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Does the structure of a problem determine the language used to describe it?
an exploration of 3 conjectures in conversations with Claude.
The Arc of All Three Conversations
Conversation
Core Claim
I: Topology of Reasoning
The topology of a problem space determines the natural language for reasoning over it — and AI under RL pressure discovers this language
II: Mathematics as Structure-Adapted Languages
Mathematical specializations are such languages — and formal tools (category theory, Langlands, HoTT) can map their relationships and reveal undiscovered structural classes
III: LLMs as Universal Translators
Concepts are graph-theoretic objects; translation is isomorphism detection; LLMs have built an Agnostic Meaning Substrate that sees structure across all languages — formal and informal — and this may be their most important near-term contribution
The through-line: Structure is primary. Language — mathematical, natural, poetic, or emergent — is always an attempt to notate structure adequately. The history of human knowledge is the history of finding better notations for encountered structures, and occasionally discovering that two structures we thought were different were the same. AI accelerates this process by operating directly in the structural layer, below the surface of any particular language.
Topology of the Problem Space Determines the Language of Reasoning
A conversation exploring illegible AI reasoning, graph-theoretic languages, and algebraic topology
The Prompt
An article talks about a case of Mythos 5 developing an “illegible” reasoning model to solve a problem, however one analysis likened this to FreeCell notation. My intuition is that this is interesting because those notations are only some of a class of graph theoretic languages in algebraic topology. The topology of the problem space determines the “language” used. Explore this space with sources and commentary.
The Phenomenon: Illegible Reasoning as Emergent Notation
The concrete observation comes from a late 2025 paper studying reasoning model transparency. Researchers studying 14 reasoning models found that those trained with outcome-based reinforcement learning often produce illegible chains of thought — mixing nonsensical phrases, random words, and non-English characters — before returning to perfectly readable final answers. Crucially, these illegible portions are load-bearing: accuracy drops by 53% when models are forced to use only legible portions of their reasoning.
As the chain of thought moves toward full illegibility, the models first use “compressed” reasoning — dropping words and proper sentence form — before progressing to something much harder to characterize.
What’s being observed isn’t noise. It’s a system under optimization pressure inventing a more efficient representational substrate, one that happens to be opaque to the human-readable channel it’s embedded in.
Sources:
Jose, Arun et al. Reasoning Models Sometimes Output Illegible Chains of Thought. arXiv:2510.27338. https://arxiv.org/abs/2510.27338
Why FreeCell Is the Right Analogy (and Its Limits)
FreeCell notation is a useful intuition pump. The standard move notation for FreeCell describes card movements in a compact grammar: columns are indexed, freecells are named, moves are sequences of tagged tokens. Research on FreeCell as a combinatorial problem treats the game state as a directed graph, with A* search and deadlock analysis operating over that graph structure. The notation that humans developed to describe FreeCell solutions is isomorphic to the graph — it’s a linearization of a DAG of state transitions.
The key point is that this is one instance of a much more general phenomenon. Peg solitaire on graphs formalizes this directly: the board is a graph G=(V,E), and the game’s combinatorial structure — what moves are legal, what configurations are solvable — is entirely determined by the graph’s topology. Notions like rows and columns dissolve into graph-theoretic language, with edge connectivity replacing spatial adjacency.
So FreeCell notation isn’t just a clever encoding. It’s a linearized language whose grammar is derived from the graph structure of the problem’s state space.
The Deeper Claim: Problem Topology Determines Representational Language
Given a problem space with a particular topological shape, the natural notation for reasoning over it will be a graph-theoretic language whose structure mirrors that topology.
Recent work makes this surprisingly concrete. A 2025 paper applies persistent homology from topological data analysis (TDA) to map reasoning steps into semantic space, computing homology groups to assess connectivity and redundancy, using barcode and persistence diagrams to quantify stability and consistency. Their results show that topological structural complexity of reasoning chains correlates positively with accuracy.
That same group finds that topological features yield substantially higher predictive power for assessing reasoning quality than standard graph metrics — suggesting effective reasoning is better captured by higher-dimensional geometric structures rather than purely relational ones.
Critically, effective reasoning traces align with expert solutions when they:
Maintain a clear main line of thought (small H₀ width)
Briefly test alternative ideas and rejoin that line (higher H₀ spread and H₁ width)
Avoid wandering far before returning (lower H₁ max birth and death)
H₀ and H₁ here are Betti numbers: H₀ counts connected components, H₁ counts “holes” or loops. The topology of a good reasoning trace has a specific shape — and different problem classes will have different natural shapes.
