more from Rudolph Carnap

Single Idea 8748

[catalogued under 6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism]

Full Idea

The logical positivists brought geometry into the fold of logicism. The axioms of, say, Euclidean geometry are simply definitions of primitive terms like 'point' and 'line'.

Gist of Idea

Logical positivists incorporated geometry into logicism, saying axioms are just definitions


report of Rudolph Carnap (Empiricism, Semantics and Ontology [1950]) by Stewart Shapiro - Thinking About Mathematics 5.3

Book Reference

Shapiro,Stewart: 'Thinking About Mathematics' [OUP 2000], p.129

A Reaction

If the concept of 'line' is actually created by its definition, then we need to know exactly what (say) 'shortest' means. If we are merely describing a line, then our definition can be 'impredicative', using other accepted concepts.