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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. |
16864 | If principles are provable, they are theorems; if not, they are axioms [Frege] |
Full Idea: If the law [of induction] can be proved, it will be included amongst the theorems of mathematics; if it cannot, it will be included amongst the axioms. | |
From: Gottlob Frege (Logic in Mathematics [1914], p.203) | |
A reaction: This links Frege with the traditional Euclidean view of axioms. The question, then, is how do we know them, given that we can't prove them. |