Combining Philosophers

Ideas for Archimedes, Niccolo Machiavelli and JC Beall / G Restall

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2 ideas

6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics

10692	Hilbert proofs have simple rules and complex axioms, and natural deduction is the opposite [Beall/Restall]
	Full Idea: There are many proof-systems, the main being Hilbert proofs (with simple rules and complex axioms), or natural deduction systems (with few axioms and many rules, and the rules constitute the meaning of the connectives).
	From: JC Beall / G Restall (Logical Consequence [2005], 3)

6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry

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.