Resources
Essential references, tools, and materials for exploring linguistic geometry
Glossary (260 terms)
Abstract Algebra
Associativity
Property of operation order independence, fundamental to linguistic composition.
Binary Operation
Function combining two elements, basic linguistic combination.
Commutator
Measure of non-commutativity, quantifying operation order effects.
Direct Product
Cartesian product with componentwise operation, combining linguistic structures.
Endomorphism
Morphism from object to itself, self-mapping in linguistic structures.
Free Group
Group with no relations beyond group axioms, unconstrained generation.
Generator
Element from which others can be derived, primitive in linguistic systems.
Homomorphism
Structure-preserving map, maintaining linguistic relationships.
Ideal
Subset closed under ring operations, modeling linguistic constraints.
Jacobson Radical
Intersection of maximal ideals, core problematic elements.
Category Theory
Arrow
Morphism between objects in a category, representing linguistic transformations.
Bifunctor
Functor of two variables, modeling binary linguistic operations.
Cartesian Closed Category
Category with products and exponentials, modeling function spaces in semantics.
Coalgebra
Dual of algebra, modeling destructors and observations in linguistic structures.
Colimit
Universal construction generalizing unions, used in semantic composition.
Comma Category
Category of morphisms with fixed domain or codomain, organizing linguistic relations.
Comonad
Structure dual to monad, modeling context-dependent computations.
Cone
Natural transformation from constant functor, used in limit constructions.
Coproduct
Categorical sum, modeling disjoint union of linguistic types.
Diagram
Functor from index category, visualizing complex categorical relationships.
Cognitive Science
Affordance
Action possibility in environment, constraint on linguistic interpretation.
Attention
Selective processing, defining focus regions in linguistic space.
Basic Level Category
Cognitively privileged taxonomic level, optimal granularity in category space.
Categorization
Grouping by similarity, partitioning cognitive-linguistic space.
Cognitive Load
Mental effort in processing, capacity constraints in working memory space.
Conceptual Blending
Integration of mental spaces, creating emergent linguistic meanings.
Conceptual Blending
Integration of multiple mental spaces, fusion operations in cognitive geometry.
Conceptual Metaphor
Understanding abstract via concrete domains, mappings between conceptual spaces.
Construal
Way of viewing and portraying situation, perspective transformation in conceptual space.
Embodied Cognition
Mind shaped by body and environment, grounding abstract space in physical space.
Embodiment
Grounding of cognition in physical experience, constraining semantic geometry.
Figure-Ground
Perceptual organization principle, structuring linguistic attention.
Gestalt
Whole greater than sum of parts, emergent properties in linguistic structures.
Image Schema
Recurring bodily pattern, fundamental structure in conceptual space.
Image Schema
Recurring bodily experience patterns, basic structures in cognitive space.
Mental Space
Cognitive construct for reasoning, local region in conceptual topology.
Mental Space
Cognitive construct for reasoning, local regions in thought space.
Prototype
Central member of category, attractor in conceptual space.
Prototype Theory
Categories organized around typical examples, radial structure in category space.
Salience
Cognitive prominence of elements, peaks in attention landscape.
Computational
Alignment
Correspondence between elements in parallel texts or structures, creating bijective mappings.
Backpropagation
Algorithm for training neural networks, optimizing paths through weight space.
Beam Search
Heuristic search exploring most promising paths, navigating probability landscapes.
Character Embedding
Vector representation of characters, mapping discrete symbols to continuous space.
Constituency Parsing
Analysis producing phrase structure trees, constructing hierarchical geometric structures.
Cosine Similarity
Measure of similarity between vectors, defining metric in embedding space.
Dependency Parsing
Analysis producing dependency trees, creating directed graph structures.
Edit Distance
Minimum operations to transform one string to another, defining metric on string space.
Embedding Space
Continuous vector space representing discrete linguistic units.
Feature Hashing
Dimensionality reduction using hash functions, projecting to lower-dimensional spaces.
Core Concepts
Grammatical Topology
Syntactic transformations visualized as homeomorphisms or fiber bundle shifts in meaning-space.
Lexostruct
A syntax tree that defines topological manifolds, where phrase structure maps to spatial deformations.
Linguistic Attractor
Words existing in semantic vector fields, pulling toward interpretive centers based on context.
