Ideas from 'Foundations of Geometry' by Bertrand Russell [1897], by Theme Structure

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5. Theory of Logic / C. Ontology of Logic / 3. If-Thenism

10053

Geometrical axioms imply the propositions, but the former may not be true

			Full Idea: We must only assert of various geometries that the axioms imply the propositions, not that the axioms are true and therefore that the propositions are true.
			From: Bertrand Russell (Foundations of Geometry [1897], Intro vii), quoted by Alan Musgrave - Logicism Revisited §4
			A reaction: Clearly the truth of the axioms can remain a separate issue from whether they actually imply the theorems. The truth of the axioms might be as much a metaphysical as an empirical question. Musgrave sees this as the birth of if-thenism.

6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry

10052

Geometry is united by the intuitive axioms of projective geometry

			Full Idea: Russell sought what was common to Euclidean and non-Euclidean systems, found it in the axioms of projective geometry, and took a Kantian view of them.
			From: report of Bertrand Russell (Foundations of Geometry [1897]) by Alan Musgrave - Logicism Revisited §4
			A reaction: Russell's work just preceded Hilbert's famous book. Tarski later produced some logical axioms for geometry.