Meaning is structure

Meaning has features, and the features have homes

Start with what meaning visibly does, before any theory is named.

Meaning has similarity. “Cat” and “kitten” sit nearer in sense than either does to “logarithm.” Anything that wears the title of a meaning-model has to encode that fact somewhere. The geometric home for similarity is distance [a numerical measure of how far apart two points sit in some space]. Closer points mean more similar things. A flat vector space with cosine similarity is one realization. A hyperbolic space with hyperbolic distance is another. What is forced is not the specific realization but the bare fact that similarity has to land somewhere, and “somewhere” is some notion of nearness.

Meaning has hierarchy. “Animal” contains “dog,” which contains “Labrador,” which contains the specific dog asleep in the next room. The structure is not a chain; it is a tree, because “animal” also contains “cat” with its own breeds, and “fish” with its own families, and so on. Trees grow exponentially. The number of leaves at depth \(n\) is the branching factor raised to the \(n\)th power. If your tree branches three ways at every node, depth ten gives you about sixty thousand leaves. The geometric home for tree-like growth is hyperbolic geometry [the curved geometry where the volume of a ball grows exponentially with its radius rather than polynomially]. Nickel and Kiela showed in 2017 that hierarchical data embeds dramatically more efficiently in hyperbolic space than in Euclidean space, because the host’s volume growth matches the data’s (Nickel and Kiela 2017). They tried to fit the WordNet hierarchy [a standard hand-curated tree of English nouns and their relationships] into a flat space and a hyperbolic one with the same number of dimensions, and the hyperbolic version won by a large margin on every metric they cared about. The 2025 HypLoRA paper (Yang et al. 2025) went further: it found that the token embeddings inside trained language models already contain power-law and tree-like structure, which the flat-space training has been trying to fit through the wrong-shaped doorway. Hierarchy is a feature meaning has, and the geometry that fits hierarchy is the geometry where exponential branching is what the volume itself does.

Meaning has correspondence. The relationship between “father” and “son” is mirrored in the relationship between “tree” and “branch.” When we recognize an analogy, we are recognizing that two structures share their connectivity even though their substances differ. The geometric home for cross-domain pattern-matching is topology [the branch of geometry that asks what features of a shape survive when the shape is bent or stretched without being torn]. Topological invariants are exactly the features that ride through transformations of the underlying material. Carlsson’s framework for topological data analysis (Carlsson 2009) and the persistent-homology results from neuroscience (Giusti et al. 2015; Reimann et al. 2017) are versions of the same move: track the shape that survives rotation, rescaling, or relabeling, and you have tracked the part of the structure that is portable across instances. Persistent homology is the technique that makes this concrete. Given a cloud of points, it asks which loops, voids, and higher-dimensional holes persist as you smoothly inflate the points into one connected blob, and the persistent ones are the genuine features rather than the noise.

Three features of meaning. Three geometric homes. I have not picked convenient mappings here. Any cognizer modeling meaning runs into similarity, hierarchy, and correspondence in turn, and each has its own geometric home regardless of substrate. This is the part of the argument I call forced in the same sense the special-relativity geometry was forced in §02. You can ignore one of the features and model meaning badly, and many meaning-models do exactly this. But the features do not stop being there because the model ignores them. They show up in the failure modes of whichever model you built. A flat-space model of a tree-shaped concept eventually crowds together the leaves it should be separating, because the volume in a flat space does not grow fast enough to give them room. A model with no topology eventually fails to recognize an analogy that would be obvious to a child. The geometry is not a stylistic choice. It is what fits or fails to fit the shape of the thing being modeled.

Containment, not analogy

The literature is full of a careless version of this move, and I need to walk past it before I can state the careful one.