Morphic Landscape
Language embedded into higher-dimensional spaces where meaning clusters form terrain-like features.
Morphological Functor
Category-theoretic mapping between morphological systems preserving derivational structure.
Phonological Manifold
Continuous surface representing the acoustic realization space of phonemes.
Semantic Field Tensor
Mathematical object encoding the multidimensional relationships between word meanings.
Semantographic Engine
System rendering meaning using graph-theoretic diagrams, vector spaces, or lambda calculus expressions.
Symbol Channel
Words and structures forming morphisms in a functional pipeline; commuting diagrams in category theory.
Syntactic Fiber Bundle
Geometric structure where syntactic categories form the base space and features form the fibers.
Dynamical Systems
Attractor Basin
Region flowing to attractor, domain of semantic convergence.
Bifurcation Point
Parameter value causing qualitative change, critical point in language change.
Chaos
Sensitive dependence on initial conditions, complexity in linguistic evolution.
Dynamical System
System evolving over time, model for linguistic change and processing.
Equilibrium
State unchanged by dynamics, stable linguistic configuration.
Fixed Point
Point mapped to itself, invariant linguistic structure.
Gradient Flow
Motion along steepest descent, optimization in linguistic space.
Hysteresis
Dependence on history, memory effects in linguistic systems.
Invariant Manifold
Subspace preserved by dynamics, stable linguistic substructure.
Julia Set
Fractal boundary of stability, complex boundary in linguistic dynamics.
Geometry
Affine Space
Space with no distinguished origin, modeling relative linguistic positions.
Curvature
Measure of deviation from flatness, quantifying complexity in semantic space.
Diffeomorphism
Smooth bijection with smooth inverse, modeling reversible linguistic transformations.
Euclidean Space
Flat space with distance metric, baseline for linguistic vector spaces.
Geodesic
Shortest path between points, optimal transformation in linguistic space.
Hyperbolic Space
Space of constant negative curvature, modeling hierarchical linguistic structures.
Isometry
Distance-preserving transformation, maintaining linguistic relationships.
Jacobian
Matrix of partial derivatives, measuring local transformation effects.
Klein Bottle
Non-orientable surface, modeling self-referential linguistic structures.
Lie Group
Smooth manifold with group structure, modeling continuous symmetries.
Historical Linguistics
Cognate
Words sharing common etymology, connected points in diachronic space.
Grammaticalization
Evolution of grammatical from lexical forms, directed paths in functional space.
Grimms Law
Germanic consonant shift pattern, systematic transformation in phonological space.
Isogloss
Geographic boundary of linguistic feature, contour lines in dialect space.
Language Family
Genetically related languages, tree structure in diachronic space.
Proto-Language
Reconstructed ancestral language, root node in linguistic phylogeny.
Semantic Shift
Meaning change over time, trajectories through semantic space.
Sound Change
Systematic phonetic evolution, continuous deformation of phonological space.
Substrate Influence
Effect of replaced language on replacing one, layered structure in contact space.
Wave Theory
Model of language change spread, wave propagation in geographic space.
Information Theory
Channel Capacity
Maximum information transmission rate, bounding communicative efficiency.
Channel Capacity
Maximum reliable transmission rate, bandwidth of communication channel.
Cross-Entropy
Divergence between distributions, distance in probability space.
Entropy
Measure of information content, quantifying linguistic uncertainty.
Hamming Distance
Number of differing positions, metric for symbolic sequences.
Hamming Distance
Number of differing positions, metric in discrete code space.
Information Gain
Reduction in entropy, measuring linguistic disambiguation.
Joint Entropy
Entropy of combined variables, measuring linguistic co-occurrence patterns.
Kolmogorov Complexity
Shortest description length, fundamental measure of linguistic complexity.
Kolmogorov Complexity
Shortest description length, compression limit in description space.
Lempel-Ziv Complexity
Measure based on compression, quantifying pattern repetition in text.
Mutual Information
Shared information between variables, measuring linguistic dependencies.
Mutual Information
Shared information between variables, intersection in information space.
Noise
Random interference in communication, perturbation in linguistic channels.
Noisy Channel Model
Communication with errors, probabilistic paths in transmission space.
Rate-Distortion Theory
Tradeoff between compression and fidelity, optimization in coding space.
Redundancy
Excess information for error correction, robustness in linguistic systems.
