8 ideas
| 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. |
| 8886 | Being a true justified belief is not a sufficient condition for knowledge [Gettier] |
| Full Idea: The claim that someone knows a proposition if it is true, it is believed, and the person is justified in their belief is false, in that the conditions do not state a sufficient condition for the claim. | |
| From: Edmund L. Gettier (Is Justified True Belief Knowledge? [1963], p.145) | |
| A reaction: This is the beginning of the famous Gettier Problem, which has motivated most epistemology for the last forty years. Gettier implies that justification is necessary, even if it is not sufficient. He gives two counterexamples. |
| 5517 | Individuals don't exist, but are conventional names for sets of elements [Buddha] |
| Full Idea: There exists no individual, it is only a conventional name given to a set of elements. | |
| From: Buddha (Siddhartha Gautama) (reports [c.540 BCE]), quoted by Derek Parfit - The Unimportance of Identity p.295 | |
| A reaction: I take this to arise from an excessively spiritual concept of a human being, which faces Descartes' problem of how to individuate non-physical minds, when they have no clear boundaries. Combine dualism with a bundle theory, and you have Buddhism. |
| 7600 | The Buddha believed the gods would eventually disappear, and Nirvana was much higher [Buddha, by Armstrong,K] |
| Full Idea: The Buddha believed implicitly in the gods because they were part of his cultural baggage, but they were involved in the cycle of rebirth, and would eventually disappear; the ultimate reality of Nirvana was higher than the gods. | |
| From: report of Buddha (Siddhartha Gautama) (reports [c.540 BCE]) by Karen Armstrong - A History of God Ch.1 | |
| A reaction: We might connect this with Plato's Euthyphro question (Ideas 336 and 337), and the relationship between piety and morality on the one hand, and the gods on the other. |
| 7601 | Life is suffering, from which only compassion, gentleness, truth and sobriety can save us [Buddha] |
| Full Idea: Buddha taught that the only release from 'dukkha' (the meaningless flux of suffering which is human life) is a life of compassion for all living beings, speaking and behaving gently, kindly and accurately, and refraining from all intoxicants. | |
| From: Buddha (Siddhartha Gautama) (reports [c.540 BCE], Ch.1), quoted by Karen Armstrong - A History of God Ch.1 | |
| A reaction: Christians are inclined to give the impression that Jesus invented the idea of being nice, but it ain't so. The obvious thought is that the Buddha seems to be focusing on the individual, but this is actually a formula for a better community. |