Axiom & Geometry Truth Lab

Epistemic Axiom Rule: In mathematics, proofs establish eternal conditional truth: "Given Axiom Set A, Theorem T is eternally proved." Change the axioms, and the valid theorems change.

EUCLIDEAN FLAT PLANE

Drag vertices A, B, C or External Point P to evaluate geodesics & parallel postulate

Live Manifold Metrics K = 0.00

Triangle Angle Sum (α+β+γ)

180.0°

Parallels Through Point P

Curvature K

Area / Angular Excess

0.0°

In Euclidean flat geometry (curvature K = 0), Euclid's 5th postulate holds: exactly 1 parallel line exists through an external point, and triangle interior angles sum to exactly 180°.

Axiomatic Switchboard

Euclid's 5th Postulate (Parallel Lines):

Deductive Logic System:

Theorem Validity in this System

Pythagorean Theorem (a²+b²=c²) VALID
Sum of Angles = 180° (α+β+γ=π) VALID
Rectangles Can Exist (4 Right Angles) VALID

Formal Deductive Proof Chain

Tracing how mathematical truth derives strictly from assumptions. Proofs do not expire with time; they hold eternally for their axiom set.

A1:

Postulates I–IV: Points, lines, circles, and right angles exist. Hilbert & Euclid Foundations

A5:

Playfair's Parallel Postulate: Given a line and point P not on it, exactly one parallel exists. Euclidean Axiom Set

L1:

Classical Logic: Law of Excluded Middle (A ∨ ¬A holds). Proof by contradiction allowed. Aristotelian First-Order Logic

∴

Theorem 180°: Sum of interior angles is constant and equal to 180°. Eternally True under {A1..A4, A5, L1}

Mathematical System Verdict Card

System: Euclidean Geometry (K=0)

Logic: Classical (Aristotelian with LEM)

Pythagoras: Valid | Angle Sum: Exactly 180° | Parallels: Exactly 1

"Is there permanent truth in Mathematics? Yes. A theorem once proven from axioms remains permanently proved for all eternity within that axiomatic frame."

Axioms, Geometry Proxies and Conditional Claims

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Changing axioms changes what the saved lab reports. Keep its default Cartesian triangle coordinates minus eighty minus fifty, eighty minus fifty, and zero seventy. The Euclidean branch reports one hundred eighty degrees. Its spherical teaching proxy takes Cartesian area nine thousand six hundred divided by four hundred as angular excess, reporting two hundred four degrees. Its hyperbolic proxy divides the same area by three hundred as a defect, reporting one hundred forty eight degrees. These are clamped display approximations, not exact geodesic angle calculations on a sphere or hyperbolic manifold. All three bars use three pixels per degree. The proxy also contains a clamp. In spherical mode the default triangle has base one hundred sixty and height one hundred twenty, giving area nine thousand six hundred. Drag the top vertex from y seventy to y ten. Height becomes sixty and area halves to four thousand eight hundred. Dividing by four hundred gives twelve degrees, but the source enforces a minimum excess of fifteen. Its displayed sum therefore changes from two hundred four to one hundred ninety five degrees, rather than one hundred ninety two. Both drawn triangles use the same coordinate scale, two pixels per source unit. The logic switch and geometry calculation are separate in this implementation. Change from classical to intuitionistic logic while retaining the same spherical geometry and default vertices. The proof-language panel changes, but the angle proxy stays two hundred four degrees. The parallel field uses zero for spherical mode, one for Euclidean mode, and nine hundred ninety nine as a sentinel displayed as infinity for hyperbolic mode. The theorem flags are assigned by the selected geometry branch; this interface does not mechanically prove an arbitrary theorem. Export preserves the selected manifold, logic, telemetry and theorem flags for inspection.