Meaning Has a Shape.
A conceptual space is a way of representing meaning as geometry. Entities become points. Properties become convex regions. Similarity becomes distance. Nothing is a symbol you have to look up, and nothing is a weight you cannot read. Everything below is live — drag it, break it, watch it reason.
What is a conceptual space?
In 2000 the cognitive scientist Peter Gärdenfors published Conceptual Spaces: The Geometry of Thought, arguing that AI had been forced into a false choice. On one side, symbolic representations — logic, ontologies, knowledge graphs — which are precise and explainable but brittle, and have no notion of one thing being a bit like another. On the other, connectionist representations — neural networks and their embeddings — which handle similarity beautifully but whose internals nobody can read.
Gärdenfors proposed a third, intermediate level. Take a domain and identify its quality dimensions: the primitive features along which things in that domain actually vary. For colour, that is hue, saturation and brightness. For people, it might be height, age, education. For a wine, body, acidity, tannin. A conceptual space is then just the Cartesian product of those dimensions — and once you have it, three things follow for free.
That third commitment — convexity — is the one that does the real work, and it is easy to miss why. A region is convex when, for any two points inside it, every point on the straight line between them is also inside it. Gärdenfors' claim is that the properties human languages actually bother to name are exactly the convex ones. There is a word for red, and every colour between two reds is red. There is no single word for “red or green but not yellow”, because that region is full of holes.
Convexity is what makes the geometry reason rather than merely store. If you know two things have a property, you get every point between them for free. That is induction, done by arithmetic.
A space of fruit, with two quality dimensions. Everything on this page is the same trick at larger scale: the spaces I have actually built have between 2 and 288 dimensions, learned from a billion words of Wikipedia rather than drawn by hand. But the commitments are identical, and they are all visible here.
Why convexity is not a technicality
Colour is the canonical example, because it is the one domain where the quality dimensions are not up for debate — hue and lightness are facts about human perception, not modelling choices. Drag the two handles below and watch what the straight line between them passes through.
Horizontal axis: hue. Vertical axis: lightness. Both are quality dimensions in Gärdenfors' original sense.
Why this matters to a machine. Convexity is not decoration — it is what makes generalisation safe. If a property is a convex region, two examples pin down everything between them and a model can extrapolate without being told. If it is not, there is nothing to interpolate along, and any region fitted to it will quietly swallow cases that do not belong. The three presets are the whole argument in three clicks: red passes the test, red → yellow shows what passing it buys you, and “not blue” shows what failing it looks like.
So what does one actually look like?
A hand-drawn fruit space is a teaching device. The hard problem — the one that kept conceptual spaces out of practical AI for fifteen years — is that nobody tells you what the quality dimensions are. Gärdenfors assumed you had them. For a domain of a million entities you do not.
The 2016 paper Steven and I wrote for ECAI attacked exactly this. The recipe: take every entity mentioned in at least ten Wikipedia articles — 1,292,702 of them — and learn a single 300-dimensional vector space in which each entity is a point and each context word is a direction. Then add the constraint that makes it a conceptual space rather than just an embedding: every entity of a given semantic type must lie in a common low-dimensional subspace, enforced by penalising the nuclear norm of the matrix spanning that subspace. Films get their own subspace. Countries get theirs. Chemical elements get theirs.
Two consequences of that constraint are worth seeing directly.
These are real outputs, not an illustration. None of the attributes shown were used to build the space — the model never saw a population figure or a birth date. It saw only Wikipedia text and Wikidata types. Yet a single direction, fitted from a handful of training examples, recovers the ordering. Across 73 attribute-and-type combinations the average Spearman correlation was 0.319, against 0.213 for the strongest prior model and 0.155 for skip-gram.
And note where it goes wrong, because that matters more than where it goes right. Bermuda, Monaco, San Marino and Barbados are all less populous than Malta but rank above it. Ptolemy lived some five hundred years after Plato and is placed before him. The geometry approximates; it does not guarantee.
