How Notations Evolve: A Historical Analysis with Implications for Supporting User-Defined Abstractions

TL;DR

This work investigates how notations and abstractions emerge, evolve, and become formalized by analyzing historical case studies across disciplines. It proposes a three-stage social framework (invention/incubation, dispersion/divergence, institutionalization/sanctification) and a three-function lens (descriptive, generative, evaluative), identifying 33 patterns that govern notation development. A key contribution is an emerging theory that dimensions of meaningful variation map to perceptual channels via grounding and linking metaphors, with rapid iteration and social processes shaping evolution. The paper concludes with design implications and a roadmap for systems that support dynamic, user-defined, incrementally formalized abstractions, including tools for governance, testing, sharing, and cross-domain notation evolution, aided by AI.

Abstract

Traditional human-computer interaction takes place through formally-specified systems like structured UIs and programming languages. Recent AI systems promise a new set of informal interactions with computers through natural language and other notational forms. These informal interactions can then lead to formal representations, but depend upon pre-existing formalisms known to both humans and AI. What about novel formalisms and notations? How are new abstractions created, evolved, and incrementally formalized over time -- and how might new systems, in turn, be explicitly designed to support these processes? We conduct a comparative historical analysis of notation development to identify some relevant characteristics. These include three social stages of notation development: invention & incubation, dispersion & divergence, and institutionalization & sanctification, as well as three functional stages: descriptive, generative, and evaluative. Within and across these stages, we detail several patterns, such as the role of linking and grounding metaphors, dimensions of meaningful variation, and analogical alignment. Finally, we offer some implications for design.

Figures (5)

• Figure 1: The 'square of formality,' revealing a hidden social dimension to formalization. Informal ($I$) becomes formal ($F$) through 'horizontal' (translation $t$ to existing form) and 'vertical' (creation $c$ of new form) movements, representing two types of formalization processes. In practice, movements across the square of formality often mix (adoption of existing notations, mixed with new ones).
• Figure 2: The evolution of SignWriting, a notation for writing sign languages originally invented by Sutton and co-designed with Deaf communities and academic researchers over decades. Tracing its lineage back to sheet music notation, we see the prominent use of linking metaphors and grounding metaphors lakoff2000mathematics throughout, including new metaphors that break fundamental aspects of the previous: music staffs gradually fade away, interpretation is drastically changed without affecting iconography, and writing shifts from horizontal rows to vertical columns. SignWriting is materially and socially institutionalized in software, educational resources, and encoded into Unicode standards. Changes now demand costly reprints, re-encodings, and mobilization of committees. Images reproduced from Sutton sutton2015historysutton1973sutton under fair use for purposes of academic analysis.
• Figure 3: Origin of Toffoli gate notation in quantum circuits, revealing a grounding metaphor to crankshaft mechanisms by Toffoli toffoli1981bicontinuous, which was then adapted and simplified by Feynman feynman1986quantum which, in turn, was adapted by Bancelo et al. barenco1995elementary. Images reproduced from original texts toffoli1981bicontinuousfeynman1986quantumbarenco1995elementary under fair use for purposes of academic analysis.
• Figure 4: A notation in incubation: Excerpts from an HCI researcher's whiteboard. "Common ground" between humans and AI is visualized through a linking metaphor to Venn diagrams. During the analogical alignment process, dimensions of variation in the target domain are discovered to be un-representable or obscured, prompting notation extension, change, or abandonment.
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