Table of Contents
Public units
| Unit | Title | Use |
|---|---|---|
| Readme | Mathematical Thinking - Readme | Follow connected mathematical work. |
| Preface | Mathematical Thinking - Preface | Understand the connected methods, their rationale, sources and limits. |
Part A - Choose and relate constructions
| § | ID & Title | Status | Keywords & Search Queries | Dependencies |
|---|---|---|---|---|
| 1 | MATH.16 - Choose a Mathematical Construction from Its Required Maps (Universal Property) | Usable, evolving | universal property; product; coproduct; pullback; function object; currying. What maps should a new object support, and how can that requirement select a construction? | MATH.2 for quotients; MATH.5 for generator extensions; MATH.7 for reversible representations; C.29 for interpretation in another subject. |
| 2 | MATH.17 - Construct Mathematical Spaces of Operations and Operations on Them | Usable, evolving | operations as objects; admissible functions; closure; higher-order operation; transformation law. How can a rule be constructed, combined or changed while retaining the consequence its use needs? | MATH.16 for function objects; MATH.1/.2 for retained steps or identification; MATH.18 for interpretation between accounts; C.29 for a working-method application. |
| 3 | MATH.1 - Build a Mathematical Structure of Composable Paths | Usable, evolving | generators; paths; endpoints; identity; associativity. Which elementary steps can be composed? When do different sequences need to remain distinct? | MATH.2 when paths will be identified; C.29 when the construction represents another subject. |
| 4 | MATH.2 - Treat Objects as the Same While Preserving Operations (Quotient) | Usable, evolving | quotient; congruence; equivalence; partial operation; refinement. Can these objects be treated as the same without losing a later operation or result? | MATH.1 when paths first need construction; MATH.6 for a counterexample to the proposed identification. |
| 5 | MATH.5 - Extend a Generator Assignment While Preserving Operations (Homomorphism) | Usable, evolving | generators; relations; homomorphism; free structure; extension. How can a choice on generators determine an operation-preserving map on everything they generate? | MATH.1 for paths; MATH.2 when an extension must descend to a quotient. |
| 6 | MATH.7 - Transport a Mathematical Structure Through a Bijection | Usable, evolving | bijection; transport; inverse map; isomorphism; domain of operation. How can a useful operation, law and answer be carried through a change of representation? | MATH.2 when identification is proposed instead of a bijection; C.29 for interpretation in another subject. |
| 7 | MATH.18 - Compare Mathematical Accounts through Interpretations | Usable, evolving | interpretation; primitive operation; preservation; reflection; round trip; equivalence. Which constructions, equations and maps transfer between two mathematical descriptions, and what can be recovered? | MATH.5 for generated interpretations; MATH.7 for bijective transport; MATH.2 for quotients; MATH.17 for transformations of operations. |
Part B - Construct and criticize arguments
| § | ID & Title | Status | Keywords & Search Queries | Dependencies |
|---|---|---|---|---|
| 1 | MATH.19 - Construct a Proof through Intermediate Claims | Stable | lemma; backward and forward reasoning; generalization; proof dependency. Which intermediate claim connects the available premises to the desired conclusion? | B.5.RA for recovery of a supplied argument; MATH.4 for induction; MATH.6 for a separating case; MATH.12 for obtaining an object from the proof. |
| 2 | MATH.4 - Construct a Witness by Induction | Usable, evolving | induction; recursive witness; base and step; representation. How can a proof supply an object for every finite input? Does the construction respect equivalent representations? | MATH.2 when a recursive construction must respect identification; MATH.12 for extracting constructions from other proof rules. |
| 3 | MATH.12 - Extract a Construction from a Proof | Usable, evolving | constructive proof; witness; function; pair; branch; finite search; computation. Which data-producing operation does a proof supply, and what is needed to execute it? | B.5.RA for an unfamiliar argument; MATH.4 for induction; C.29.2/.3 for formulation or execution questions. |
| 4 | MATH.6 - Refute a Mathematical Claim with a Countermodel | Usable, evolving | counterexample; countermodel; quantifiers; finite scope; encoding. What concrete structure refutes the claim? What does an unsuccessful bounded search leave unresolved? | B.5.RA if the claim’s argument needs recovery; MATH.2 when the counterexample defeats an identification. |
| 5 | MATH.20 - Bound a Mathematical Unknown by Comparable Constructions | Stable | bound; inequality; enclosure; relaxation; attainability; residual and error. Which comparison can answer the question before the whole unknown is obtained? | MATH.19 for an intermediate inequality; MATH.6 for a failed bound; MATH.21 for convergent approximation; FPF for choosing further work. |
Part C - Change a construction and develop its theory
| § | ID & Title | Status | Keywords & Search Queries | Dependencies |
|---|---|---|---|---|
| 1 | MATH.11 - Construct an Invariant from Transformation Rules | Usable, evolving | invariant; transformation rule; preservation equation; coefficient; reachability. How can a preserved expression be constructed and used to obtain a formula or exclude a target? | MATH.2 for identification; MATH.7 for transport; C.29 for a consequence about the modeled subject. |
| 2 | MATH.13 - Derive a Consequence from a Symmetry | Usable, evolving | symmetry; uniqueness; fixed point; orbit; conservation; numerical update. What does a transformation preserve, and which conclusion actually follows from that symmetry? | MATH.8 for the full orbit construction; MATH.9 for compatible choice; MATH.10 for admissible variation. |
| 3 | MATH.8 - Generate a Solution Family by Symmetry | Usable, evolving | group action; solution orbit; stabilizer; repetitions; representatives. Which solutions can be generated from one solution, and how much of the solution set does this cover? | MATH.13 for an unresolved symmetry consequence; MATH.9 when one compatible representative is required. |
| 4 | MATH.9 - Determine Whether and How a Choice Rule Can Respect Symmetry | Usable, evolving | equivariant choice; stabilizer; symmetry obstruction; additional data. Can one allowed answer be chosen consistently with symmetry? What can replace an impossible choice? | MATH.8 for solution orbits; MATH.13 for an earlier consequence or uniqueness question. |
| 5 | MATH.10 - Improve a Mathematical Candidate or Derive a Necessary Condition by Admissible Variation | Usable, evolving | admissible variation; improving change; stationary point; boundary minimum; constraint; first variation. Which change is allowed, can it improve the candidate, and what condition follows? Does that condition establish an optimum? | B.5.RA if the variational argument needs recovery; C.29 when a mathematical variation represents a subject change. |
| 6 | MATH.21 - Construct an Object through Convergent Approximations | Stable | limit; completeness; compatible approximation; uniform convergence; error control. How can finite approximations construct an object while retaining the next operation? | MATH.20 for bounds; MATH.19 for convergence and interchange arguments; MATH.2 for classes of representations; computational methods for effective obtaining. |
| 7 | MATH.22 - Change Axioms and Trace Their Consequences | Stable | axiom change; theory; interpretation; model; independence; proof repair. Which constructions and consequences survive when assumptions change? | MATH.18 for interpretations; MATH.19 for replacement proofs; MATH.6 for countermodels; MATH.23 for a further conjecture. |
| 8 | MATH.23 - Develop a Conjecture by Changing a Construction | Stable | conjecture; construction variation; proof and refutation; generalization; next problem. How can a change or obstruction yield a precise useful claim and an attainable next operation? | MATH.19 for proof construction; MATH.6 for refutation; MATH.22 for theory change; B.5.QD/C.40.CD for continued inquiry. |