Mathematical Axiom & Pattern Discovery Workbench

Axiom & Rule Builder Invented Rules

Formal System Preset

Target Domain

Selected Axioms (Arbitrary Constraints)

Natural Number Succession Ax-1

Peano induction: every natural number has a unique next successor.

Multiplicative Closure Ax-2

Products of elements yield elements belonging strictly to the set.

Alternate Parity Rule Ax-3

Odd and even parity strictly alternate along succession paths.

Commutative Symmetry Ax-4

Binary operations satisfy a * b = b * a invariantly.

Deduction Rules (Inference Engine)

Mod-n Congruence Op-1

Equivalence partitioning by remainder cycles.

Exponential Scaling Op-2

Repeated multiplication creating power series trees.

Contradiction Elimination Op-3

Law of excluded middle for non-constructive assertions.

Philosophical Grounding: As Dr. Andrew Winkler (NYU Courant) observed, mathematical axioms and rules of deduction are arbitrary choices that define a formal game. You create the rules arbitrarily.

Formal Dependency & Invariant Network

Interactive D3 Force Graph (Drag nodes to inspect)

Invented Axiom / Rule

Discovered Invariant

Domain Core

Graph active: 3 axioms, 2 inference operators mapping to 4 inevitable invariants.

Discovery Matrix & Telemetry Discovered Truths

Arbitrary Axioms

Discovered Invariants

Consistency Score

88.5%

Deduction Rules

Primary Synthesis

While axioms are invented game rules, prime distribution invariants emerge inevitably regardless of initial notation.

Emergent Invariants (Platonic Discovered Truths)

The Resolution: Is math created or discovered? As the Courant & UIUC logicians conclude: Mathematics is both. Humans invent the formal rules and notation; but once those rules are set, their logical consequences and structural invariants exist inevitably and are genuinely discovered.