Balisage Paper: Abstraction, Paradox, and the End of History

Introduction

Abstraction is the unifying ideology for the triumphalist history of the computer and information sciences. It plays a leading role in how we tell the story of the evolution of programming languages, software engineering, data management systems, and document text encoding, exemplifying the search for generality, efficiency, and elegance that characterizes these fields.

Yet it has not gone unnoticed that abstraction and formalization may also be problematic. There are the romantic poets, of course: May God us keep / From Single vision & Newton’s sleep! (Blake); we murder to dissect (Wordsworth). And from the perspective of information history, Bonnie Mak observes that our modern information infrastructures are configured to retrain uncertainty, tolerating neither incommensurability nor imagination well (Mak 2026). But caution has also been expressed from within the information sciences. In that regard, a special place must be given to William Kent’s renowned Data and Reality (Kent 1978). More recently, the applied ontology community has focused on how formal ontologies often mishandle notions such as identity, unity, time, and space (Guarino & Welty 2002, Borgo et al. 2022).

We continue that conversation here, arguing that these difficulties are not merely local defects to be repaired with better abstractions, but are recurrences of paradoxes that have accompanied abstraction since antiquity. Furthermore, the resolutions adopted in working systems are not really resolutions, but, in some sense, workarounds. Familiar concepts are replaced with surrogates that are serviceable for the most part, but which can also create real-world problems. Most importantly, perhaps, the stakes have changed: our formal models, superior in some respects but flawed in others, are part of the foundation for the lights-out automation that runs the world we now live in.

We will not address the larger philosophical issues here. Some theorists have argued that no formalization can ever carve the world at its joints, because there are no objective joints to be discerned. Others, such as Brian Cantwell Smith, see real joints, but only as a mutual achievement of mind and world (Smith 1996). And a variety of other philosophical perspectives are available. Nor do we take up at length the momentous significance of LLM-based AI, which seems to turn some of these issues inside-out. We will here only provide a frame for that discussion here.

Abstraction I: The Logical Level

Consider the familiar periodized history of data management systems. Data storage before the emergence of the relational model in the 1970s is typically characterized as flawed because end users and application programs interacted directly with physical storage structures. This limited the functionality of these systems and generated various liabilities. Codd discusses these limitations in his classic paper introducing the relational model (Codd 1970), a simple abstraction that represents data independently of how it is stored and maps that representation to storage structures.

The mediating abstraction is a relation, a tabular set of tuples. Software applications and end users interact with stored data only indirectly, via a simple language of columns, rows, and values that references the relations. This general architecture was institutionalized in the 1975 Interim Report of the ANSI/X3/SPARC Study Group on Data Base Management Systems (ANSI/X3/SPARC 1975).

Among other things, this approach provides physical and logical data independence: changes in storage methods may be made without any effect on preexisting access subroutines and queries, and data attributes may be added or removed without affecting those subroutines and queries. In addition, it is no longer necessary for end users and application programmers to understand how data is stored; inconsistencies can be more easily avoided; inferencing and validation are more straightforward; and system design, documentation, reformatting, and interoperability are simplified and disciplined.

The descriptive markup story runs along similar lines, although here the focus is primarily on abstracting away from intended processing rather than from storage methods (Goldfarb 1981, ISO 8879:1986). The abstract model is an ordered rooted tree with an accompanying formal grammar as a schema. The distinctive communicative components of the document are represented as labeled nodes. Processing is mapped to node types. As with the relational model, the advantages, grounded in abstraction and indirection, are substantial and varied (Coombs et al. 1987).

Abstraction II: The Conceptual Level

To some in the database community, however, it appeared that relational databases still recorded and organized data, rather than representing how things are in the world. The values in a relation are strings, not things; the juxtapositions of adjacent cells or column groupings are paratactic, not predicational. Any connection with the world is left to human interpretation. Table-talk (SQL) is about rows, columns, and values, not things (like people), properties (like being German), or relationships (like being the supervisor of an employee). The explicit features of the relational model – tuples, sets, foreign keys, and so on – are natural neither to our common-sense conceptual scheme nor to the domain problems we are attempting to address. Moreover, relations are just one possible abstraction; alternative abstract models can record the same information. But what methods or languages do we have to express what different but equivalent logical abstractions have in common in any particular case? How can we provide representational continuity when we change from one logical model to another?