The careless version runs: the universe is non-Euclidean, therefore meaning embeddings should be non-Euclidean. That is an analogy. Object A has property X, so the model of A should have property X, because A and its model resemble each other. The reasoning is bad. The non-Euclidean structure of spacetime arose from particular physical pressures: the invariance of the speed of light, the curvature induced by mass, the gauge structure of the standard model. Those pressures do not transfer to language modeling. A reader who left §02 thinking “spacetime is hyperbolic, so embeddings should be hyperbolic” was reading the section wrong, and I am not endorsing that move.

The argument I am making is containment.

The universe is the geometric reality meaning inhabits. Meaning occurs in this universe, done by structural beings (brains, ecosystems of brains, machines built by brains) inside structural reality. Meaning does not float free of the host. Whatever shape meaning has is a shape of something that lives inside the host that is trying to predict the universe, and the host has regimes: Euclidean local, hyperbolic where exponential branching is the natural growth, Hilbert-space-like where degrees of freedom multiply combinatorially, topological in whatever survives the various transformations the host runs over its contents. When we model meaning, we are not picking a geometry by analogy with the universe’s. We are picking a geometry that approximates the regime of the host meaning it happens to be trying to model.

The distinction changes what evidence counts. An analogy argument can be defeated by pointing out the analogy is shallow: the model of A does not need to share property X with A just because we noticed a resemblance. A containment argument cannot be defeated that way, because containment is not an inference from one structure to another. It is a statement about where meaning lives. If meaning is a process inside the universe, the shape meaning takes is whatever shape the universe accommodates for that kind of process. The four-program convergence in §04 is then not a coincidence and not a methodological artifact. It is what containment looks like from the inside, observed in four different examples.

This is the load-bearing move of the section. For the engineering argument the paper is making, what matters is the structural relationships between meaning and host, regardless of which metaphysical reading the reader prefers. The structural picture sits one level above the metaphysical disputes. A reader committed to mathematical-universe-hypothesis maximalism, an ontic structural realist, a sophisticated substantivalist about spacetime, and a relationalist can each take the containment claim home with their own gloss on what “the universe is structural” amounts to. The §04 convergence is the data. The containment framing is a way of reading what the data is showing, and it is compatible with several deeper readings of what existence is.

The features-to-homes argument is forced by the features themselves. The §04 convergence is the empirical fact. Containment is the philosophical proposal that ties the two together, and it is the part a reader is most welcome to push back on. A reader who declines containment can still take much of the paper home; the §07 architecture sketch rests on the geometric features being real, not on any particular reading of why. But the section’s strong claim is the containment claim, and it is a claim, not a derivation.

Distance from Tegmark, position relative to structural realism

The position has neighbors in the philosophy literature, and I owe the reader a clear map of which neighbors I am close to and which I am explicitly not.

The closest distant neighbor, the one a careful reader is most likely to confuse this position with, is Max Tegmark’s mathematical universe hypothesis, often abbreviated MUH [the claim that every consistent mathematical structure is realized as a physical universe; ours is one such structure among an enormous ensemble] (Tegmark 2008). Tegmark’s argument runs roughly like this: the only way to take seriously a physical reality independent of human description is to identify physical reality with mathematical structure. Once you do that, there is no principled reason to grant physical existence to one consistent structure (ours) and withhold it from all the others. So all of them exist. Our universe is one structure in an ensemble whose size is the size of “all consistent mathematical structures.”

The position here does not make that claim. I am asserting that this universe has whatever structure it has, and that meaning shares that structure because meaning happens inside this universe. I am not asserting that every other consistent mathematical structure exists physically. The narrower claim does not require the broader one. The §04 convergence and the containment move are equally compatible with “this universe is structural and the only universe” and with “one structural universe among many.” This paper picks neither. It is about modeling meaning, not about the cardinality of existence.