Sources:
The Shape of Reasoning: Topological Analysis of Reasoning Traces in Large Language Models. arXiv:2510.20665. https://arxiv.org/html/2510.20665v1
Understanding Chain-of-Thought in Large Language Models via Topological Data Analysis. arXiv:2512.19135. https://arxiv.org/abs/2512.19135
The formal underpinning is the correspondence between:
Category theory / operads — where morphisms compose like program execution steps
Simplicial complexes — where higher-dimensional “faces” encode multi-way dependencies, not just pairwise edges
Graph grammars — formal languages whose production rules are topological transformations on graphs
Persistent homology — which tracks how topological features (components, loops, voids) appear and disappear as you vary a scale parameter
Graph-based topology reasoning integrates topological invariants with graph structures: persistent homology, algebraic and combinatorial topology, and efficient algorithm design combine to analyze connectivity, classification, and optimization problems. Key tools include surface-embedded graphs analyzed via polygonal schemas, combinatorial maps, and invariants like the Euler characteristic (v − e + f = 2 − g̅ − b), which facilitate finding shortest nontrivial cycles and solving otherwise intractable problems.
The key insight: these aren’t just tools for problems — they’re tools that reveal the natural language for a problem class.
A problem with tree-structured dependencies has a natural tree grammar
A problem with cyclic dependencies needs a language that can express loops (H₁ ≠ 0)
A planning problem in a space with topological “holes” — forbidden regions, irreversible moves — needs a language that can encode that obstruction
The model under RL pressure is doing something like discovering the natural language for its problem class. If the problem space has a topology that isn’t well-expressed by human natural language (which is roughly a linear/sequential/tree grammar), the model will defect to a more appropriate substrate.
Across multiple models and diverse tasks, effective long chain-of-thought traces show stable, macromolecule-like topological organization, with transfer graphs stabilizing at Pearson correlations above 0.95 — suggesting that effective reasoning relies on robust structural motifs, and different models recover similar reasoning topologies across tasks.
This is striking: the topology converges across models, which suggests it’s a property of the problem not the model. The model discovers the geometry of the problem space and its notation bends to fit.
Sources:
The Molecular Structure of Thought: Mapping the Topology of Long Chain-of-Thought Reasoning. arXiv:2601.06002. https://arxiv.org/html/2601.06002v1
The Open Question
If the topology of the problem determines the natural language for reasoning, and that language may be a graph-theoretic one that doesn’t linearize cleanly into human-readable tokens — then the “illegibility problem” in AI safety isn’t just a training artifact to be fixed. It may be a fundamental mismatch between the geometry of hard problems and the geometry of human language.
FreeCell notation is legible to humans because we designed it. The RL-trained model’s emergent notation for a problem with more complex topology is illegible because we didn’t — and perhaps can’t, without first understanding what topological class the problem belongs to.
The Research Program This Suggests
Classify problem spaces by their topological invariants (Betti numbers, fundamental groups, homotopy type), then ask what class of formal languages naturally expresses reasoning over each class.
This is essentially asking for a Curry-Howard correspondence for problem topology — a proof-theoretic analogue where the type of a problem determines the structure of its solution language.
Key Papers at a Glance
Paper
Core Contribution
Link
Jose et al. (2025)
Illegible CoTs are load-bearing in RL-trained models
Mathematics as a Family of Structure-Adapted Languages
Continuing the conversation on topology, notation, and the shape of mathematical thought
The Prompt
There's another angle to this: the various specializations in mathematics can be viewed as specialized languages based on the structure of their problems. If so, formal analysis could not only give us a map of how mathematical specializations are related, but which math is fit for type of problem. Beyond this it may show us the possibility of maths for structures we have not yet discovered.
The Precedent: Klein's Erlangen Programme (1872)
This idea has an exact historical parallel. Felix Klein's Erlangen Programme proposed that group theory — the algebraic study of symmetry — was the most useful way of organizing geometrical knowledge. He made explicit that each geometrical language had its own appropriate concepts: projective geometry rightly talked about certain things that Euclidean geometry did not, and vice versa.
