Why Analogies Break: Projection and Residue¶

A conceptual paper for the Encyclopedia of Abstractions project

Abstract¶

Everyone knows analogies break down. Almost no one can say where or why with any precision, and the usual explanation — that we carelessly carry over surface details that don't belong — captures only the shallow half of the phenomenon. This essay argues for the deeper half: that even a transfer which shares nothing but genuine relational structure, and shares all of it that was ever checked, will still break when pushed. The reason is not contamination but incompleteness. An abstraction is a projection: it keeps a specified set of relations and discards everything else. Two situations that instantiate the same abstraction are guaranteed to share that projected structure and — if they are genuinely different situations — guaranteed to differ everywhere the projection was silent. Pushing an analogy means drawing inferences that lean on the discarded part. So the breakdown is not a flaw in the analogy; it is the projection's residue reasserting itself, and it occurs precisely at the edge of what was shared.

None of this is a new discovery — structure-mapping theory and the philosophy of scientific analogy have circled it for decades. The claim here is narrower and, we think, more useful: an explicit catalog of abstractions changes what kind of understanding of the phenomenon is available. Without a named inventory of the pure structural cores, a situation cannot be cleanly factored into "the shared abstraction" plus "the rest," and every breakdown looks like the same fog. With one, the boundary becomes a thing you can point at.

1. The familiar half-truth¶

Tell someone an atom is like a solar system and, if they know a little physics, they will tell you why the analogy is bad: electrons are not little planets, they do not travel in neat ellipses, there is no gravity doing the holding, and so on. This is the ordinary understanding of how analogies fail, and it is not wrong. It is just shallow. Every one of those objections is a complaint about surface leakage — features of the source domain (planets, ellipses, gravity) that were imported into the target where they never belonged. The lesson drawn is the familiar one: strip away the incidental resemblances, keep the real correspondence, and the analogy is safe to the extent that a genuine structural parallel remains.

That lesson is true and incomplete. It suggests that the danger lives entirely in the surface, and that a transfer purified of surface — one that maps only deep relational structure, and maps it correctly — would be secure. It would not be. The purified analogy breaks too. Understanding why is the whole point of this essay, and it requires seeing that "the structure transfers" is a claim with a hidden quantifier.

2. Necessary but never sufficient¶

Here is the hidden quantifier. When we say two situations "share the same structure," we always mean they share a structure — a particular set of relations we have singled out and checked. A thermostat and a national economy both instantiate feedback: something senses a state, compares it to a target, acts to close the gap, and the action changes what gets sensed next. That shared loop is real, and an analogy resting on it is licensed to make loop-level inferences in both domains: that a disturbance will provoke a corrective response, that the corrective response feeds the next measurement, that the system can hunt or settle.

But a thermostat and an economy are not otherwise the same object. The thermostat corrects in milliseconds through one crude channel; the economy corrects through millions of agents over months, with long and uncertain lags. That difference in timing is not part of the feedback schema — the loop structure says nothing about how fast the loop turns. So an analogy that pushes from "both are feedback loops" to "the economy will therefore correct promptly, the way the thermostat does" is drawing on structure the shared schema never contained. And it will mislead badly, because feedback with long lags behaves qualitatively differently from feedback that is near-instant: it overshoots, it oscillates, it can go unstable exactly where the fast loop is placid.

Worked case — feedback, pushed until it breaks. Map the thermostat onto a central bank fighting inflation. The shared projection holds cleanly: the sensor is the inflation reading; the target is the 2% goal; the actuator is the interest rate; the return arrow is that a rate change moves inflation, which is what gets read next. Every one of those correspondences is real, so the analogy licenses the loop-level claim — there is a correcting loop that pushes inflation back toward target. Now push it one step further: "so, like the thermostat, a rate hike will bring inflation down promptly." That inference is not in the projection. Raise rates and, for many months, little visible happens; a policymaker reasoning from the thermostat sees no response and hikes again, and again — and when the stacked-up effects finally land all at once, inflation overshoots downward into recession, prompting cuts that overshoot the other way. The result is exactly the stop-go oscillation that long-lag feedback is known for and that the instantaneous thermostat never shows. The shared structure never failed. The analogy broke precisely at the timing — a quantity the feedback projection discards, and therefore residue. And the edge was locatable in advance: the loop transferred; the lag did not.