Nobody set these numbers. The nuclear-norm penalty decides how many dimensions each semantic type needs, and the answers are quietly sensible. Candy is a thin concept and lands in 2 dimensions. Human is enormous and needs 288. And note that house, with 2,762 instances, claims more dimensions than railway station, with 4,120 — because houses vary along more axes than railway stations do. That is the model reporting the conceptual complexity of a category, not its popularity.
Reasoning that fills its own gaps
Every ontology and every knowledge graph is incomplete. SUMO has plenty to say about chess, darts and poker, and nothing at all about checkers. CYC says an ice cream shop cannot be a restaurant; Wikipedia says it is one. A logic engine handles neither situation: it cannot invent the missing fact, and it falls over on the contradiction.
In our AAAI 2017 paper — with Zied Bouraoui — we made the space do the inductive half of the job. The move is simple. Assume that, along each quality dimension, the instances of a concept are Gaussian-distributed. A subconcept is then a narrower Gaussian sitting inside its parent's. To ask whether some entity belongs to subconcept B given that it belongs to parent A, compare the two densities at its coordinate:
Positive means the entity looks more like a B than like a generic A. With so few known instances, the Gaussians' own parameters are uncertain, so we do not point-estimate them — we integrate them out under a normal-inverse-χ² prior. That is a small detail with a large consequence, and it is the seed of everything I am doing now.
Three sibling subconcepts, known to be mutually disjoint by the ontology. Their known instances sit at coordinates 1–3, 4–6 and 7–9 along a single quality dimension. Move the candidate entity and watch which concept claims it.
Watch the region where two scores are both positive. Pure induction would put the candidate in two disjoint concepts at once — a contradiction. The ontology knows they are disjoint, so the deductive layer repairs the inductive one by keeping only the highest-confidence conclusion. Induction proposes; logic vetoes. Neither could draw this boundary alone.
What conceptual spaces can and cannot do
I have spent ten years on this. That is long enough to be fond of it and long enough to know precisely where it fails. Both columns are load-bearing — click any item.
Interpolate meanings that were never stored
A point halfway between red and yellow is orange — not by lookup, but because that is where it sits. Give the space two known instances of a property and it hands you every case between them. A symbolic knowledge base has to be told each one separately.
Rank by properties nobody ever recorded
Most Wikipedia articles never state that a mountain is tall. But tall gets mentioned in the articles about the mountains where tallness is the notable thing — so the direction fitted to that word points at exactly the tall ones. Figure 3 above is this effect: population, founding year and birth date all recovered from text that never listed them.
Generalise from three examples
Give the model a handful of films directed by Spielberg, without telling it that “directed by Spielberg” is the property in question, and ask for more. Because the property is a region, a few points pin it down. In the ECAI evaluation this induction task reached 0.883 MRR and 0.609 P@5, against 0.756 and 0.490 for the strongest baseline.
Explain themselves without a post-hoc story
Every conclusion reduces to a coordinate, a direction and a distance. There is no saliency map, no attention heatmap, no attempt to reverse-engineer a decision after the fact. The representation is the explanation, which is the entire reason this line of work is usable in defence and intelligence settings.
Repair contradictions instead of choking on them
When induction proposes two conclusions that the ontology declares disjoint, confidence scores break the tie (Figure 5). A pure logic engine has no way to prefer one of two contradictory derivations; a pure neural model has no notion that they conflict.
Answer queries a graph has no edge for
“Famous tourist sites in France” is not a relation in any knowledge graph. In our SIGIR 2017 work, MEmbER learned entity embeddings under max-margin constraints so that query terms became separating hyperplanes — letting a text query be interpreted directly as a region of the space rather than matched against documents.
Represent concepts that are not convex
This is the theory's own boundary, not a bug. “Not blue” has no region (Figure 2). Neither does weekend, which is two disjoint stretches of a week. Wittgenstein's game — board games, ball games, solitaire, war games — is the classic case of a category with family resemblance but no common core. Force a convex region around it and you sweep in everything in between.