This narrowing also dissolves the standard objection to Tegmark. George Ellis’s 2009 critique (Ellis 2009) is the sharpest version of the worry, and it deserves to be stated at full strength before any reply. Ellis grants that mathematics is unreasonably effective in describing the physical world; that part is not in dispute. What he argues is that Tegmark conflates two different domains in identifying them. The first is the observable world that physics is responsible to: the world you can probe with apparatus, measure with clocks and rulers, and check against. The second is the Platonic mathematical world that mathematics is responsible to: the world of consistent structures, theorems, and proofs. Physics, on Ellis’s reading, is the discipline that holds mathematical models accountable to observation. To identify physical reality with mathematical structure, Tegmark has to throw out the observational content of physical theories as “baggage” around a mathematical core. Ellis argues this is incoherent: the external-reality claim was supposed to be about the observable world in the first place, and a structure-only ontology has nothing to observe with and nothing to be observed.

Ellis’s critique lands on Tegmark and does not transfer here, because the structure-versus-observation split Ellis assumes is exactly what the containment view denies. Observation is not extra-structural content laid on top of a mathematical skeleton. Observation is itself structural. It is the phenomenology of a constrained slice observer, which §03 set up directly. When a physicist looks at an apparatus, the looking is a structural process inside the universe. The data the apparatus produces is structural. The match between mathematical model and observed measurement is a match between two structures, both inside the host. There is no separate observation-domain on one side and a separate structure-domain on the other; there is a single structural reality, and observation is a process inside it. If a reader finds that move unconvincing, the section’s strongest claim degrades to “structural realism applied to language modeling,” which is still defensible. It is just less ambitious than the version with observation folded in.

The closer neighbors are the structural realists. John Worrall’s 1989 paper (Worrall 1989) introduced the move. Worrall noticed a striking pattern in the history of physics: when one theory replaces another, the entities the old theory posited often get discarded outright. Caloric (the imagined fluid that was supposed to carry heat from one body to another) is gone. Phlogiston (the imagined substance released when things burned) is gone. The aether of §02 (the imagined medium that light was supposed to wave through) is gone. But the structural relationships those theories captured between observable quantities tend to survive into the new theory in a modified form. Maxwell’s equations were originally written down for an aether-based theory of electromagnetism. The aether got discarded in 1905, but the equations themselves survived almost untouched into relativistic field theory. Worrall’s diagnosis: what science tracks across theory change is structure, not the underlying entities. Theory change preserves the structural skeleton while replacing the metaphysical clothing.

James Ladyman and Don Ross sharpened Worrall’s move into ontic structural realism [the claim that structure is not merely what science best tracks but what fundamentally exists; the underlying entities are either nothing over and above the structural relations, or nothing at all] in their 2007 book Every Thing Must Go (Ladyman and Ross 2007). Their argument runs through philosophy of physics. Quantum mechanics, on their reading, gives us particles whose individuality is ambiguous to the point of being meaningless: two electrons in a singlet state cannot be told apart even in principle, and the formalism behaves as if there is no fact of the matter about which is which. General relativity gives us a spacetime whose points have no identity independent of the relations the metric imposes on them. Both theories, they argue, push toward the conclusion that relations are more fundamental than any objects standing in them. The position this paper takes can fairly be described as ontic structural realism applied to the modeling of meaning, sitting at the more aggressive end of that tradition.

The most recent kin is Colin Hamlin’s 2026 paper in Synthese (Hamlin 2026). Hamlin develops what he calls a Universal Theory of Structure, abbreviated UTS, by holding ontic structural realism and Tegmark’s MUH together at once. Both views, taken seriously, run into what Hamlin calls the collapse problem: the worry that they erase the distinction between physical structures and merely mathematical structures, leaving no principled reason to call one of them “real” and the other “abstract.” Hamlin’s move is to say the collapse is not a bug but the feature, and that the collapsed view, where physical and mathematical structures are not different kinds of thing at all, is the correct one. I am not following him into that move. My position remains narrower: this universe has structure, meaning shares it because it is a model of it, and the cardinality of the rest of existence is left open. But Hamlin is the closest published kin to this section’s philosophical core, and a reader who wants the maximally ambitious version of structure-as-existence has a developed account of it in his work.

