↩ ExploRational · A Catalogue of Divergent Rationality
explorational · a divergent move
An analogy from next door refines what you already think; an analogy from a distant world can reframe it entirely — or decorate it with nonsense. What separates the borrowed structure that carries from the borrowed costume that doesn't?
Kepler, needing a force that could reach from the sun to the planets across empty space, reasoned from the one thing that already did: light streams from a source, weakens with distance, crosses a void — perhaps the moving power did too. Rutherford's atom was a solar system in miniature — heavy centre, light orbiters, an attraction binding them — magnificently fertile and wrong in auditable ways: planets do not jump between orbits, and an orbiting charge should radiate its energy away and spiral inward. And this morning a consultant called the company a garden, everyone smiled, and nothing was predicted.
Dedre Gentner's structure-mapping theory names the difference. An analogy is a mapping of relations, not of things: the sun resembles a nucleus in no attribute — not size, not heat, not charge — yet the relational skeleton (the heavy one attracts the light one, which orbits it) maps exactly. An analogy's worth is which relations carry; the discipline is the audit: the explicit list of what transfers and what does not. The instrument below performs that audit by brute force, on domains small enough to check by hand.
the signature instrument
Two toy domains, each a graph: entities with surface attributes, relations, and higher-order relations — relations between relations, like causes(attracts, orbits). The engine enumerates every entity correspondence, scores structure and charm separately, and lays the loan beside the audit. Untick any relation and everything recomputes; hover anything to light up its counterpart.
| Pair | Surface (charm) | Structure (cargo) |
|---|
scored fresh from the shipped definitions at the current systematicity weight — edits above don't touch this table
relations, not things
Attributes describe things; relations describe how things bear on one another — and only relations are portable. "The nucleus is tiny, the sun is huge" blocks the analogy if you read attributes; "each holds a lighter partner in orbit by attracting it" is the analogy. Gentner's rule: a correspondence is judged by structural consistency — the same relations binding the corresponding objects in the same roles — and the surface resemblance of the things counts for nothing. The instrument meters charm anyway, because charm is what usually does the choosing.
The second rule is systematicity. An isolated shared relation is a coincidence; a connected cluster held together by a higher-order relation — attracts causes orbits — is a mechanism, and mechanisms predict. The scoring rewards depth over count: one deep shared cause outweighs ten scattered echoes. Distance, then: a near analogy shares attributes and relations, and refines; a distant one shares relations only, and reframes. The surface score is a distance meter, read backwards.
what to try
Read the inversion table. Office ↔ garden wins on charm (6 shared attributes to the atom's 0) and comes last on structure (1.0 to the atom's 10.0 at the default weight). The two columns sort in opposite orders — the pairing that feels most apt carries the least machinery.
In solar ↔ atom, untick causes(attracts, orbits): structure falls 10.0 → 7.0. Restore it, then untick lies far from instead: 10.0 → 9.0. One deletion either way — triple the damage when it was the higher-order spine of a cluster.
Drag systematicity to 0. Clusters count as loose facts: the atom's lead over the circuit shrinks from 10.0 − 6.0 to 6.0 − 4.0. The slider is the credit mechanism gets over coincidence; at zero, an analogy is a tally.
Open your own and rename the debate column's advances to commands. One verb, honestly meant, and the causes-cluster snaps in: 1.0 → 5.0. Whether the rename is honest is the question — the engine only prices it.
the audit is the discipline
The move has two stages; the second is the craft. First the echo: the atom is like a solar system — cheap, fast, merely salient. Then the mapping: name the correspondence entity by entity, run every relation across it, and split the result into two lists. The carried list is what the metaphor genuinely predicts in the target. The not-transferred list is where the loan is void — every place the metaphor keeps talking and means nothing.
Rutherford's audit is the historical argument that the second list is the valuable one. The carried cluster organized his scattering data; the remainder — the electron radiates, the orbits come in fixed steps — was the coastline of the physics the analogy could not reach; working that remainder became quantum mechanics. A distant analogy is not finished when it feels right; it is finished when its failure list is written down.
model
The scoring is disclosed in full. Every injective entity correspondence is enumerated, including partial ones — 7 candidates for the atom's two-by-two entities, 34 for three-by-three — and each is scored twice. A relation carries when the target holds a relation with the same label and arity whose bindings all agree; a higher-order relation carries only if the relations it connects carry too. Structure = 1 point per carried relation, plus the systematicity weight × (2 per carried higher-order relation, plus 1 each time a carried higher-order relation stands on it). Charm = shared attributes between paired entities. Ties on structure break toward charm, then toward more pairs — where structure has no opinion, charm decides, openly.
Two toy honesties. Relations match by literal label: deciding that advances and commands name the same relation is your modelling act, and the engine only prices it. And real structure-mapping — Falkenhainer, Forbus & Gentner's SME, 1989 — handles hundreds of relations without brute force; these graphs stay small enough to enumerate, so no step is hidden. The sizes are toy. The shape of the audit is not.
the move ↔ the engine
| The engine | The move |
|---|---|
| an entity | a thing in either world — sun, seedling, claim — with its surface attributes, which charm and prove nothing. |
| a relation | how things bear on each other; the only cargo an analogy can actually ship. |
| the correspondence | the loan the analogy proposes: this stands for that, everywhere, or the deal is off. |
| carried relations | what the metaphor genuinely predicts in the target — belief you may now hold at a discount. |
| the not-transferred list | where the loan is void; read it before spending, because the metaphor won't stop talking there. |
| systematicity | why one deep shared mechanism beats ten surface echoes — clusters predict, coincidences pose. |
how this opening fails
risk · novelty for its own sake
The move's named risk. A source chosen for exotic charm, with no consistent relations, produces what the office garden produces: a warm feeling and an empty loan list. The engine names it: charm bar full, one weak verb carried, no higher-order relation crossing. A bare loan list means you have decorated the problem, not reframed it; the pleasure of the metaphor is real, and it is the entire product.
risk · the audit is not optional
A deep, consistent mapping can fail anyway, wherever the target holds relations the source never had. The solar-system atom carried the whole force-and-orbit cluster and still broke: radiates energy as it orbits sat in the target's remainder, outside the loan. The remainder is not a footnote — it is where the analogy's expiry date is printed.
Borrow the relations, audit the remainder — the distance is where the reframe lives.
can you use it?
RECOGNITION — Which is a distant analogy? A: comparing this quarter to last quarter. B: modeling a stalled project on a traffic jam — a wave that moves backward through cars that are each moving forward — then checking which relations actually transfer. C: comparing your startup to a similar startup.
B. It borrows structure from a remote domain and audits the mapping. A and C are near analogies — same domain, refining rather than opening.
THE NEAREST NEIGHBOR — Random entry also imports something foreign. What separates distant analogy from it?
Choice. A distant analogy is a deliberately selected remote source whose structure you map and audit; random entry injects an unchosen element precisely because you would never have picked it. One is targeted, one is a forced surprise.
PRODUCTION — Your team keeps missing deadlines. Borrow structure from a non-work system (an ecosystem, a kitchen, a river) and name one relation that transfers — and one that does not.
A version: a restaurant kitchen — tickets pile when one station is slower than the flow (transfers: the bottleneck sets the pace); but a kitchen can stop seating (does not transfer: our work arrives whether we're ready or not). Yours works if you audited a relation that fails as well as ones that fit.