2020) applies this directly to vocational rehabilitation contexts, showing how different institutional actors construct different social representations of labour market inclusion. The framework is relevant to the challenges I observed at SCÖ, where "data science" meant radically different things to different stakeholders.
But social representations theory describes the process of representation construction, not the structure of representations themselves. It explains how groups come to hold different understandings; it doesn't provide a formalism for expressing what those understandings structurally consist of. You can say "this group anchors rehabilitation in medical recovery while that group anchors it in employability" - but you can't express the structural difference as a mathematical object.
Prototype Theory and Grounded Cognition
Eleanor Rosch's prototype theory (1975, 1978) showed that natural categories have graded membership - robins are more typical birds than penguins - and that categorisation operates through similarity to central exemplars rather than through necessary and sufficient conditions. Wittgenstein's family resemblance anticipated this, but Rosch provided the empirical programme.
Barsalou's grounded cognition (2008) takes this further, arguing that concepts are not abstract symbols but are grounded in perceptual and motor experience. Conceptual processing involves partial reactivation of the sensorimotor states that occurred during actual experience with category members. Larsson (2017) draws on both in analysing how legal categories resist the neat boundaries that formal systems require.
These frameworks capture genuine features of concept structure - typicality effects, graded membership, perceptual grounding - but none provides a single mathematical formalism that unifies them.
What a Geometric Formalism Provides
Peter Gärdenfors's conceptual spaces framework (2000) provides the missing piece: mathematical infrastructure that turns these cognitive insights into formal, computable structure.
In Gärdenfors's framework, a conceptual space is defined by quality dimensions - the fundamental respects in which things can vary. Concepts are regions (specifically, convex regions) in these spaces. Similarity is distance. Prototypes are central points of regions. These are mathematical objects - vector spaces with defined metrics, where operations like distance calculation, region comparison, and dimensional analysis are formally specified.
The framework makes "different stakeholders represent the domain differently" formally precise. Different conceptual spaces means different quality dimensions - different vector spaces. The structural mismatch between a clinician's representation and an engineer's representation can be expressed as a difference in dimensionality, a rotation in shared dimensions, or a difference in how regions are carved out. Raubal (2004) showed how conceptual spaces can be formalised to bridge to computational ontologies.
But the deeper relevance for this series is that vector spaces are also the mathematical foundation of how machine learning systems represent meaning. Word embeddings (Mikolov et al., 2013), transformer attention mechanisms, graph neural network embeddings - all represent meaning as positions in continuous vector spaces where similarity is distance. Gärdenfors published this as cognitive science in 2000. The convergence is not coincidental: continuous geometric spaces turn out to be how you represent structured similarity, whether in neurons or in neural networks.
The consequence: if a clinician's understanding of "rehabilitation" is a region in a conceptual vector space, and a machine learning model's representation of "rehabilitation" is a position in a learned embedding space, then alignment between human and computational understanding becomes a question of geometric alignment between spaces - formally precise in a way no other framework in this survey can match.
Gärdenfors doesn't replace the other frameworks - each does work that geometric formalism cannot, and the series draws on them where they're needed. Boundary objects are fundamentally about distributed cognition; the SCÖ case involves activity system contradictions; social representations help explain why concept mapping produced regressive effects. But for the specific question this series is building toward - how to bridge human and computational representation of services - the geometric formalism is the necessary foundation.
The next post develops this in detail: Gärdenfors's quality dimensions, concepts as convex regions, the connection to state spaces, and what happens when you try to construct shared conceptual space in practice.
References
Barsalou, L. W. (2008). Grounded cognition. Annual Review of Psychology, 59, 617-645.
Blomkvist, J. (2014). Representing Future Situations of Service: Prototyping in Service Design. PhD thesis, Linköping University.
Clemmensen, T. & Kaptelinin, V. (2016). Making HCI theory work: An analysis of the use of activity theory in HCI research. Behaviour & Information Technology, 35(8), 608-627.
Cornet, V. & Voida, S. (2017). Activity theory analysis of heart failure self-care. Proceedings of the 2017 CHI Conference on Human Factors in Computing Systems.
Engeström, Y. (2002). Expansive learning at work: Toward an activity theoretical reconceptualization. Journal of Education and Work, 14(1), 133-156.
Gärdenfors, P. (2000). Conceptual Spaces: The Geometry of Thought. MIT Press.
Hutchins, E. (1995). Cognition in the Wild. MIT Press.
Larsson, S. (2017). Conceptions, Categories and Embodiment. Springer.
Mikolov, T., Chen, K., Corrado, G. & Dean, J. (2013). Efficient estimation of word representations in vector space. Proceedings of ICLR 2013.
Moscovici, S. (2001). Why a theory of social representations? In K. Deaux & G. Philogène (Eds.), Representations of the Social (pp. 8-35). Blackwell.
Raubal, M. (2004). Formalizing conceptual spaces. In Formal Ontology in Information Systems. IOS Press.
Rosch, E. (1978). Principles of categorization. In E. Rosch & B. B. Lloyd (Eds.), Cognition and Categorization. Lawrence Erlbaum.
Spinuzzi, C. (2019). "Trying to predict the future": Third-generation activity theory and the challenge of anticipation. Mind, Culture, and Activity, 27(1), 4-20.
Strindlund, L. (2020). The Social Dynamics of Labor Market Inclusion. Linköping University.
Zenker, F. & Gärdenfors, P. (Eds.) (2015). Applications of Conceptual Spaces: The Case for Geometric Knowledge Representation. Springer.