Unit 1 Critical Notes: From Pasch’s Eight Grundsätze to Hilbert Group II¶
Scholarly & Constructive Commentary on Vorlesungen (1882), § 1
Direct Lineage to Hilbert’s Axioms II 1–3 (1899): Seventeen years after Pasch’s Vorlesungen, David Hilbert (Grundlagen § 3,
geometor/hilbert) compressed Pasch’s Grundsätze I–VIII into his Group II linear order axioms (II 1: symmetry $A\text{--}B\text{--}C \implies C\text{--}B\text{--}A$ and collinearity; II 2: density + prolongation $A\text{--}C\text{--}B$ and $A\text{--}B\text{--}D$; II 3: at most one of three collinear points lies between the other two), noting in 1902 that E. H. Moore proved Hilbert’s original four-point Theorem 4 redundant once Pasch’s planar Axiom II 4 is assumed!Why Spherical Geometry Fails Grundsatz III & VI: On a sphere (Riemannian elliptic geometry), great-circle segments prolonged sufficiently far return to their starting point, so $C$ can lie between $A$ and $B$ along one arc while $A$ lies between $B$ and $C$ along the complementary arc—violating Grundsatz III and breaking Euclid’s proof of the Exterior Angle Theorem (
Elements I.16)!Exact Symbolic Implementation: In
geometor.staudt.projective,is_between(A, B, C)verifies exact collinearity via zero_test of the $2 \times 2$ cross product and tests strict open positivity $0 < t < 1$ in $(C - A) = t (B - A)$ over exact SymPy rationals and quadratic radicals.