ifc-0199
15.3 A hierarchy of mathematical construction
Mathematical success is not a single endpoint. At least five levels should be kept separate:
Parameter recovery fits constants inside a supplied law.
Family selection chooses among supplied presentations.
Candidate declaration extension introduces a typed operation or property missing from the seed language.
Candidate presentation construction creates a variable-size family of objects, maps, and equations, evaluated up to an explicit equivalence.
Admitted persistent theory construction makes later definitions, proofs, experiments, or decisions possible and demonstrably easier.
The first two levels can be scientifically useful without being transformationally creative. The third crosses the declared seed-language boundary, but the new declaration may still be a familiar operation reconstructed from a strongly informative probe. The fourth asks the system to determine the size and arrangement of a candidate extension. Admission at either level still requires the applicable independent checks. Only the fifth begins to approach the field-building sense of mathematical creativity that motivates this book. The experiments below move through the middle levels; none demonstrates the fifth at historical scale. Two abbreviated evidence families recur below. Infinitesimal Probes of Creativity (IPC) studies hidden mathematical presentations; the AI-Feynman-inspired family (AF) studies symbolic-law construction.
|
Evidence family |
Strongest supported construction |
Boundary not crossed |
|---|---|---|
|
Lea |
Reusable Lean declarations induced by an explicit structural obstruction |
Semantic persistence and open-ended concept formation |
|
IPC |
A variable-size latent presentation, scored up to permutation and scaling |
Reliable proposal-side abstention outside the registered doctrine |
|
AF |
An unnamed executable generator inside a typed semantic grammar |
Open proposal coverage once the candidate vocabulary is removed |
|
DIAL-URL |
A behavior-functor expression and an oriented comparison cell inside a supplied constructor and repair grammar |
A new primitive, distributive law, or persistent downstream learning theory |
|
Cellular automata and AGENTIC-MATH–0 |
Proposed tests of macroscopic ontology and integrated reuse |
No empirical result is reported here |
This map is an evidential ordering, not a leaderboard. The families use different worlds, observers, proposal languages, and admission contracts. Their results show where particular construction interfaces work or fail; they do not establish a common percentage of mathematical creativity.