Notice what happened. The shared structure did not fail. Every loop-level inference it licensed remained true. The analogy failed only when it was pushed to an inference the shared structure never underwrote. This is the general shape, and it is worth stating flatly:

Structural sharing is necessary but never sufficient for a transfer. An analogy is sound exactly for the inferences its shared structure licenses, and unsound the moment it is pushed to inferences that depend on structure lying outside what was shared. Because any two genuinely distinct situations carry more structure than any single abstraction captures, there is always such an outside — and so every real analogy has a breaking point.

3. Projection and residue¶

The cleanest way to see why the "outside" is guaranteed to exist is to be precise about what an abstraction is. An abstraction is a projection. Out of the full, tangled structure of a situation — its relations at every level, its quantities, its materials, its context — an abstraction keeps a specified subset of relations and throws the rest away. To recognize that a situation instantiates feedback is to project it onto the sense–compare–act–return schema and ignore everything the schema is silent about. Projection is not a defect of abstraction; it is the entire value of it. An abstraction that kept everything would be a duplicate, and a duplicate is useless for thought, because you could not carry it to a second situation. (We will see in §6 that the limit cuts the other way too: if two situations shared everything, there would be nothing left to project away, and what looked like a flawless analogy would turn out to be no analogy at all — just two instances of one abstraction.)

Whatever the projection discards, we will call the residue. The residue is not noise and not error. It is simply the structure of the particular situation that lies outside the shared core — the economy's lags, the thermostat's crude on/off channel, the specific materials and magnitudes and adjacent relations that make each instance the concrete thing it is. Note the factoring: it is the instance that carries both parts. An instance's total structure divides, under a given projection, into the kept relations and the residue; the projection itself is only ever the kept part. And the residue is not only extra relations lying beyond the schema — it includes the quantities running under the shared relations themselves: how fast the loop turns, how large the load is, how strong the coupling. A projection that keeps a relation does not thereby keep its magnitudes. Every instance of an abstraction has a residue, because every instance is more than the abstraction. (The lone exception is discussed in §6.)

This gives the principle its name and its mechanism:

The projection–residue principle. An abstraction is a projection that keeps some relations and discards the rest. Two instances of one abstraction necessarily share the projected structure and necessarily differ in their residues. An analogy transfers the projection. Pushing an analogy — extending its inferences — is the act of reaching past the projection into the residue, where nothing was ever shared. That is where, and why, it breaks.

(Several names for this were on the table while drafting: necessary but never sufficient, the incompleteness of structural transfer, the residue principle. We have used projection–residue because it names both halves of the mechanism — what is kept and what is left behind — and because it points at the operation, projection, that does the keeping.)

4. The breakdown is not random — it is at the edge of the projection¶

The projection–residue principle does more than explain that analogies break. It says where. An analogy is exactly as sound as its shared projection and no sounder; it fails at the first inference that depends on residue. So the breaking point is not a mystery to be discovered by trial and error — it is the boundary of the shared structure, and if you know that boundary, you know in advance the class of inferences the analogy can and cannot bear.

This is why the ordinary "analogies are imperfect" is so unsatisfying as guidance. It is true but it gives you no coordinates. It tells you to be careful without telling you of what. The projection–residue view replaces the warning with an address: the analogy holds across the abstraction's relations and breaks wherever the domains carry structure the abstraction discards. To evaluate an analogy is, on this view, a locatable task: find the projection the two cases share, and then ask whether the inference you want to make lives inside that projection or outside it.

