What Is a Model?
Structure, representation, and purpose
Consider two representations of a railway. The first is a timetable: every train, platform, arrival, and departure for the day. The second is the Underground map — a few colored lines and dots, with distances and directions frankly falsified. Hand a stranger both and ask how to get from Paddington to Bank, and the diagram makes the network legible at a glance. But the contrast is not that one object is a dataset and the other a model. A timetable can function as a model for scheduling, just as a map can function as data for another model. The difference is the structure each preserves for a particular use.
That question is the hinge of this whole volume. All empirical knowledge is model-mediated — I argued that in Maps, Models, and Understanding, and I will not re-run the argument here. That chapter owns the epistemology: why there is no view of the territory except through some map, and why the map’s distortions are the source of its power rather than a defect in it. This chapter owns the anatomy. Granting that cognition, science, and control all run on models, what is a model, mechanically? What has to be true of a structure before it earns the name?
Structured Representation
At the most general level, a model is a purpose-relative structured representation of some domain — a structure whose internal relations preserve the distinctions and relations needed for a particular task, while omitting or suppressing others. A model need not resemble what it represents. It need not look like it, be made of the same stuff, or share its geometry. The Underground map resembles London in almost no respect a surveyor would recognize; it preserves connectivity — which line reaches which station, where the interchanges are — and throws away geography wholesale. That is not a compromise forced by the printer. It is what makes the map usable. A model is defined by two choices: what to preserve and what to discard.
This definition is deliberately catholic about format. Newtonian mechanics is a model; so is a climate simulation, a probability distribution over disease states, and a child’s expectation that dropped things fall. Cybernetics extends the term to the tuning of a thermostat and the pattern of synaptic weights in a visual cortex. That extension is useful, but it is an as-if structural sense: it need not imply that the system explicitly represents anything to itself. What unifies these uses is not their medium but the mapping they support. Ask what mapping a candidate supports, for whom or what, and over what domain it holds.
From which the central discipline follows: adequacy is always relative to a purpose. A model is never adequate full stop, only adequate for something. The Tube map is adequate for planning a journey and useless for estimating walking distance between two stations that sit adjacent on the diagram and a mile apart in the city. Newtonian mechanics is adequate for launching a satellite and inadequate for the timing chips in the satellite’s GPS receiver, which need relativity. A model earns its status by enabling appropriate action or explanation within its domain of use, and forfeits it the moment it is carried past that domain’s edge. This is the same conditionality that governs all truth, specialized to representations: the model is where the conditions live.
Implicit and Explicit
Nothing in that account requires a model to be written down, or even to be the kind of thing that could be. Most models in the world are not.
An explicit model is one constructed deliberately and represented symbolically: a system of equations, a simulation, a diagram, a stated theory. These are the models science refines — objects you can inspect, criticize, and hand to someone else. But a model can also be implicit: realized in the physical structure of a system rather than in any representation the system holds of itself. An enzyme pathway regulates a cell’s chemistry according to regularities it has evolved to exploit, without anywhere representing those regularities. A neural circuit encodes the statistics of the sensory world in its wiring without expressing them as propositions. The circuit is the model; there is no separate copy of it stored elsewhere for the organism to consult.
The distinction matters because it stops us from looking for models in the wrong place. To ask whether a bacterium “has” a model, in the sense of holding a description it could recite, is to ask the wrong question. The model is in the doing — in the structure that reliably maps conditions to appropriate responses. Explicitness is a feature of some models, prized because it makes them shareable and revisable. It is not part of the definition.
The Generative Line
Return now to the timetable and the map. A dataset records or organizes observations. A model preserves relations that support some task. These roles overlap: a timetable is a dataset that can answer scheduled-time questions, while a network map can itself become input data. The useful distinction is therefore not between two exclusive kinds of object but between two roles an object can play.
A powerful model often yields expectations about how a state will evolve or how a system will respond to an intervention not yet tried. It supplies a generative mechanism: a set of relations that maps inputs to outputs, including inputs never yet seen. A bare list of observations has no such capacity on its own. A timetable may answer questions licensed by its tabular structure, but it cannot infer how a closure changes the network unless closure relations have been encoded. A network map can support that inference because it preserves connectivity.
This is why a model is not made better merely by making it bigger. Adding rows to the timetable does not add a network topology. Generative structure differs from the observations from which it may be distilled, and confusing the two — treating a rich dataset as though it automatically supplied understanding — is one of the standing errors of the data age. Generativity is strong evidence of understanding because it tests whether a representation carries structure beyond memorized cases. It is not, by itself, a definition of either model or understanding.