Redundancy
Excess information for error correction, robustness in message space.
Shannon Entropy
Information content measure, uncertainty in message space.
Source Coding
Efficient representation of information, minimal encoding in code space.
Linear Algebra
Basis
Linearly independent spanning set, fundamental directions in linguistic space.
Characteristic Polynomial
Polynomial whose roots are eigenvalues, encoding matrix properties.
Determinant
Scalar measuring matrix invertibility, volume scaling in transformations.
Eigenspace
Space of eigenvectors for given eigenvalue, invariant subspace in linguistics.
Frobenius Norm
Matrix norm from element squares, measuring transformation magnitude.
Gram Matrix
Matrix of inner products, encoding geometric relationships.
Hermitian Matrix
Complex matrix equal to conjugate transpose, modeling symmetric relations.
Inner Product
Generalization of dot product, measuring linguistic similarity.
Jordan Form
Canonical matrix representation, revealing transformation structure.
Kernel
Null space of linear map, elements mapped to zero.
Linguistics
Ablaut
Systematic vowel alternation in related words or morphemes, fundamental to understanding morphophonological geometry.
Adjacency Pair
Two-part conversational structure where the first part requires a specific type of second part, forming a geometric relation in discourse space.
Affix
Morpheme attached to a stem to modify meaning, operating as a function in morphological space.
Allomorph
Variant form of a morpheme, representing points in the same equivalence class in morphological topology.
Allophone
Variant pronunciation of a phoneme, forming a continuous manifold in acoustic space.
Anaphora
Reference back to an earlier expression, creating directed edges in discourse graphs.
Antecedent
Expression to which an anaphor refers, serving as the source node in reference networks.
Argument Structure
Pattern of semantic roles required by a predicate, defining a vector space of thematic relations.
Aspect
Grammatical category expressing temporal structure of events, mapping to topological properties of time intervals.
Autosegmental Phonology
Theory treating phonological features as autonomous tiers, creating multi-dimensional geometric representations.
Logic
Axiom
Foundational assumption, base element in formal linguistic systems.
Biconditional
Logical equivalence, creating bidirectional implications in semantic networks.
Completeness
Property that all valid formulas are provable, ensuring linguistic coverage.
Decidability
Existence of algorithm determining truth, computational boundary of linguistic analysis.
Entailment Relation
Logical consequence, creating directed structure in proposition space.
First-Order Logic
Logic with quantification over individuals, modeling basic semantic relations.
Gödel Numbering
Encoding of formulas as numbers, bridging symbolic and numeric representations.
Higher-Order Logic
Logic allowing quantification over predicates, capturing complex linguistic patterns.
Intension
Function from possible worlds to extensions, navigating modal semantic space.
Judgment
Assignment of semantic value, point in truth-value space.
Mathematics
Abelian Group
Group where the operation is commutative, fundamental to understanding symmetries in linguistic structures.
Adjoint Functor
Pair of functors with a natural isomorphism between their hom-sets, modeling semantic transformations.
Affine Transformation
Geometric transformation preserving collinearity and ratios, used in modeling syntactic movement.
Algebraic Topology
Study of topological spaces using algebraic methods, applied to semantic and syntactic structures.
Automorphism
Structure-preserving mapping from an object to itself, modeling linguistic recursion.
Banach Space
Complete normed vector space, used in modeling infinite-dimensional semantic spaces.
Betti Number
Topological invariant counting holes of different dimensions, applied to semantic networks.
Bifurcation
Qualitative change in system behavior, modeling grammaticalization and language change.
Borel Set
Member of the smallest σ-algebra containing all open sets, used in probabilistic semantics.
Bundle
Topological space that locally looks like a product space, modeling feature geometries.
Morphology
Agglutination
Morphological process adding discrete affixes, creating linear chains in morpheme space.
Allomorphy
Variation in morpheme realization, forming equivalence classes in morphological space.
Blocking
Prevention of regular forms by irregular ones, creating lexical constraints in derivational space.
Circumfix
Discontinuous affix surrounding a stem, creating non-local dependencies in word structure.
Cliticization
Attachment of weakly stressed elements, occupying intermediate position in morphosyntactic space.
Compounding
Combination of multiple roots, creating product structures in lexical space.
Conversion
Category change without affixation, representing zero-derivation in morphological space.
