3 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. |
| 14248 | We could accept the integers as primitive, then use sets to construct the rest [Cohen] |
| Full Idea: A very reasonable position would be to accept the integers as primitive entities and then use sets to form higher entities. | |
| From: Paul J. Cohen (Set Theory and the Continuum Hypothesis [1966], 5.4), quoted by Oliver,A/Smiley,T - What are Sets and What are they For? | |
| A reaction: I find this very appealing, and the authority of this major mathematician adds support. I would say, though, that the integers are not 'primitive', but pick out (in abstraction) consistent features of the natural world. |
| 4377 | Intellectualism is an excessive emphasis on reasoning in moral philosophy [Burnyeat] |
| Full Idea: Intellectualism, a one-sided preoccupation with reason and reasoning, is a perennial failing in moral philosophy. | |
| From: Myles F. Burnyeat (Aristotle on Learning to be Good [1980], p.70) | |
| A reaction: But Kant's reply would be that while there is much more to moral behaviour, the only part which matters in morality is the reasoning part. And Socrates' view (ignorance is evil) is not obviously wrong. |