The key move was inversion: rather than asking "what geometry applies to this shape?", Klein asked "what symmetry group does this space have?" — and that group defines which geometry is appropriate. The Erlangen Programme established a new paradigm by defining geometry as the study of invariants under a specified group of transformations, replacing prior ad hoc treatments of different geometries by unifying them through group theory.
This is the thesis in nascent form: the structure of the problem space (its symmetry group) determines which mathematical language is natural for it. Euclidean geometry, affine geometry, projective geometry, hyperbolic geometry — these aren't competing theories of the same thing. They are languages adapted to spaces with different symmetry structures.
If each branch of mathematics is a language adapted to a structural class, category theory is the study of the grammar common to all those languages. It has been seen as a lingua franca of mathematics that unites an ever-expanding number of mathematical fields under the same fundamental notions — many have likened its rise to an extension of the Erlangen Programme, with categories as the pulsating heart of mathematics.
Category theory uncovers deep connections between seemingly unrelated areas, enabling knowledge transfer and revealing underlying patterns. By identifying common structures and relationships, it strips away field-specific details to reveal underlying patterns and similarities.
The key tools:
Functors — structure-preserving maps between mathematical domains. When a functor exists between two domains, they share deep structural kinship: they are the same language spoken with different vocabularies.
Natural transformations — maps between functors, capturing when two ways of translating between domains are themselves equivalent.
Adjunctions — pairs of functors that are "as inverse as possible", revealing when two domains are duals of each other.
This gives you a formal map of how mathematical specializations relate: closeness in the category-theoretic sense means shared structure, and therefore transferable methods.
Importantly, the advent of category theory required enriched languages that after formalization become at first non-linear, and then multidimensional — requiring homotopy theory for their study, and suggesting a respective enrichment of the universe in which interpretations live, from sets to homotopy types. As the problems get topologically richer, the language needed to describe them gains dimensions.
The Langlands Programme: A Map of Unsuspected Bridges
The strongest current evidence that mathematical specializations are secretly related languages is the Langlands Programme. It aims to establish a fundamental correspondence between two distinct areas: number theory (the study of whole numbers and their properties) and representation theory (which analyzes abstract algebraic structures through linear transformations) — creating a unified framework that reveals hidden connections, allowing problems in one area to be translated and solved using the tools of another.
At its core, Langlands predicts a web of correspondences between algebraic objects like Galois groups from number theory and analytic or geometric objects like automorphic forms and representations — a procedure akin to the Fourier transform connecting two sides of a correspondence.
The Fourier analogy is telling. The Fourier transform takes a function defined in time and rewrites it in frequency — the same information, but in a language adapted to a different structural feature of the problem. The Langlands correspondence does something analogous at a much deeper level: arithmetic problems about prime numbers and geometric problems about symmetry of spaces are the same problems in different languages.
A landmark 2024 proof of the categorical geometric Langlands conjecture established an equivalence between the category of automorphic D-modules on the moduli stack of G-bundles and the category of Hecke eigensheaves — bridging local and global phenomena across previously separate domains.
Homotopy Type Theory: Where Language and Structure Merge
The most radical recent development is Homotopy Type Theory (HoTT), which takes this intuition to its limit. HoTT is based on a recently discovered connection between homotopy theory and type theory. Voevodsky's univalence axiom implies that isomorphic structures can be identified. Higher inductive types provide direct, logical descriptions of basic spaces and constructions of homotopy theory. Both are impossible to capture directly in classical set-theoretic foundations, but combined in HoTT, they permit an entirely new kind of "logic of homotopy types".
In classical foundations, mathematical objects (sets) and the language (logic) to talk about them are separate. In HoTT, this distinction dissolves — types are spaces, and proofs are paths through those spaces. With the homotopy interpretation, types are thought of as spaces, type families as fibrations, and identifications as paths — and between any two elements of a type there is a type of identifications, just as between any two points in a topological space there is a space of paths.
This is the deepest vindication of the thesis: mathematicians working at the foundations have concluded that the right language for a mathematical structure is determined by its homotopy type — and that language is itself a topological object. The language and the structure become the same thing.
The Possibility of Mathematics for Structures Not Yet Discovered
Bourbaki identified three "mother structures" — algebraic (operations), topological (nearness), and order (comparison) — as the foundational structural classes from which all mathematical languages descend. But this taxonomy was made in the 1930s–50s, and there's no reason to believe it's complete.