Responding to these concerns, among others, models at a higher level of abstraction were proposed. The seminal model, still commonly used, is Peter Chen’s entity-relationship model (Chen 1976). The entity-relationship model identifies explicitly the entities, properties, and relationships that are only implicit in the rows, columns, and values in the relational model. So, just as models like the relational model added a layer of abstraction on top of storage models, conceptual data models added a layer of abstraction on top of those logical data models, allowing system-design decisions to be made independently of possible logical-level models. This greatly facilitated design, allowing designers to focus on things and relationships rather than values and relations, and supported algorithmic generation of sets of normalized relations. Conceptual models also allow wider participation in system design, as no knowledge of features specific to the relational model (or any other logical data model) is necessary, only a common-sense, informal understanding of the needs of users and organizations. One might even say that conceptual models democratized systems development.

Although characteristically represented with diagrams, conceptual data models were explicitly defined in first-order predicate logic (plus identity). Entity types represented monadic predication, relationships represented dyadic (or polyadic) predication, and cardinality was expressed with quantifiers and identity. Such formal constructs could, however, remain behind the scenes as designers operated with their common-sense understandings of things, properties, and relationships. NIAM, IDEF1X, and UML class diagrams are other examples of such late-20th-century conceptual data modeling systems.

Early conceptual data models were primarily applied, as intended, to the development of relational databases. However, as conceptual models became more wide-ranging in application, they came to be better described as ontologies with upper-level categories (such as physical object, event, property, string, number), and often including axioms and executable rules that formalize a detailed understanding of the domain of interest.

Entity-relationship models are awkward for XML documents and there is no commonly used conceptual modeling system in the document markup community. Nevertheless, some markup theorists argued that markup languages would also benefit from conceptual models (Cover 1998). An effort along these lines that is well-known to the Balisage markup community is the Béchamel project led by Michael Sperberg-McQueen. Béchamel represented an XML document as a set of predicate logic statements that could be processed according to inference rules or axioms that expressed our understanding of the assertions licensed by the document’s markup (Sperberg-McQueen et al. 2000, Renear et al. 2002, Sperberg-McQueen et al. 2002, Dubin et al. 2003). For a survey of proposed conceptual models for XML, see Nečaský 2006.

Conceptual data models and ontologies appeared to be, finally, allowing us to describe the world and not simply organize data. It was sometimes objected that a representation of the world had been sufficiently implicit in logical-level models all along, and that, far from providing a missing semantics, conceptual data models were really just more syntax. From that point of view the more abstract models seemed unnecessary, and perhaps just another doomed attempt to externalize human thought and finally resolve the symbol grounding problem (Harnad 1990).

Nevertheless, this higher level of abstraction has turned out to be extremely valuable in many ways. Conceptual models and ontologies did seem to function at a level that approximates human understanding, and to make our assumptions about a domain logically explicit and actionable. It appeared that with conceptual models and ontologies we really had, finally, reached peak abstraction and were reaping the benefits.

Paradox I: Ancient Origins

Yet there is a problematic, and perhaps ominous, perspective on this history.

The relationship between abstraction, paradox, and scientific progress has been a common – if sometimes suppressed – theme in the history of science. Scientific thought as it emerged in antiquity is itself an abstraction agenda. It is an effort to generalize and deepen our grasp of the world around us by formalizing our understanding and creating abstractions that represent how things seem to be.

However, in philosophical traditions across time and cultures, abstraction and formalization often led to paradox as well as to scientific advance. Early examples include puzzles about motion, space, and time; identity and change; self-referential statements; the identity of groups and collections; vague predicates; and various unexpected failures of transitivity. In the West this began perhaps as early as Parmenides and Heraclitus, and was made vivid and notorious in the aporiai of Zeno and the seven paradoxes of Eubulides of Miletus, but puzzles of this sort are found throughout history and around the world (Sorensen 2003, Mak & Renear 2023).

For the modern mind, the paradox-generating nature of abstraction is perhaps best exemplified by Bertrand Russell’s famous antinomy of naïve set theory, in which formalizing the simple concept of a collection of things that share a property leads quickly to a contradiction (the set of all sets that are not elements of themselves is both an element of itself and not an element of itself). Gottlob Frege, believing he had completed the logical foundations for arithmetic, anxiously wrote back to Russell that this observation undermined mathematics itself (der Grund, auf dem ich die Arithmetik sich aufzubauen dachte, ins Wanken gerät).