The philosopher Mary Hesse gave this a vocabulary long ago.^[1] In her analysis of analogy in science, the positive analogy is the set of properties known to be shared, the negative analogy the properties known to differ, and the neutral analogy the properties not yet sorted into either. Sound analogical inference lives in the positive analogy; danger lives in mistaking neutral analogy for positive — reasoning as if an unexamined correspondence were a shared one. In the vocabulary here, the positive analogy is the shared projection, and the negative and neutral analogies are the residue: the part known to differ and the part not yet checked. Pushing an analogy is, almost always, quietly promoting a piece of neutral analogy to positive without having earned it.

One amendment keeps this from libeling the push itself, and it is Hesse's own deeper point.^[1] In her account the neutral analogy is not primarily where danger lives — it is where a model's predictive power lives: pushing past the checked core is how a model generates genuinely new hypotheses at all. Push the wave model of sound onto light and you predict interference and diffraction — confirmed spectacularly — and, in the same motion, a medium for the waves to travel in: the ether, disconfirmed just as spectacularly. Same push, same neutral analogy; only experiment could sort the two. So the edge of the projection does not mark where reasoning must stop; it marks where its epistemic status changes. Inside the projection an inference is licensed and may be concluded. Past the edge it is conjecture — often the most valuable conjecture available, because the shared structure makes it a directed guess rather than a blind one — but it must be sent to the evidence, not to the conclusions. The failure mode this essay anatomizes is not pushing; it is pushing and concluding. And when a tested push fails, the failure is itself a finding: it localizes a piece of residue, telling you exactly where the target's mechanism departs from the source's.

5. What the encyclopedia changes¶

If Hesse named the parts sixty years ago, and Gentner's structure-mapping theory^[2] has for four decades distinguished the relational structure that transfers from the surface attributes that do not, then in what sense is any of this enabled by the Encyclopedia of Abstractions? Not in the sense of discovery. The components are old. What is new is realizability — the phenomenon becomes something you can operate on rather than merely assent to.

The reason is factoring. To apply the projection–residue principle to a concrete analogy, you must be able to separate the situation into "the shared abstraction" and "the residue." But absent an explicit inventory of the pure structural cores, that separation is exactly what you cannot do cleanly. (Domain experts do perform it — a physicist knows precisely where the atom–solar-system analogy stops paying — but they perform it implicitly, on an internalized inventory, inside the domains where they have paid for that internalization. The claim here concerns everyone else and everywhere else: the factoring made explicit, portable across domains, and available before expertise.) You feel that the thermostat and the economy are "somehow alike," you sense the analogy straining somewhere, and you have no way to say precisely which relations were the shared ones and which were the local extras. Every breakdown blurs into the same undifferentiated "analogies are imperfect," because you lack the coordinate system in which a breakdown has a location.

A catalog of named abstractions supplies that coordinate system. When feedback is an explicit object — with a stated set of load-bearing relations and, on each entry's page, a documented account of what it is not — the factoring becomes possible. You can say: here is the shared projection (these relations), here is where this instance carries more (its lags, its channels, its magnitudes), and therefore here is the boundary the analogy must not be pushed past. The encyclopedia converts a vague felt-sense into addressable structure. That is the different kind of understanding it enables: not the knowledge that analogies are incomplete, which everyone has, but the ability to say of a given analogy exactly where its completeness ends.

There is a second, quieter contribution. Because the catalog documents the "what it is not" of each entry — the near-misses and the boundary cases — it makes the residue partly pre-mapped. Some of the most common ways a given structure's residue reasserts itself have already been charted. The catalog does not eliminate the need to think; it hands you a map of the terrain where thinking usually goes wrong.

6. The limiting case: where analogy becomes identity¶

Push the principle to its limit and something clarifying happens. Suppose two situations shared all their structure — every relation, at every level, with every quantity matched. Then there would be no residue, and by the projection–residue principle the analogy between them could never break. But look at what that situation actually is: it is no longer an analogy at all. Two objects with identical structure at every level are simply two instances of one abstraction. The "analogy" has collapsed into recognition — the perception that here are two members of the same class.

