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2 ideas
| 13936 | Questions about numbers are answered by analysis, and are analytic, and hence logically true [Carnap] |
| Full Idea: For the internal question like 'is there a prime number greater than a hundred?' the answers are found by logical analysis based on the rules for the new expressions. The answers here are analytic, i.e., logically true. | |
| From: Rudolph Carnap (Empiricism, Semantics and Ontology [1950], 2) |
| 8748 | Logical positivists incorporated geometry into logicism, saying axioms are just definitions [Carnap, by Shapiro] |
| 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'. | |
| From: report of Rudolph Carnap (Empiricism, Semantics and Ontology [1950]) by Stewart Shapiro - Thinking About Mathematics 5.3 | |
| 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. |