Handle relational structure on their own
A conceptual space tells you how tall a building is and which buildings resemble it. It does not tell you who designed it. Relations between entities are not distances, and every model in this line has needed a separate knowledge-graph component bolted alongside the geometry to carry them.
Do genuine logical composition
Negation, quantification, nested scope. “No student passed every module” has no natural rendering as a region. Regions intersect and they contain; that is the whole of the logic you get for free. Anything richer has to come from a description logic sitting on top — which is exactly how we structured the AAAI work.
Tell you what the dimensions are
Gärdenfors' theory assumes the quality dimensions are given. For colour they genuinely are. For “insurgent activity” or “maritime vessel” they are not, and getting them is most of the work. The earlier approach — multidimensional scaling over a distance matrix — is quadratic in the number of entities and simply cannot reach a million of them. Ours can, but the dimensions it finds are learned approximations, not perceptual primitives.
Guarantee that the geometry is right
Spearman 0.319 is a strong result and also a modest number. Monaco outranks Malta on population; Ptolemy is placed before Plato. Directions recover orderings, approximately, from statistical traces in text. Treat a single answer as authoritative and you will eventually be badly wrong.
Say how sure they are
The deepest limitation, and the one I am now attacking. A point is a point: it carries no width. A vague report and a confirmed identification land in the space as identical objects. Region boundaries are drawn crisply where the underlying concept is genuinely fuzzy. For anything with consequences, “here is the answer” is not enough — you need “here is the answer, and here is how much to trust it.”
Conceptual spaces and large language models
The question I get asked most is some version of “hasn't this been made obsolete?” It is a fair question and it has a more interesting answer than yes or no, because the two approaches do not fail in the same places. An LLM will fluently tell you something false and sound equally confident either way. A conceptual space cannot tell you anything it has no dimension for — but when it does answer, you can read off exactly why.
What follows is not a rhetorical contrast. Every row below points at a published result, and where the evidence cuts against conceptual spaces I have said so.
| Capability — click a row for the evidence | Conceptual spaces | LLMs | In one line |
|---|
Two of those rows are worth seeing as numbers, because they are the ones people find most surprising. In 2023 Steven's group at Cardiff asked directly whether an LLM can learn conceptual space dimensions — ranking foods by sweetness, objects by mass, and so on. The result splits cleanly in two, and neither half is the one people expect.
GPT-4 is absent: it could not rank the items.
Scale wins on the left-hand side of the world and loses on the right. On objective physical magnitudes GPT-4 is close to perfect — 99.1% on both height and size. On a genuinely perceptual dimension it is not merely worse, it is absent: ChatGPT and GPT-4 could not produce a ranking of the 590 items at all, so the authors excluded them from that experiment. The best perceptual result came from a fine-tuned DeBERTa at ρ = 69.1, ahead of the 175-billion-parameter GPT-3 davinci at 55.0 and far ahead of the smaller curie at 36.0 — from a model two to three orders of magnitude smaller. Source: Chatterjee, Gajbhiye & Schockaert, EMNLP 2023.
The part I did not expect
Here is the finding that reframed this whole comparison for me. If Gärdenfors is right that natural concepts occupy convex regions, then a neural network that has learned a domain well ought to show convexity in its own latent space — without anyone imposing it. In 2025 a group at the Technical University of Denmark tested exactly that across image, text, audio, activity and medical data, and found approximate convexity nearly everywhere.
The stronger result is what convexity predicts. How convex a class region is after pretraining correlates with how accurate the model becomes after fine-tuning — and with few-shot recall. A criterion proposed in 2000 from cognitive science, on entirely non-computational grounds, turns out to forecast how well a deep network will learn.
The test is the one from Figure 2, applied to a learned representation: take two points known to be in the same class, and check whether the point between them is still in that class. Drag the slider from an early layer through to a fine-tuned one. The score is counted live from the geometry you are looking at.
