5 ideas
| 13007 | Archimedes defined a straight line as the shortest distance between two points [Archimedes, by Leibniz] |
| Full Idea: Archimedes gave a sort of definition of 'straight line' when he said it is the shortest line between two points. | |
| From: report of Archimedes (fragments/reports [c.240 BCE]) by Gottfried Leibniz - New Essays on Human Understanding 4.13 | |
| A reaction: Commentators observe that this reduces the purity of the original Euclidean axioms, because it involves distance and measurement, which are absent from the purest geometry. |
| 13230 | Particular essence is often captured by generality [Steiner,M] |
| Full Idea: Generality is often necessary for capturing the essence of a particular. | |
| From: Mark Steiner (Mathematical Explanation [1978], p.36) | |
| A reaction: The most powerful features of an entity are probably those which are universal, like intelligence or physical strength in a human. Those characteristics are powerful because they compete with the same characteristic in others (perhaps?). |
| 13229 | Maybe an instance of a generalisation is more explanatory than the particular case [Steiner,M] |
| Full Idea: Maybe to deduce a theorem as an instance of a generalization is more explanatory than to deduce it correctly. | |
| From: Mark Steiner (Mathematical Explanation [1978], p.32) | |
| A reaction: Steiner eventually comes down against this proposal, on the grounds that some proofs are too general, and hence too far away from the thing they are meant to explain. |
| 13231 | Explanatory proofs rest on 'characterizing properties' of entities or structure [Steiner,M] |
| Full Idea: My proposal is that an explanatory proof makes reference to the 'characterizing property' of an entity or structure mentioned in the theorem, where the proof depends on the property. If we substitute a different object, the theory collapses. | |
| From: Mark Steiner (Mathematical Explanation [1978], p.34) | |
| A reaction: He prefers 'characterizing property' to 'essence', because he is not talking about necessary properties, since all properties are necessary in mathematics. He is, in fact, reverting to the older notion of an essence, as the core power of the thing. |
| 7492 | Early societies are based on community, and modern societies on association [Tönnies, by Watson] |
| Full Idea: Pre-modern societies are based on community (Gemeinschaft), whereas modern societies are based on association (Gesellschaft). | |
| From: report of Ferdinand Tönnies (Community and Association [1887]) by Peter Watson - Ideas Ch.32 | |
| A reaction: A very interesting distinction. The modern term implies contracts, and it strikes me as an extremely accurate description of modern liberal democracies. There is very little sense of community, but a strong sense of innumerable contracts that bind us. |