What is “wrong” meaning, if meaning is structure?

The standard worry about identifying meaning with structure is that the identification dissolves correctness. If a cognizer’s [the thing doing cognition] internal model is a structure, and meaning is structure, then any internal structure is meaning of something, and the cognizer is incapable of being wrong. The “anything goes” objection.

The objection has a clean answer once the containment framing is in place.

A cognizer is itself a structure inside the host. Its internal model is another structure, the representational machinery considered as a system of relations. The cognizer’s model is correct about a target to the extent that the structure of the model matches the structure of the target. Correctness is structure-to-structure fit. It is not arbitrary; the host’s relations either are or are not preserved by the model.

What is configuration-dependent is which target the cognizer is modeling and which structural features matter for its purposes. A bee modeling nectar sources cares about the spatial structure of flower distributions and not about bee parliaments, because there are no bee parliaments. A human modeling a conversation cares about social-relational structure (who said what to whom, what was implied, how the listener took it) and not about the quantum-mechanical phase of every electron in the room. Different cognizers, configured differently by genetics, environment, and training, fit different aspects of the host. They get different “correct” models because they are modeling different sub-structures, not because correctness has dissolved into preference.

In the configuration-independent limit, where a cognizer’s purposes shrink to nothing in particular and the question is whether the model fits the host as such, there is a single target: the structure of the universe itself. A model is substrate-correct in this limit when it preserves the structural relations of the host. Science is the long-running project of approaching this limit asymptotically. We do not get there; the match is always partial, and there is no view from nowhere we could check the match against. But “more substrate-correct” is a meaningful comparative even when the asymptote is unreachable.

I want this commitment visible, because it is the place where my position smuggles the most. The substrate-correct-in-the-limit move is not a derivation. It is a structural-realist commitment of the same kind Worrall and Ladyman-Ross make, and I am inheriting it from them rather than re-earning it here. A reader who rejects structural realism wholesale will reject the move, and will be entitled to. I flag the dependence rather than let it pass disguised.

What this gives back is the resolution of the “anything goes” worry. Different configurations of cognizer give rise to different “correct” models, each correct relative to its configuration. No model is correct in isolation from a configuration, and no model is wrong simply because some other model exists. But there is a structural-realist asymptote all of them are approaching, and approach is comparable. The identity claim does not collapse into relativism. It anchors at the substrate.

The reach of the claim

Here is what the paper does claim. Meaning has structural features (similarity, hierarchy, correspondence) with natural geometric homes (distance, hyperbolic geometry, topology). Cognizers modeling meaning must respect those features or their models break in predictable ways. The universe is the structural host inside which meaning happens. Meaning’s shape is the host’s shape because meaning is a process inside the host. Observation is itself structural. Correctness is structure-to-structure fit, with substrate-correct as the configuration-independent limit.

Here is what the paper does not claim. That every consistent mathematical structure exists physically. That ours is one structure among many. Substantivalism or relationalism about spacetime. Any specific reading of quantum mechanics. The existence of a view from nowhere that would let us check our models against reality directly. The position is consistent with several resolutions of each, and declines to choose. The structural picture sits one level above those metaphysical disputes, and that is by design. A reader who wants a metaphysical commitment can pick one and bring it back; the engineering argument the paper is making does not need it.

The paper is doing one job: arguing that models of meaning should be built after the structure of the universe meaning inhabits, rather than after the linear-algebra defaults that fit a flat geometry on a GPU. That argument does not depend on settling what existence is. It depends on the four-program convergence being real, the features of meaning being real, and the containment relation between meaning and host being a more honest framing than analogy.

The next section turns from the philosophical core to the engineering implication. If meaning shares the host’s regimes, the architecture for modeling meaning should span those regimes rather than collapse them into a single tractable space. That is where the paper goes from “what is the picture” to “what is the build.”