Derivational Paradigm
Set of related derived forms, forming a network in morphological space.
Exponence
Realization of morphosyntactic features, mapping abstract features to concrete forms.
Fusion
Morphological process combining multiple features in single form, creating feature bundling.
Network Theory
Adjacency Matrix
Matrix representation of graph edges, encoding linguistic network structure.
Betweenness Centrality
Measure of node importance in paths, identifying key linguistic connectors.
Clustering Coefficient
Measure of node neighbor connectivity, quantifying local linguistic coherence.
Degree Distribution
Probability distribution of node degrees, characterizing network topology.
Eigenvector Centrality
Measure of node influence, identifying important linguistic elements.
Flow Network
Graph with capacity constraints, modeling information flow in language.
Graph Laplacian
Matrix encoding graph structure, fundamental to spectral analysis.
Hub
Highly connected node, central element in linguistic networks.
Isomorphism
Structure-preserving bijection between graphs, identifying equivalent networks.
Jaccard Index
Similarity measure for sets, quantifying lexical overlap.
Optimization
Admissible Heuristic
Heuristic never overestimating cost, ensuring optimal linguistic search.
Barrier Function
Function preventing constraint violation, enforcing linguistic boundaries.
Convex Optimization
Optimization over convex sets, guaranteeing global linguistic optima.
Dual Problem
Related optimization providing bounds, alternative linguistic perspective.
Epigraph
Set of points above function graph, geometric view of optimization landscape.
Feasible Region
Set satisfying all constraints, valid linguistic solution space.
Gradient Descent
Optimization following steepest descent, iterative linguistic improvement.
Hessian
Matrix of second derivatives, curvature of optimization landscape.
Interior Point Method
Optimization through feasible region interior, avoiding linguistic boundaries.
Karush-Kuhn-Tucker
Optimality conditions for constrained problems, linguistic equilibrium conditions.
Phonology
Acoustic Phonetics
Study of physical properties of speech sounds, mapping to continuous acoustic manifolds.
Articulatory Phonetics
Study of speech sound production, defining trajectories in articulatory space.
Coarticulation
Overlapping articulation of adjacent sounds, creating smooth paths in gestural space.
Distinctive Feature
Minimal unit distinguishing phonemes, forming basis vectors in phonological space.
Feature Geometry
Hierarchical organization of phonological features, creating tree-structured feature spaces.
Formant
Resonant frequency in speech, defining coordinates in acoustic space.
Gestural Phonology
Theory treating speech as coordinated gestures, modeling articulation as dynamical systems.
Harmonic
Integer multiple of fundamental frequency, creating harmonic series in frequency space.
Intonation
Pitch patterns in speech, forming contours in prosodic space.
Juncture
Phonetic boundary between linguistic units, marking discontinuities in speech flow.
Pragmatics
Conversational Implicature
Implied meaning beyond literal content, additional dimensions in meaning space.
Cooperative Principle
Gricean maxims governing conversation, constraints on communicative geometry.
Deixis
Context-dependent reference, anchoring language to coordinate systems.
Face-Threatening Act
Utterance challenging social face, perturbations in politeness space.
Illocutionary Force
Intended effect of speech act, defining vectors in pragmatic force fields.
Indexicality
Context-dependent meaning, creating variable mappings in semantic space.
Perlocutionary Effect
Actual effect of utterance on hearer, resultant vector in social space.
Presupposition
Background assumption required for utterance felicity, defining pragmatic base space.
Relevance Theory
Cognitive approach to pragmatics, optimizing information geometry.
Speech Act
Utterance performing an action, mapping words to force vectors in communicative space.
Probability Theory
Bayesian Network
Probabilistic graphical model, representing linguistic dependencies.
Conditional Probability
Probability given evidence, contextual likelihood in language.
Distribution
Assignment of probabilities, statistical landscape of linguistic events.
Expectation
Average value, central tendency in linguistic distributions.
Frequentist
Interpretation based on long-run frequency, empirical approach to language statistics.
Gaussian Process
Stochastic process with normal distributions, smooth random functions in linguistics.
Hidden Markov Model
Stochastic model with hidden states, capturing latent linguistic structure.
Independence
Lack of probabilistic dependence, orthogonality in linguistic features.
Joint Distribution
Probability over multiple variables, complete statistical description.