Compression and Generalization
Where does generative structure come from? Often from compression. A model that captures a regularity can represent an unbounded range of cases in a compact form: a small set of Newtonian equations replaces the individual trajectories of projectiles. Compression can buy generalization when the compressed regularity tracks structure that persists outside the sample. Good compression and good generalization often travel together, but neither guarantees the other: a system can memorize compactly yet generalize badly, or generalize through a representation that is not minimal.
The contrast case sharpens this. A lookup table — a fixed list pairing each anticipated condition with a stored response — performs no compression. It can answer only for states it has already been given, and for a novel state it is silent. A compressed model, having thrown away the individual cases in favor of the structure behind them, extends smoothly to cases it was never shown. This is why complex environments reward compression so heavily: when the space of possible situations is astronomically larger than any list could enumerate, only a structure that generalizes can keep up. Compression is not strictly required for something to be a model — as we are about to see, the degenerate uncompressed case still counts — but in any environment large enough to matter, it is the difference between a model that copes and one that stalls the first time the world does something new.
The Interpretive Layer
There is one more use of “model” that has to be separated out cleanly, because conflating it with the others causes no end of trouble. When we deal with agents, models appear a second time — not inside the agent, but in us. To attribute beliefs, desires, and intentions to a system is to build a model of that system at a conceptual level: an interpretive scheme that predicts its behavior by treating it as if it wanted things and represented the world. These interpretive models are not components of the agent. They are our tools for making sense of it, and they live in a different representational layer than whatever regulatory machinery the agent runs on.
Keeping the two layers apart is the whole game when we come to belief. A system’s own internal model — the structure that maps its conditions to its actions — is one thing. The model we build of that system, in the vocabulary of belief and desire, is another. A thermostat has the first and does not need the second; we may still find it convenient to say it “wants” the room at twenty degrees. Beliefs, I argue in What Beliefs Are, are not contents an agent stores but features of the interpretive model an observer attributes — including the model each agent keeps of itself. That is a claim about the second layer. It does not compete with the claim that the agent runs on an internal model of the first kind; the two coexist, at two levels, and most confusion about machine belief comes from collapsing them.
The Minimal Case
Push the definition down to its floor and you reach the lookup table — and the floor turns out to be inside the house, not outside it. The uncompressed condition-to-action list is a degenerate model, but it is a real one.
The traditional specimen is the Sphex wasp. Provisioning her nest, the wasp drags a paralyzed cricket to the burrow’s threshold, leaves it there, enters to inspect the nest, and then hauls the cricket in. Move the cricket while she is inside, and on emerging she may drag it back to the threshold and inspect the nest again. The anecdote usefully illustrates a rigid action pattern, although real wasp behavior is more variable than the textbook caricature suggests. A small state-transition policy can approximate the sequence; that approximation does not establish that the animal carries no additional state or representation.
The wasp is not alone. Bacterial chemotaxis uses short-timescale comparison of chemical concentration to bias switching between swimming and tumbling. Fixed-action patterns in birds and fish respond strongly to releasing stimuli. Plant tropisms convert local gradients into differential growth. Small policies can work where relevant distinctions are few and stable; none of these examples licenses the claim that the organisms are literally lookup tables.
Calling these tables models adopts the broad cybernetic convention discussed above. Conant and Ashby’s Good Regulator Theorem motivates that convention under specific formal assumptions; it does not establish that every successful controller contains a semantic representation. I develop the theorem and its limits in Control Requires Models. Here the table is a model by the chapter’s functional convention: it preserves the distinctions a task requires and discards others. It need generate no expectations beyond fixed entries, compress nothing, and generalize not at all. It is the boundary case that prevents useful properties of sophisticated models from being smuggled into the definition.
And it is a model without beliefs. The wasp regulates without instantiating anything resembling a propositional attitude; her internal structure is the first-layer kind, not the second. Reading beliefs into her is our interpretive overlay, and a poor fit at that. The lookup table thus marks both the lower bound of modeling and the cleanest illustration of the anatomy: a structural model with no interpretive layer, no generativity, no compression — real, adequate, and utterly minimal. Everything richer, from the child’s grasp of falling objects to the physicist’s field equations to whatever large language models are doing, is built by adding back what the wasp does without: compression, generativity, the capacity to generalize past what was ever encountered. Those additions are the subject of the chapters that follow. The floor is a model. A mind builds more on top of it.