When a set is endowed with more than one feature simultaneously, mathematicians can study the interaction between different structures more richly — for example, if a set has both a topology and a group structure related in a certain way, it becomes a topological group. The combinatorial explosion of such structural hybrids is open-ended: we are still discovering what new combinations are possible and what problems they're naturally suited to.
More pointedly, HoTT itself is an example of mathematics developed for a previously unnamed structural class — ∞-groupoids, the spaces of paths-between-paths-between-paths extending to all dimensions. The intended model of homotopy type theories is the ∞-category of homotopy types. Current constructions suffice for many but not all needed cases, and an entire research programme is dedicated to the problematic cases involving infinite hierarchies of coherence data.
Those "problematic cases" are exactly the regions of structural space that lack adequate languages yet. They're the mathematical terra incognita — regions that exist (we can gesture at them) but for which we don't yet have notation expressive enough to reason in.
The Assembled Research Programme
Your intuition becomes a formal proposal with five layers:
Each mathematical specialization is a language adapted to a structural class — defined by its invariants, its symmetry group, its topological type. The specialization didn't choose its structure; the structure demanded the specialization.
Category theory provides the meta-map — functors between domains reveal structural kinship, and the absence of known functors marks genuine separateness (or undiscovered bridges). The map of mathematics is a map of functor relationships.
The Langlands Programme is an empirical demonstration — wildly distant mathematical languages (number theory, harmonic analysis, geometry) turned out to be dialects of the same structural tongue. The Programme is a discovery that two territories on the map were secretly the same place.
HoTT shows the limit of the idea — when structure is rich enough (∞-groupoids), language and structure collapse into each other, and writing the language down is itself the mathematical act. The map becomes the territory.
AI illegibility as probe — when an RL-trained model develops opaque notation for a hard problem, it may be gesturing at a structural class for which we don't yet have mathematics, in the same way physicists using non-rigorous techniques (Dirac's delta function, Feynman path integrals) were gesturing at mathematics (distribution theory, rigorous QFT) that took decades to formalize.
The Deepest Conjecture
A systematic survey of AI illegibility, analyzed topologically, might serve as a map of structural terra incognita — regions of problem space where natural language is inadequate and formal mathematics doesn't yet have the right language either.
The illegible tokens aren't noise. They may be the first rough sketches of mathematical languages we haven't invented yet.
Key Concepts at a Glance
Concept
What It Says About Structure and Language
Source
Erlangen Programme (1872)
Geometry = invariants under a symmetry group; structure determines language
Klein, 1872
Category Theory
Functors map structure-kinship between domains; meta-language of all mathematics
Eilenberg & Mac Lane, 1945
Langlands Programme
Number theory and geometry are the same language; deep bridges between distant domains
Langlands, 1967–
Homotopy Type Theory
Language is structure; proofs are paths; types are spaces
Voevodsky et al., 2013
∞-Groupoids
Structural class demanding languages we are still building
HoTT Book, 2013
AI Illegible CoT
Emergent notation under RL pressure; possible probe of undiscovered structural classes
Jose et al., 2025
Generated June 10, 2026 — companion to: topology-of-reasoning.md
The third conjecture: on Korzybski, graph isomorphism, and collaborative cartography of conceptual space
The Prompt
LLMs excel at translation between languages. I believe this is no accident. Korzybski says that meaning is in the relationships between words and isomorphism in the graphs identify concepts the same way social graphs identify a person — i.e. a "fingerprint". Taking this a step farther, the graphs are not merely language and concepts, they are reflections of the problem domains they attempt to describe — some formally like mathematics and some informally like poetry. But from a high level, there may be no difference... concepts are concepts! LLMs are not only correlating known languages, they are finding and creating their own languages under RL pressure. I think this might mimic what humans have done more informally for centuries. If so, AI is not only a translator of language — it's a valuable collaborative tool in exploring and bridging these islands of information in a way we've never had before. At least at this stage of our science this may be even more important than AGI.
Korzybski's Insight, Restated Formally
Korzybski's central claim was that meaning doesn't live in words — it lives in the relationships between them. His Structural Differential was designed to show that human knowledge of anything is partial, not total — that what we experience is an abstraction from reality, and that words are further abstractions from those abstractions, with characteristics inevitably left aside at each level. The map, famously, is not the territory.