Kernel Density
Non-parametric density estimation, smooth approximation of linguistic distributions.
Psycholinguistics
Cohort Model
Theory of spoken word recognition, pruning paths in phonological space.
Connectionist Model
Neural network approach to language, distributed representations in weight space.
Eye-Tracking
Method measuring reading patterns, trajectories through text space.
Garden Path Sentence
Syntactically ambiguous construction, bifurcation points in parsing space.
Lexical Access
Word retrieval from mental lexicon, navigation through lexical topology.
N400 Component
ERP response to semantic anomaly, perturbation in neural semantic space.
Priming
Activation spreading through semantic networks, wave propagation in meaning space.
Semantic Priming
Facilitation through meaning relations, gradient descent in semantic space.
Tip-of-the-Tongue
Partial lexical retrieval state, local minima in retrieval landscape.
Working Memory
Cognitive buffer for language processing, finite-dimensional processing space.
Semantics
Compositionality
Principle that meaning is determined by parts and their combination, fundamental to semantic geometry.
Denotation
Relation between expressions and their referents, mapping linguistic forms to semantic space.
Entailment
Logical relation where one statement necessarily follows from another, creating directed edges in logic space.
Event Semantics
Framework treating events as basic entities, populating event manifolds in semantic space.
Extension
Set of entities to which an expression applies, defining regions in semantic space.
Felicity Conditions
Contextual requirements for appropriate utterance use, constraining pragmatic space.
Frame Semantics
Theory organizing word meanings around conceptual structures, creating semantic networks.
Generalized Quantifier
Set-theoretic interpretation of determiners, operating on semantic lattices.
Hyperintensionality
Semantic distinctions finer than logical equivalence, requiring higher-order geometric structures.
Implicature
Meaning conveyed indirectly, creating additional dimensions in communicative space.
Sociolinguistics
Accommodation Theory
Convergence/divergence in interaction, dynamic adjustment in style space.
Code-Switching
Alternation between languages or varieties, path switching in linguistic space.
Diglossia
Two varieties with distinct functions, bifurcated sociolinguistic space.
Enregisterment
Process linking forms to social meanings, mapping linguistic to social space.
Idiolect
Individual language variety, point in infinite-dimensional speaker space.
Indexical Order
Levels of social meaning, hierarchical structure in indexical space.
Language Ideology
Beliefs about language and speakers, constraint fields in social space.
Linguistic Variable
Feature varying across speakers, dimension in sociolinguistic space.
Register
Language variety for specific context, discrete regions in stylistic space.
Sociolect
Language variety of social group, clustering in sociolinguistic space.
Syntax
Binding Theory
Constraints on the interpretation of pronouns and anaphors, creating geometric constraints in reference space.
C-command
Structural relation in syntax trees, defining a partial order on the geometric tree structure.
Cartographic Syntax
Approach mapping fine-grained functional structure, creating high-dimensional syntactic spaces.
Case Theory
System of morphological marking on nominals, forming discrete points in grammatical space.
Constituent
Syntactic unit forming a single node in phrase structure, defining regions in tree geometry.
Control
Relation between a verb and the subject of its complement, creating directed paths in syntactic networks.
Cyclic Derivation
Syntactic computation proceeding in phases, modeling as iterative transformations.
Deep Structure
Abstract syntactic representation before transformations, the initial state in derivational geometry.
Dependency Grammar
Syntactic framework based on head-dependent relations, creating directed graph structures.
Derivational Morphology
Word formation through affixation, modeled as paths through morphological space.
Topology
Base Space
Underlying topological space in a bundle, representing core linguistic structure.
Boundary
Set of limit points not in the interior, marking linguistic category edges.
Closed Set
Set containing all its limit points, modeling complete linguistic categories.
Compact Space
Space where every open cover has finite subcover, ensuring bounded linguistic structures.
Connected Space
Space that cannot be separated into disjoint open sets, modeling unified concepts.
Continuous Function
Map preserving topological structure, modeling smooth linguistic transformations.
Covering Space
Space mapping locally homeomorphically onto another, unfolding linguistic ambiguities.
Dense Set
Set whose closure is the entire space, modeling productive linguistic patterns.
Embedding
Homeomorphism onto its image, preserving linguistic structure in larger space.
Fiber
Preimage of a point under projection, representing linguistic features at fixed position.