(counted live, this figure)
Read the two published figures on the right as the point of this section. They come from Tětková et al., Nature Communications, 2025, who measured convexity in RoBERTa, BERT, ViT, DINOv2, wav2vec2 and others. The animation left of them is my illustration of their measurement procedure, not a plot of their results.
So the honest summary is not that one approach beat the other. It is that the geometry turned out to be there all along, unlabelled, inside the models that displaced it — and that the property Gärdenfors singled out as the mark of a natural concept is the same property that predicts whether a network will generalise.
Which leaves the single row in that table where both columns say no: knowing how confident to be. An LLM's stated confidence degrades under the very post-training that makes it useful. A classical conceptual space has no confidence to state at all — a hedged report and a confirmed one land on the same coordinate. On explaining itself the geometry already wins; on calibration neither side has an answer, and that gap is the whole of what I am building now.
Ten years on one idea
This did not start as a fashionable topic and it never became one. I began working on conceptual spaces with Steven Schockaert at Cardiff in 2016, at a point when the field's entire attention was on making embeddings bigger rather than making them mean anything. The line has run continuously since.
Entity Embeddings with Conceptual Subspaces as a Basis for Plausible Reasoning
ECAI · The HagueThe first scalable construction. 1.29 million Wikipedia entities, semantic types as low-dimensional subspaces enforced by nuclear-norm regularisation, salient properties recoverable as directions. arXiv →
Plausible Reasoning based on Qualitative Entity Embeddings
IJCAI · New York · invitedThe qualitative reading of the same machinery, presented to the reasoning community rather than the representation-learning one. Proceedings →
Inductive Reasoning about Ontologies Using Conceptual Spaces
AAAI · San FranciscoWith Zied Bouraoui. Gaussian concepts, a normal-inverse-χ² prior, and inductive completion of both the ABox and the TBox — with the description logic repairing the inductive conclusions where they conflict. This is the paper Figure 5 comes from, and the first Bayesian move in the line.
MEmbER: Max-Margin Based Embeddings for Entity Retrieval
SIGIR · TokyoInterpretability made the training objective rather than a post-hoc analysis: query terms become separating hyperplanes learned jointly with the entity positions, so a text query can be read straight off the geometry.
Regions, relations and the limits of directions
CoNLL · ACL · COLINGModelling context words as regions rather than vectors; unsupervised distributional relation vectors; and a hard second look at relation induction in word embeddings, which found the standard evaluation considerably more forgiving than it should be.
PICASO — conceptual spaces, made properly Bayesian
Alan Turing Institute · MoDEvery point becomes a distribution. The last limitation in the ledger above, taken seriously.
PICASO: giving conceptual spaces a Bayesian model
Read the right-hand column above and one entry sits underneath all the others: a point cannot tell you how sure it is. That is fine for fruit. It is not fine for an intelligence analyst holding a report that says “a vehicle, possibly military, near the border”, because a classical conceptual space will place that at a single crisp coordinate and lose the vagueness that was the most important thing in the sentence.
PICASO — Probabilistic Conceptual Spaces Sensemaking under Uncertainty — replaces every point with a probability cloud. A vague description produces a wide Gaussian; a confirmed identification produces a tight one. Regions stop being crisp and start being distributions, which is what Gärdenfors' own treatment of vagueness always implied but the computational models never delivered. The seed was already in the AAAI 2017 paper's normal-inverse-χ² prior; PICASO makes it the whole architecture.
It is funded by the Alan Turing Institute's Defence & Security Programme for the UK Ministry of Defence, I am the Principal Investigator, and the core engine is built with my PhD student Ubaid Azam. It matches deep graph neural networks on link prediction while running at the cost of an embedding lookup — and, unlike any of them, reports calibrated uncertainty natively.
Read the primary sources
The foundational text is Peter Gärdenfors, Conceptual Spaces: The Geometry of Thought (MIT Press, 2000). For the full list of my work, see all publications.