28 Concepts by Type
The map’s master diagram has one node per chapter, colored by which of five parts that chapter belongs to — Foundations, Objects, Structures, Theories, Applications. That’s why it can look sparse in a category like Objects: most objects (a vector, a matrix, a random variable) are discussed inside a structure or theory chapter rather than getting a page of their own, so they never became their own node. This page is the finer-grained companion: every concept in the glossary, classified individually, whether or not it has a chapter to itself.
A caveat worth stating rather than hiding: not everything below is cleanly an object, a structure, a theory, or an application — some entries are properties a structure can have (convex, bijective), maps between structures (homomorphism, linear map), or theorems about a theory (Bayes’ rule). Forcing those into one of the four boxes would be exactly the kind of false precision SPEC.md’s terminology discipline (§11) warns against, so the Notes column says so explicitly instead. The five section headings below match the map’s legend exactly; the notes carry the nuance the legend can’t.
28.1 Foundations
The substrate the object/structure/theory/application distinction is itself built on top of — logic, sets, functions, proof. Deliberately outside that four-way split, not a missing fifth box.
| Concept | Notes | Chapter |
|---|---|---|
| Axiom | a starting statement, not a claim needing proof | Axioms and Definitions |
| Bijective / injective / surjective | properties of a function | Relations and Functions |
| Equivalence relation | a relation with three specific properties | Relations and Functions |
| Function | a map — see What Is a Mathematical Structure? for why maps matter as much as objects | Relations and Functions |
| Logic (proposition, proof) | the inference rules everything else is checked against | Logic and Proof |
| Mathematical induction | a proof technique, introduced alongside its first real use | Discrete Mathematics |
| Relation | the general notion underlying equality, order, and functions alike | Relations and Functions |
| Set | the most primitive collection — almost everything else is built from it | Sets |
| Theorem | a statement derived from a theory’s axioms | Axioms and Definitions |
28.2 Objects
Specific instances — the level What Is a Mathematical Structure? uses “the matrix \(\begin{pmatrix}1&2\\0&1\end{pmatrix}\)” to illustrate. A general vector, a general matrix, and a general polynomial don’t get their own rows below — they’re the motivating examples inside Vector Spaces, not separate named concepts.
| Concept | Notes | Chapter |
|---|---|---|
| Basis | a specific spanning, linearly independent set of vectors | Vector Spaces |
| Derivative | a specific linear map (the best local approximation to \(f\)) | Differentiation |
| Dimension | a number attached to a vector space | Vector Spaces |
| Eigenvalue / eigenvector | a specific vector-and-scalar pair a linear map leaves undirected | Linear Algebra |
| Embedding | a specific learned vector | Machine Learning |
| Entropy / KL divergence / mutual information | numbers computed from a distribution | Machine Learning |
| Expectation | a number computed from a random variable | Probability |
| Number systems (\(\mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C}\)) | the site’s one dedicated Objects chapter | Number Systems |
| Random variable | a specific function \(\Omega \to \mathbb{R}\) | Probability |
| Span | a specific set — all linear combinations of a given collection | Vector Spaces |
| Zero divisor | a specific ring element | Rings and Fields |
28.3 Structures
Sets equipped with operations or relations satisfying a fixed axiom list — What Is a Mathematical Structure?’s own subject, plus the maps between structures of the same kind (homomorphism, linear map, functor), grouped here rather than given a sixth category of their own.
| Concept | Notes | Chapter |
|---|---|---|
| Abelian group | a group with one extra property (commutativity) | Groups |
| Affine map | a map between affine spaces (linear map + translation) | Affine Spaces |
| Affine space | points and displacements, kept structurally apart | Affine Spaces |
| Category | the maximal abstraction of the homomorphism pattern | Advanced Topics |
| Euclidean space | affine space + inner product | Euclidean Geometry |
| Field | a ring where every nonzero element has a multiplicative inverse | Rings and Fields |
| Functor | a map between categories | Advanced Topics |
| Group | one operation, four axioms | Groups |
| Hilbert space | a complete inner-product space | Advanced Topics |
| Homomorphism / isomorphism | maps between structures of the same kind | What Is a Mathematical Structure? |
| Inner product / norm | angle/orthogonality, and the length it induces | Inner Product Spaces |
| Linear map | a map between vector spaces; a matrix once bases are fixed | Linear Algebra |
| Mathematical structure | the general pattern this whole section instantiates | What Is a Mathematical Structure? |
| Metric space | a set with a distance function | Metric Spaces |
| Open set | the basic building block a topology is made of | Topological Spaces |
| Probability space | outcomes, events, and a probability measure | Probability |
| Ring | two operations, related by distributivity | Rings and Fields |
| Topological space | a set with a designated collection of open sets | Topological Spaces |
| Vector space | addition and scaling | Vector Spaces |
| Properties a structure can have: completeness, orthogonal, discrete/dense, convex, continuous | not structures themselves — see each one’s own chapter | — |
28.4 Theories
The body of results about a structure or family of problems — plus the named theorems and techniques that body of results actually produces.
| Concept | Notes | Chapter |
|---|---|---|
| Bayes’ rule | a theorem, not a structure | Probability |
| Combinatorics | counting subsets/functions of finite sets | Discrete Mathematics |
| Continuous | classically \(\varepsilon\)–\(\delta\); topologically, preimages of opens | Limits and Continuity |
| Convex / critical point | properties/objects Optimization is built around | Optimization |
| Differential Equations (ODE / PDE, dynamical system) | equations relating a function to its own derivatives | Differential Equations |
| Differentiation | the theory built on the derivative | Differentiation |
| Discrete / dense | properties distinguishing \(\mathbb{N},\mathbb{Z}\) from \(\mathbb{Q},\mathbb{R}\) | Discrete Mathematics |
| Euclidean Geometry | affine + inner-product structure, assembled into geometry | Euclidean Geometry |
| Functional analysis, measure theory, differential geometry, category theory | preview-depth only, per SPEC.md §21 |
Advanced Topics |
| Fundamental theorem of calculus | a theorem, not a structure | Integration |
| Gradient descent | an algorithm, not a structure | Optimization |
| Independence | a property/relation between random variables | Probability |
| Integration (Riemann sum) | the theory built on the integral | Integration |
| Linear Algebra | the theory of vector spaces and the linear maps between them | Linear Algebra |
| Probability | measure theory, wearing a specific interpretation | Probability |
| Real analysis | the rigorous foundation under Limits and Continuity | Limits and Continuity |
28.5 Applications
Domains that borrow a theory to solve a problem that isn’t, itself, mathematics.
| Concept | Notes | Chapter |
|---|---|---|
| Backpropagation | the chain rule, not separate mathematics | Machine Learning |
| Machine Learning | the site’s one written Applications chapter | Machine Learning |
| Manifold hypothesis | Euclidean Geometry’s manifold sketch, doing real work | Machine Learning |
| Physics | named throughout (Newton’s laws as ODEs, symmetry and invariance) but not yet its own chapter | — (planned) |
28.6 Reading this table against the map
Where a row’s category surprised you, that’s usually the point: a derivative is an object (a specific linear map), not a theory, even though Differentiation — the theory about derivatives in general — is a whole chapter. That gap between “the theory” and “an instance the theory produces” is exactly what What Is a Mathematical Structure? spends a full chapter making explicit, and it’s the same distinction the map’s five-color legend is tracking at chapter granularity.