3 ideas
| 18189 | ZFC could contain a contradiction, and it can never prove its own consistency [MacLane] |
| Full Idea: We have at hand no proof that the axioms of ZFC for set theory will never yield a contradiction, while Gödel's second theorem tells us that such a consistency proof cannot be conducted within ZFC. | |
| From: Saunders MacLane (Mathematics: Form and Function [1986], p.406), quoted by Penelope Maddy - Naturalism in Mathematics | |
| A reaction: Maddy quotes this, while defending set theory as the foundation of mathematics, but it clearly isn't the most secure foundation that could be devised. She says the benefits of set theory do not need guaranteed consistency (p.30). |
| 3425 | Reduction has been defined as deriving one theory from another by logic and maths [Nagel,E, by Kim] |
| Full Idea: Ernest Nagel defines reduction as the possibility of deriving all laws of one theory by logic and mathematics to another theory, with appropriate 'bridging principles' (either definitions, or empirical laws) connecting the expressions of the two theories. | |
| From: report of Ernest Nagel (The Structure of Science [1961]) by Jaegwon Kim - Philosophy of Mind p.213 | |
| A reaction: This has been labelled as 'weak' reduction, where 'strong' reduction would be identity, as when lightning is reduced to electrical discharge. You reduce x by showing that it is y in disguise. |
| 9425 | Lewis later proposed the axioms at the intersection of the best theories (which may be few) [Mumford on Lewis] |
| Full Idea: Later Lewis said we must choose between the intersection of the axioms of the tied best systems. He chose for laws the axioms that are in all the tied systems (but then there may be few or no axioms in the intersection). | |
| From: comment on David Lewis (Subjectivist's Guide to Objective Chance [1980], p.124) by Stephen Mumford - Laws in Nature |