This is the fixed point of the whole picture, and it is exactly what an abstraction is. An abstraction is the structure at which sharing is total by construction, because it is defined as nothing but that shared structure. Two instances of feedback, considered strictly as feedback, cannot disagree about feedback — the disagreement only appears when you step outside the projection and start comparing the instances as the fuller things they are. So the relationship between analogy and abstraction is this: an abstraction is the limit at which an analogy becomes an identity and can no longer fail, and every real-world analogy is a partial approximation to some abstraction, sound across the shared core and breaking wherever the approximation is pushed past it. To reason well by analogy is, in effect, to reason toward the abstraction the two cases share and to stop at its edge. (The catalog's prime abstractions are simply these shared cores made explicit and portable — the cross-domain ones singled out and named.)

A wrinkle is worth naming here, because it looks like a counterexample to the limit and is not. Two situations can share a relational projection completely and still behave differently, if the same relations run under different quantities — because quantity can cross a qualitative threshold. A short bridge and a long bridge share the structure of a loaded beam, yet the long one sags disproportionately and may fail where the short one is fine, because bending stress rises faster than span. But this is not the limiting case breaking; it is a case that never reached the limit. The limit demands total sharing — every relation and every quantity — and here the magnitudes were never matched. Magnitude is residue (§3): the quantitative species of it, which a relational projection discards even when it keeps every relation. Rightly read, the wrinkle is a reminder that residue comes in a quantitative species as well as a relational one, and that the quantitative species is a distinct failure mode for transfers. (This is the domain of scale — itself a prime abstraction in the catalog — and it is why "structurally isomorphic but in a different regime" is a real and separate way for a transfer to break: not a failure of the shared relations, but of the unshared quantities beneath them.)

7. Two flavors of residue: structural and framed prime abstractions¶

The companion essay "Structural and Framed Primes — A Typology for Cross-Domain Portability of Abstractions" distinguishes prime abstractions by how much of their home domain travels with them. A structural prime abstraction carries a clean relational skeleton and little else; a framed prime abstraction carries a portion of its home domain's interpretive framing — assumptions about agency, purpose, value — baked into the abstraction itself. The projection–residue principle interacts with that distinction in a way that matters, because the two kinds hide their residue in different places.

For a structural prime abstraction, the shared projection is a bare skeleton, and the residue is mostly the domain's mechanics: its timings, its magnitudes, its material specifics, whether an agent happens to be present. Breakdown is correspondingly legible. "Both are feedback loops, but this one has long lags" is a difference you can see once you look for it, because the projection and the residue are cleanly separable — the skeleton is thin and obvious, and everything else is plainly extra.

For a framed prime abstraction, the separation is treacherous, because the abstraction has already smuggled interpretive commitments into what you take to be the structure. When you transfer it, you are not carrying a bare skeleton; you are carrying a skeleton wearing part of a costume — and the costume can fail to fit the new domain even at what looks like the structural level. Consider transferring a reward-shapes-behavior schema from animal training to managing people. The relational core does transfer: attach a payoff to an action and you get more of the action. But the schema, drawn from its home domain, quietly assumes a responder who does not reinterpret the reward. People do reinterpret it — they game the metric, they optimize the payoff rather than the goal — and so the analogy breaks not on a timing or a magnitude but on an assumption you never knew you were importing, because it felt like part of the structure. The residue of a framed prime abstraction includes commitments mistaken for structure. That makes framed prime abstractions both more useful (they carry more) and more dangerous (they carry more silently), and it makes their breakdowns harder to anticipate and more instructive to study. If one wanted a single reason the structural/framed typology earns its keep, this is it: the two kinds fail differently, and knowing which kind you are holding tells you what sort of residue to watch for.

