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2 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. |
| 5202 | Maths and logic are true universally because they are analytic or tautological [Ayer] |
| Full Idea: The principles of logic and mathematics are true universally simply because we never allow them to be anything else; …in other words, they are analytic propositions, or tautologies. | |
| From: A.J. Ayer (Language,Truth and Logic [1936], Ch.4) | |
| A reaction: This is obviously a very appealing idea, but it doesn's explain WHY we have invented these particular tautologies (which seem surprisingly useful). The 'science of patterns' can be empirical and a priori and useful (but not tautological). |