Mathematics Beta

Axiom-dependence is overstated. Core mathematical results are remarkably stable across axiom systems. You’re cherry-picking edge cases.

The continuum hypothesis is not an edge case — it’s a fundamental question about the size of infinity that axioms cannot resolve.

Refuted. Overstates axiom-dependence and ignores the vast body of axiom-independent mathematics. The constructivist case needs stronger foundations.

Social processes of validation don’t determine mathematical truth. The four-color theorem was true before computers verified it.

The four-color theorem’s acceptance required the mathematical community to expand its definition of ‘proof’ to include computer verification — a social decision.

Validated with revisions. The social-technology framing is productive. Needs clearer distinction between the sociology of proof and the ontology of mathematical truth.

Notation facilitates discovery but doesn’t create mathematical truth. Calculus existed conceptually before Leibniz’s notation.

Leibniz’s notation enabled manipulations (chain rule as fraction-like operation) that were literally unthinkable in Newton’s fluxion notation. The notation constitutively shaped the mathematics.

Partial. Continued growth from the social-technology line. The constitutive role of notation is well-argued. Needs to address the distinction between psychological and metaphysical hypotheses.