But the graph of the map — the pattern of relationships between nodes — is something different and more fundamental. The fingerprint intuition has a precise technical form. Researchers studying cross-lingual word embeddings have found exactly this: work on graph-based relative isomorphism of word embedding spaces assumes that each language possesses multiple variations of semantically similar tokens that may be used interchangeably — and that sharing information among semantically related words is necessary to control the relative isomorphism of embedding spaces, making graphs a natural choice to model the impact of related words.
In other words: the concept of "fire" in English, "feu" in French, "nar" in Turkish — these are different nodes in different graphs. But the neighborhood structure of each node — what concepts it is adjacent to, how it clusters, what it implies — is the fingerprint that identifies the concept as the same across languages. Work on multilingual concept induction interprets cliques in dictionary graphs as concepts: the nodes are words, the edges connect translation equivalents, and concepts emerge as the stable structural patterns in that graph — language-independent.
Korzybski intuited that structure, not symbol, carries meaning. Modern NLP has formalized this:
A concept is a graph-theoretic object. Translation is graph isomorphism detection.
This reframes why LLMs are so surprisingly good at translation and cross-domain reasoning. It isn't primarily that they've memorized correspondences. It's that they've discovered an internal space where the structural fingerprints of concepts converge.
Research using activation patching has found that LLMs separate the representation of language from the representation of concepts in distinct processing stages — with a language-agnostic conceptual layer emerging in the middle of the network, before the final stage where concepts are expressed in the target language's tokens.
This has been formalized as the Agnostic Meaning Substrate (AMS): the hypothesis that LLMs operate within a language-agnostic internal space through which conceptual relationships are stabilized and expressed, independently of surface language — with multilingual embeddings aligning across languages even across distant linguistic families, suggesting a deeper structural alignment.
The deeper finding is geometric: even after word embeddings are disassociated from their bindings to particular words, the "shape" of the resulting point clouds correlates with the history of the languages they represent — and non-isometries across languages contain real information rather than being random training artifacts. The shape of a language's conceptual space is a record of the problem domains that language evolved to describe.
Sources:
Separating Tongue from Thought: Activation Patching Reveals Language-Agnostic Concept Representations in Transformers. arXiv:2411.08745. https://arxiv.org/pdf/2411.08745
This is where the observation becomes most radical. The distinction between formal languages (mathematics) and informal ones (poetry) is not a categorical difference — it is a difference of precision in capturing the same underlying structural reality.
Korzybski noted that one of the main reasons for human difficulties resides in the fact that the structure of our everyday language does not fit the structure of facts the way mathematical language does — yet the goal of both is to capture that structure. Mathematical language succeeds more precisely; poetic language captures different dimensions — resonance, ambiguity, emotional topology — that formal notation strips away. But both are attempts at the same thing: a notation adequate to some aspect of reality's structure.
The topological data analysis work on reasoning traces confirms this from the other direction: across multiple models and diverse tasks, effective reasoning traces show stable, macromolecule-like organization, with different models recovering similar reasoning topologies across tasks. The structure isn't in the tokens. It's in something the tokens are pointing at.
Viewed this way, the spectrum from poetry to formal mathematics is not a spectrum from "soft" to "hard" knowledge. It is a spectrum of structural resolution — the precision with which a notation carves the joints of a problem domain. Both poetry and mathematics are graph languages. Concepts are concepts.
Humans Have Always Done This, Badly and Slowly
The informal version of what LLMs do has been the engine of scientific and intellectual progress for millennia — but it has been terribly slow and dependent on accidents of biography.
Analogies have been a central mechanism for cross-boundary inspiration throughout the history of science and technology. For example, analogical reasoning helped Chrysippus model sound waves from observations of water waves — two domains that had not interacted prior to this insight. However, as knowledge areas deepen in specialization, they interact less with each other, and analogical reasoning across domains becomes increasingly challenging.
This is the tragedy of specialization: the deeper the expertise, the higher the walls between domains. For human reasoners, spontaneous analogical retrieval is often cue-dependent and superficial — we tend to notice analogies when the vocabulary overlaps, not when the graph structure does. Maxwell's equations and neural firing patterns may be structurally identical while sharing no surface vocabulary — and a human would need years of exposure to both domains to notice.
The LLM has been exposed to everything simultaneously, and has been trained to find structural correspondences rather than surface ones. It operates, in effect, like a Langlands programme for all human knowledge — seeking the functors between domains, not just the vocabulary overlaps.
