This essay proposes a conceptual duality for understanding analogy in cognition and its implications for artificial intelligence: analogy operates as the native analog primitive of thought—fluid, proportional relational correspondence—while coordinate-based representations (Cartesian grids, vector embeddings) impose a digital scaffolding that enables manipulation but constrains organic discovery.
At the heart of this framing lies a productive tension drawn from analog hardware itself: analog systems routinely incorporate discrete structural components—resistors, capacitors, inductors as fixed, countable elements—without losing their proportional, fluid character. This apparent oxymoron serves as first-principles evidence that analogy discovery in cognition likely embeds discrete mechanical primitives within an overall relational, emergent process.
Analogy, at its core, is proportional correspondence: a structure-preserving mapping of relations across domains, often without surface similarity. It is inherently fluid and index-free—relations glue locally and emerge globally, detached from any pre-imposed addressing or grid. This aligns directly with the etymological root of "analog" (ἀνάλογος: according to ratio or proportion).
In contrast, coordinate space—whether Cartesian axes, high-dimensional vector embeddings, or fixed-basis projections—imposes discrete addressing. Positions become countable and manipulable via linear operators; transformations require projection onto a shared frame. Even nominally continuous embeddings digitize analogy by forcing organic streams into addressable points, enabling scale but introducing brittleness on distant, cross-domain leaps.
This duality is not merely metaphorical. It reflects a foundational tension: analogy thrives in proportional, emergent correspondence (the analog engine of thought), while coordinate projection digitizes it for computability (the digital harness that makes it tractable but constrained).
Analog computation has long been misunderstood as requiring strict continuity. Historical and philosophical analyses show otherwise: analog systems can—and often do—rest on discrete structural primitives without contradiction.
"Analog representation is about covariation, not continuity... A series of single-ohm resistors, along with a device to bypass the unneeded ones, would count as an analog representation. The number is still represented by the amount of resistance, but... the resistance varies in unit steps." — David Lewis (1971), extended in Maley (2009, 2020)
Corey J. Maley refines this taxonomy: analog representation involves proportional covariation with the target, distinct from mere continuity or discreteness. Resistors, capacitors, and inductors—discrete, fixed components—implement analog modeling via their magnitudes (ohms, farads, henries) in circuits that solve differential equations proportionally.
This hardware reality provides a template for cognition: if analog systems tolerate discrete building blocks inside fluid proportion, then analogy discovery likely embeds discrete relational primitives (micro-agents, codelets, binding operations) within emergent, proportional resonance. Hybrid architectures like Copycat (Hofstadter) and DUAL/AMBR (Kokinov) already implement this: discrete symbolic elements interact in continuous activation landscapes to yield fluid analogical insight.
Current AI paradigms project domains into coordinate-bound manifolds (embeddings as points in ℝ^d), approximating analogy via metric alignment. This yields powerful but brittle results: relational regularities emerge from scale, yet true cross-domain leaps falter without deep structural preservation.
A more analog-native approach might prioritize proportional gluing—e.g., sheaf morphisms for local consistency, continuous-variable dynamics for resonance, or morphological computation where discrete interactions yield fluid emergence—over imposed coordinates. The hardware oxymoron suggests discrete primitives (relational operators, binding sites) could enable scalable, discovery-oriented analogy without flattening the cloud into grids.
This framing synthesizes existing threads—Maley's discrete-in-analog defense, Damper's analog computation for analogical reasoning, Gentner's structure-mapping, Hofstadter's fluid analogies—into a novel lens: analogy as analog primitive, coordinate space as digital imposition, with discrete primitives enabling proportional fluidity.
It invites testable hypotheses: Can analog-inspired substrates (neuromorphic, fluidic, or hybrid) outperform coordinate-bound models on distant-analogy benchmarks? Do cognitive signatures of insight (entropy drops, relational coherence spikes) align with discrete-primitive models embedded in continuous flows?
By drawing first principles from analog hardware's own internal analogies, we gain a reflexive tool for rethinking how minds—and future machines—discover meaning.