The load-bearing analogy

"A company is like a family." "The atom is like a solar system." "The brain is like a computer." Each of these carries some structure across and drops some, and the trouble is always the silent dropping — the relation the analogy quietly leaves behind while you keep reasoning as though it came along. "We're a family here" imports authority and belonging, then hopes you won't notice that a family can't lay you off.

Mathematics has a precise ladder for how faithful a mapping is. Does it preserve the relations — send connected things to connected things (a homomorphism)? And is it reversible without loss, so nothing on either side is invented or ignored (an isomorphism)? Analogies are homomorphisms of thought: they preserve some structure, rarely all of it. And most arguments go wrong in exactly one way — by treating a lossy homomorphism as a lossless isomorphism, importing conclusions across a mapping that never carried them.

Two small structures sit side by side — a source you understand and a target it is meant to illuminate. Each is a set of nodes joined by labelled relations. Tap a source role, then tap where it maps (or drag). The checker then verifies, relation by relation, which edges the mapping preserves, which it breaks, and which structure in the target it can't even see — then tells you which inferences you may legitimately import.

A mapping sits somewhere on a ladder of faithfulness. The bottom rung is a mere correspondence — a pairing of things that respects nothing in particular. One rung up is a homomorphism: every relation in the source lands on a real relation in the target. Structure survives, but the map may still be lossy — it can collapse distinct things together, or leave whole regions of the target untouched. The top rung is an isomorphism: a homomorphism that is reversible, one-to-one and onto, whose inverse is also a homomorphism. Nothing is invented and nothing is dropped; the two structures are, for every purpose that the relations capture, the same structure wearing two names.

Every analogy is a claim about where on this ladder its mapping sits — usually an unstated claim, and usually an overstatement. In the checker, rock–paper–scissors → C₃ reaches the top: it really is one structure in two costumes. map → territory settles honestly in the middle — everything the map says is true, and the territory keeps a thousand things the map never carried. The rest are vibes with a fidelity score.

Map family onto company and read the broken edges. Start on the default scenario. Authority and duty come across green; unconditional care and permanent belonging snap red. That red pair is precisely what "we're a family here" borrows the warmth of while the company keeps its one relation the family never had — at-will termination, glowing gray, unseen by the analogy.

Import an inference and watch it flagged. In the import test, push "the head owes unconditional care." It comes back unsupported — the relation it rests on was dropped. Push "the head may direct those below" and it is licensed, because authority survived. Same analogy, two inferences, opposite verdicts: the honesty is never in the metaphor as a whole, only in the particular relation your conclusion needs.

Run the site's own metaphor through the audit. Switch to the self-audit tab — "a habit is an attractor basin," one of this project's own pictures. You may import "a small lapse self-corrects." You may not import "the basin is fixed; you're trapped." Then scramble any scenario's roles and watch fidelity collapse: most mappings are worse than the obvious one, which is why the obvious one feels like truth.

Analogies mislead not by what they map but by what they quietly don't. The visible transfer is honest advertising — "a company is a bit like a family, in these ways." The damage is done by the unmapped relation you reason across anyway, because the mapped ones lent the whole comparison an air of completeness. You accepted authority and belonging; the absence of care travelled in on their coattails, unexamined, and now you feel disloyal for updating your résumé.

So here is how to argue with an analogy, in two moves. First, name the mapping — say exactly which source node each target node is playing, out loud, because a mapping left implicit can be silently switched mid-argument. Second, check the one relation your conclusion needs — the single edge the inference stands on. Is it green? Then carry the conclusion. Is it red, or gray, or absent from the source entirely? Then the conclusion was smuggled, however beautiful the rest of the correspondence. Most bad arguments from analogy are one unchecked edge wearing the credibility of five checked ones.