7 ideas
19369 | Lull's combinatorial art would articulate all the basic concepts, then show how they combine [Lull, by Arthur,R] |
Full Idea: Lull proposed a combinatorial art. He wanted to reconcile Islam and Christianity by articulating the basic concepts that their belief systems held in common, and then inventing a device that would allow these concepts to be combined. | |
From: report of Ramon (Ars Magna [1305]) by Richard T.W. Arthur - Leibniz 2 Intro | |
A reaction: Leibniz's Universal Characteristic was an attempt at continuing Lull's project. Lull's plan rested on Aristotle's categories. |
13472 | Hilbert aimed to eliminate number from geometry [Hilbert, by Hart,WD] |
Full Idea: One of Hilbert's aims in 'The Foundations of Geometry' was to eliminate number [as measure of lengths and angles] from geometry. | |
From: report of David Hilbert (Foundations of Geometry [1899]) by William D. Hart - The Evolution of Logic 2 | |
A reaction: Presumably this would particularly have to include the elimination of ratios (rather than actual specific lengths). |
9546 | Euclid axioms concerns possibilities of construction, but Hilbert's assert the existence of objects [Hilbert, by Chihara] |
Full Idea: Hilbert's geometrical axioms were existential in character, asserting the existence of certain geometrical objects (points and lines). Euclid's postulates do not assert the existence of anything; they assert the possibility of certain constructions. | |
From: report of David Hilbert (Foundations of Geometry [1899]) by Charles Chihara - A Structural Account of Mathematics 01.1 | |
A reaction: Chihara says geometry was originally understood modally, but came to be understood existentially. It seems extraordinary to me that philosophers of mathematics can have become more platonist over the centuries. |
18742 | Hilbert's formalisation revealed implicit congruence axioms in Euclid [Hilbert, by Horsten/Pettigrew] |
Full Idea: In his formal investigation of Euclidean geometry, Hilbert uncovered congruence axioms that implicitly played a role in Euclid's proofs but were not explicitly recognised. | |
From: report of David Hilbert (Foundations of Geometry [1899]) by Horsten,L/Pettigrew,R - Mathematical Methods in Philosophy 2 | |
A reaction: The writers are offering this as a good example of the benefits of a precise and formal approach to foundational questions. It's hard to disagree, but dispiriting if you need a PhD in maths before you can start doing philosophy. |
18217 | Hilbert's geometry is interesting because it captures Euclid without using real numbers [Hilbert, by Field,H] |
Full Idea: Hilbert's formulation of the Euclidean theory is of special interest because (besides being rigorously axiomatised) it does not employ the real numbers in the axioms. | |
From: report of David Hilbert (Foundations of Geometry [1899]) by Hartry Field - Science without Numbers 3 | |
A reaction: Notice that this job was done by Hilbert, and not by the fictionalist Hartry Field. |
3460 | Superactors and superspartans count against behaviourism [Putnam, by Searle] |
Full Idea: Putnam proposed the superactor/superspartan objection to behaviourism. | |
From: report of Hilary Putnam (Brains and Behaviour [1963]) by John Searle - The Rediscovery of the Mind Ch. 2.II | |
A reaction: This is a beautiful compression of the obvious counterexamples, which are behaviour-wth-no-experience, and experience-with-no-behaviour. Presumably, though, Spartans are disposed to go 'aagh!' when they get home, and there are no 'super' actors. |
19371 | Nine principles of God: goodness, greatness, eternity, power, wisdom, will, virtue, truth and glory [Lull, by Arthur,R] |
Full Idea: Lull restricted himself to only nine 'absolute principles' of God: goodness, greatness, eternity, power, wisdom, will, virtue, truth and glory | |
From: report of Ramon (Ars Magna [1305]) by Richard T.W. Arthur - Leibniz 2 'Combinatorics' | |
A reaction: Leibniz responded that God's perfections are infinite in number, and thus beyond human comprehension. Lull cut them down to nine, because he was designing a sort of conceptual logic that employed them. |