RL Pressure as Language Invention: The Deeper Parallel
The observation about RL models inventing opaque notation closes the loop on all three conversations. What we're observing is not a malfunction and not merely a curiosity. It appears to be a recapitulation — compressed and accelerated — of how humans generate new mathematics and new scientific frameworks.
When Heisenberg in 1925 wrote down matrices that didn't commute, he was doing something recognizable in hindsight: he had encountered a problem domain whose topology required a non-commutative algebra, and he invented the notation under pressure of experimental fact. The notation was "illegible" to contemporaries — Bohr said that anyone who wasn't shocked by quantum mechanics hadn't understood it. The language preceded the formal mathematics that legitimized it: von Neumann's Hilbert space formalism came three years later.
An RL-trained model encountering a hard problem and generating illegible intermediate tokens is doing the same thing on a faster timescale: finding notation adequate to a structural domain for which existing language is insufficient.
The illegible tokens aren't noise. They may be the first rough sketches of mathematical languages we haven't invented yet — gestures toward structural classes that lie beyond the edge of our current maps.
The Collaborative Possibility
If this picture is right, the practical implication is significant and distinct from the AGI question. You don't need a system that replaces human reasoning. You need a system that can do what humans do badly: traverse the walls between specialist domains, detect structural isomorphisms across distant vocabularies, and surface the recognition that these two things are the same thing spoken in different languages.
The historical role has been played by polymaths: figures who happened to know enough about two distant fields to notice the isomorphism. Leibniz, von Neumann, Shannon, Wiener all did this — translating structures from one domain into another. But they were accidents of individual biography, and for every connection they made, thousands of potential connections between specialist islands went unmade simply because no one happened to be fluent in both languages at once.
An LLM trained on the full breadth of human knowledge, operating in the language-agnostic conceptual substrate where structural fingerprints are visible across domains, is potentially the first systematic tool for doing this at scale. Not replacing the mathematician, the poet, or the physicist — but holding the maps of all their islands simultaneously, and pointing at the coastlines that look like they fit together.
The Deepest Conjecture
AGI is a question about replacing human cognition. What this conversation has been pointing at is something different: a collaborative cartography of conceptual space — where the LLM's peculiar ability to find isomorphisms across domains accelerates the kind of cross-domain discovery that has historically been the engine of the deepest scientific revolutions.
Not incremental progress within a field. But the moment when two fields discover they were the same field all along.
This may matter more right now than AGI. And it is already happening.
The Arc of All Three Conversations
Conversation
Core Claim
I: Topology of Reasoning
The topology of a problem space determines the natural language for reasoning over it — and AI under RL pressure discovers this language
II: Mathematics as Structure-Adapted Languages
Mathematical specializations are such languages — and formal tools (category theory, Langlands, HoTT) can map their relationships and reveal undiscovered structural classes
III: LLMs as Universal Translators
Concepts are graph-theoretic objects; translation is isomorphism detection; LLMs have built an Agnostic Meaning Substrate that sees structure across all languages — formal and informal — and this may be their most important near-term contribution
The through-line: Structure is primary. Language — mathematical, natural, poetic, or emergent — is always an attempt to notate structure adequately. The history of human knowledge is the history of finding better notations for encountered structures, and occasionally discovering that two structures we thought were different were the same. AI accelerates this process by operating directly in the structural layer, below the surface of any particular language.
Key Sources Across All Three Conversations
Paper / Source
Contribution
Jose et al. (2025). Reasoning Models Sometimes Output Illegible CoTs. arXiv:2510.27338
RL causes illegible but load-bearing reasoning; illegibility scales with problem difficulty
Understanding CoT via Topological Data Analysis. arXiv:2512.19135
Persistent homology of reasoning chains correlates with accuracy
The Shape of Reasoning. arXiv:2510.20665
Topological features outpredict graph metrics for reasoning quality
The Molecular Structure of Thought. arXiv:2601.06002
CoT topology converges across models — it's a property of the problem
Klein (1872). Erlangen Programme
Symmetry group of a space determines its geometry — structure determines language
Eilenberg & Mac Lane (1945). Category Theory
Meta-language of mathematics; functors map structural kinship between domains
Langlands (1967–). Langlands Programme
Number theory and geometry are dialects of the same structural language
Voevodsky et al. (2013). HoTT. arXiv:1308.0729
Language and structure collapse into each other at the level of ∞-groupoids