8. Honest positioning¶

It would be a disservice to overclaim. The parts of this picture are not original to the encyclopedia. Gentner's structure-mapping theory^[2] already holds that analogy maps systems of relations rather than attributes, that "good" analogies are the ones whose higher-order relational structure maps (the systematicity principle), and that candidate inferences projected by an analogy can be false.^[3] Hesse's positive/negative/neutral analysis^[1] already locates analogical danger in the neutral analogy. The observation that abstraction is selective, and that what it selects away can matter, is as old as the study of abstraction.

What is offered here is a synthesis and a sharpening, plus one genuinely enabling ingredient. The synthesis: the ordinary "surface leakage" account and the deep "structural incompleteness" account are usually run together, and separating them — insisting that a fully purified, correctly mapped analogy still breaks, and for a different reason than a surface-contaminated one does — is where most people's understanding is thin. The sharpening: the projection–residue framing gives the breakdown an address rather than a warning. The enabling ingredient: an explicit catalog of abstractions is what turns the address from a metaphor into a usable coordinate — without it, the factoring the principle demands is not something a reasoner can actually carry out. The contribution is not "here is a new fact about analogy." It is "here is the machinery that lets an old fact about analogy be operated rather than merely known."

9. Why it matters, and a note on teaching¶

If the crown-jewel skill of cross-domain reasoning is recognizing that a familiar structure is present in an unfamiliar place, then its indispensable twin is knowing where that recognition stops paying — where the shared projection ends and the residue begins. A person who can do the first but not the second is worse off than one who can do neither, because they will transfer confidently and be wrong in ways they cannot see.^[4] The projection–residue principle names that second skill precisely: it is the ability to locate the edge of the projection and refuse to reason past it.

This has a direct consequence for how the material should be taught, which the companion teaching proposal (Teaching Abstract Reasoning Directly) develops at length and which we only flag here. Teaching people that analogies break is nearly worthless — they already believe it. Teaching them to find the edge is the thing, and it is best taught actively: give a learner a transfer whose structure genuinely holds, have them confirm it, then have them push it until it breaks, and lead them to see that it broke not because the shared structure failed but because they reached into the residue. Nor is the drill mere pedagogical hygiene: it is the discovery protocol of model-based science in miniature (§4) — push the analogy, hold the push as conjecture, and let the breakage teach you the residue. A learner running it is rehearsing Young and Michelson–Morley at desk scale. Done on a structural prime abstraction it teaches the clean case; done on a framed one, the insidious. The recognition and the boundary are two halves of a single competence, and a curriculum that trains only the first is training people to be confidently wrong.

References¶

[1] Hesse, Mary B. Models and Analogies in Science. Notre Dame: University of Notre Dame Press, 1966. Introduces the positive / negative / neutral analogy trichotomy — properties known to be shared, known to differ, and not yet sorted (a distinction Hesse adapts from J. M. Keynes's A Treatise on Probability). Staged as a dialogue between a Duhemist, for whom models are dispensable scaffolding, and a Campbellian, with whom Hesse sides: the neutral analogy is the source of a model's predictive power — both the risk anatomized in §4 and the engine of conjecture credited there. ↩

[2] Gentner, Dedre. "Structure-Mapping: A Theoretical Framework for Analogy." Cognitive Science 7, no. 2 (1983): 155–170. Founds structure-mapping theory: analogy maps relations between objects rather than object attributes, with the mapped relations selected by systematicity — the preference for systems of relations governed by higher-order relations. ↩

[3] Gentner, Dedre, and Arthur B. Markman. "Structure Mapping in Analogy and Similarity." American Psychologist 52, no. 1 (1997): 45–56. Develops structural alignment as the common process beneath analogy and similarity, including the projection of candidate inferences from base to target — inferences that can be false and must be independently evaluated. ↩

[4] Holyoak, Keith J., and Paul Thagard. Mental Leaps: Analogy in Creative Thought. Cambridge, MA: MIT Press, 1995. A multiconstraint theory of analogy across problem solving, decision making, and discovery, with sustained attention to how analogical inference goes wrong. ↩