Some of these early paradoxes may seem to have been resolved. For instance, the paradoxes of time and motion and related Enlightenment concerns about infinitesimals are believed by many to be sufficiently addressed by the Cauchy–Weierstrass theory of limits, and the paradoxes of naïve set theory sufficiently addressed by axiomatic set theory (ZF) or by type hierarchies. But other puzzles (and perhaps even those as well) seem to be with us still. Contemporary ontology engineers, as well as philosophers, are unsure about exactly how to formalize meaningful but intrinsically vague predicates such as middle-aged (sorites), sameness across material change (the ship of Theseus), elusive subjects of predication (Aristotle’s paradox of increase and Abelard’s puzzle about group membership), among others. To be sure we need not think of all of these in terms of imminent paradox; they are perhaps simply enduring, and perhaps permanent, difficulties for successful formalization.

Paradox II: Modern Challenges

Abstraction considerably improves the performance of our digital information systems. Toward that end, through ontologies and conceptual models, we attempt to make our understanding of the world tractable by eliminating idiom, metaphor, vagueness, ambiguity, logical fiction, and copredication. This is how we take advantage of the specific affordances that formalization provides, such as semantic compositionality, logical inference, existential instantiation, deterministic parsing, ontological specificity, and algorithmic transparency. These features are all valuable to information management, yet none are easily or unproblematically provided by ordinary language.

However, it is difficult to develop formal versions of common-sense concepts that match our common-sense beliefs in all contexts. It is notable that our puzzles and paradoxes seem to involve notions that are fully serviceable in their ordinary applications, but nevertheless lead to difficulties when slightly formalized versions are applied to unanticipated cases. Efforts to resolve these difficulties are generally attempts at better formalizations, that is, formalizations that address the problems initially raised by formalization. These are perhaps not resolutions exactly, but rather workarounds. They are alternative concepts that provide the advantages of formalization and abstraction, but are themselves problematic when the difference between the common-sense concept and the formal surrogate that is intended to model that concept leads to unexpected consequences.

And so on.

We note that long-standing efforts in analytic philosophy to provide conceptual analyses of familiar notions like knowledge, cause, communication, and personal identity also continue to fail in the face of increasingly creative counterexamples (Gettier 1963, Zagzebski 1994). This has led some philosophers to conclude that formal conceptual analysis of such concepts is either impossible, or must be substantially revised as a method (Williamson 2007, Cappelen 2012). Another response, appearing outside as well as inside philosophy, is to shift attention away from concepts and toward practices. Information history, for instance, examines cultural practices around information and provides insights germane to our concerns (Mak & Renear 2023). Indeed, it could be said that traditional information science itself usefully prioritizes a social perspective (Egan & Shera 1952). Some help in understanding the interplay between common-sense concepts and their formal surrogates might be found in these quarters.

Why It Matters

Francis Fukuyama famously described the apparent success of liberal democracy as the end of history, intending a Hegelian interpretation of both history and end. It was a happy thought, if a brief one. The imminent success of formal ontologies was a similar happy thought. And the desired impedance match between such ontologies and our common-sense conceptual scheme is proving as elusive as the triumph of liberal democracy (let alone Hegel’s Golgotha of absolute Spirit). The optimistic expectation that it is achievable is, perhaps, similarly dangerous.

Humans themselves may be abstraction machines, but if so we are selective abstraction machines. For the most part, when we navigate the world around us, we make liberal use of idiom and metaphor, tolerate vagueness, deliberately leave ambiguities unresolved, traffic in concepts we cannot fully explain, and so on. Logical fictions abound in our discourse, and if serviceable noun phrases imply dubious existential instantiations, or common-sense beliefs harbor hidden contradictions, well, no matter – we can usually just look the other way. And in practice there is typically little evidence of resulting difficulties. No harm, no foul. We see such locutions as merely façons de parler, and any resulting paradoxes as just riddles unrelated to the urgencies of the current project. We know how to proceed with the work before us, and we get on with it. Resolution of the problem is a simple matter of programming; we can keep the model as is and deal with any difficulties in the software.

