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.
18 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.
- 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.
- Constraint — Limits possibilities to guide outcomes.
- 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.
- 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.
- Symmetry — Invariance under transformation.
- Vector Space — A collection closed under linear combination, where adding and scaling are coherent.