Foundational Mathematical Structures

Primes that supply the basic vocabulary of structured space: sets, relations, and functions organize elements; basis, dimension, vector space, and linear (in)dependence describe generating structure; symmetry, invariance, duality, and measure describe what a transformation preserves or assigns size to.

23 primes in this family — primes that sit near one another in abstraction space (k-means over structural-signature embeddings). Each is shown with its short description.

• Anisotropy — Make a property or response depend on direction, orientation, or axis in the carrier, so rotating the same probe changes the measured relation.
• Basis — A minimal independent generating set — the smallest collection from which every element of a space can be produced, with no member derivable from the others.
• Cartesian Product — All combinations of independent dimensions, one choice per axis.
• Coaxiality — Two or more spatial entities are coaxial when their constitutive axes coincide, allowing placement, rotation, and radial offset to be reasoned about relative to one common line.
• Constraint — Limits possibilities to guide outcomes.
• Coordinate-free — Define and reason about an object through intrinsic relations whose truth is invariant under every admissible change of coordinates, basis, chart, or component representation.
• Data Structure — An arrangement of information that makes some operations cheap at the structural cost of others.
• Dimension — Degrees of freedom in a system.
• Duality — Complementary perspectives.
• Frame of Reference — Observational perspective.
• Function (Mapping) — Relates inputs to outputs.
• Fuzzy Set — A fuzzy set makes belonging graded by assigning every candidate element a membership degree between zero and one, while retaining crisp sets as the endpoint-valued special case.
• Invariance — Properties unchanged under transformation.
• Linear Combination — Scale each of several objects by a weight and add them together.
• Linear Independence — No member of a collection is reproducible as a weighted sum of the others.
• Measure — An additive rule that assigns non-negative size to subsets of a space.
• Network — Models interactions between components.
• Relation — Describes associations or dependencies.
• Scale — Properties change with size.
• Set and Membership — Groups and categorizes elements.
• Subadditivity — A combined whole never evaluates above the sum of its separately evaluated parts, so decomposition supplies a guaranteed upper bound.
• Symmetry — Invariance under transformation.
• Vector Space — A collection closed under linear combination, where adding and scaling are coherent.