Friedrich Waismann, even though a member of the Vienna Circle, influentially insisted that our open-textured concepts would always resist complete formalization (Waismann 1945) – a notion more famously associated with Wittgenstein (concepts are based on elusive family resemblances, not necessary and sufficient conditions), and since developed in a wide variety of forms in the literature on concepts, categories, and classification (e.g., Rosch 1978, Lakoff 1987).

The practical problem with depending on formal surrogates is that the processing conceptual models support is increasingly taking place without human oversight. A formalization might work well enough in the intended domain, but then in an unanticipated case fails with no one present to notice. In the past, our paradoxes were primarily of academic interest; now they are embedded in the systems that organize and sustain the world we live in (Bowker & Star 1999), atavistically creating new sources of risk and unreliability. The renowned jurist H. L. A. Hart was deliberately applying Waismann’s notion of open texture when he held that it is the responsibility of a judge to resolve the penumbra of meaning surrounding our concepts (Hart 1961). But in a world of lights-out automation the judge is at home, sound asleep.

Strangely, the models that seemed to function, and were intended to function, at the same level of abstraction as our common-sense conceptual scheme seem to have first approximated but then overshot their objective. Is a paradox-free formalization of our common-sense conceptual scheme possible? Or does understanding the world force us to choose between logical fiction and logical paradox? Is that yet another tradeoff in the information sciences?

The failure of paradox-free abstraction is, in any case, one more reason for keeping a human in the loop. As humans, we know how to navigate a world of metaphor, idiom, vagueness, ambiguity, logical fiction, and latent paradox. We have been doing it, successfully, for a long time.

Looking Ahead: Deep Learning and LLM-based AI

The problems described above have been apparent for a while. Today there is a new development. Deep learning has accomplished what traditional logic-based AI could not: the successful navigation of salad bars and roundabouts. And, most dramatically, LLM-based AI, exploiting deep learning, allows humans to present their queries and instructions in unregimented natural language — and then receive apt and accurate responses that are also in ordinary natural language.

LLM-based AI has little use for the familiar logical abstractions of conceptual modeling and ontologies, and consequently appears to avoid their attendant vulnerability to paradox. It accepts ordinary-language prompts and responds in kind. The vagaries of ordinary language are still latent in both the prompts and the responses, but they do not present as impending problems. In fact, and stranger still, our interactions with LLM-based AI are free, on both sides, to benefit from the convenient vagueness, ambiguities, and meaningful fictions of ordinary language – and perhaps even tone and style as well. Is it here, then, with the sudden re-emergence of the vernacular, that we really do, at last, have an impedance match between human and machine?

Or have the issues and dangers simply been hidden behind an unaccountable simulation? Even if much human cognition is itself connectionist in its fundamental nature, humans are still accountable in ways that LLMs are not: humans can ask for surveyable reasons and actual provenance, and humans can provide them (Smith 2019).

That is a story for another time – and might be, some would say, the true end of history.

Envoi: On the Occasion of the Final Balisage Conference

Over the last three decades, I have discussed many of these challenges, puzzles, and paradoxes in papers or conversations at Balisage. My reprise and elaboration here will be recognized by present company as characteristic of my contributions to this conference (no eye-rolling please!). This is fitting. For good or ill, most of what I have written, said, or thought would not have happened without Balisage.

More importantly, Balisage has had a unique and profound impact on our profession and on the technologies of communication that are so essential to the welfare of society. The impact of our collective work will continue and expand. But we will miss the sharp, wonderfully un-formalized fellowship of this annual gathering, which has been so important to all of us. To Balisage then: Ave atque vale.

Acknowledgements

These thoughts owe much to collaborations with many people, especially Michael Sperberg-McQueen, who gave me a (transitive) identity, and also Dave Dubin, Claus Huitfeldt, Karen Wickett, and Bonnie Mak. I am also indebted to conversations with many others at Balisage and at the School of Information Sciences, University of Illinois Urbana-Champaign, and elsewhere. Nevertheless, everyone mentioned here or implicated should be held blameless.

The notion that when the march of abstraction reached its peak we would face the re-emergence of ancient paradoxes was first suggested in presentations at the National Information Standards Organization (NISO) Forum on Research Data in 2012 and the Oxford University e-Research Centre in 2014.