6 ideas
| 18470 | Maybe truth-making is an unanalysable primitive, but we can specify principles for it [Smith,B] |
| Full Idea: The signs are that truth-making is not analysable in terms of anything more primitive, but we need to be able to say more than just that. So we ought to consider it as specified by principles of truth-making. | |
| From: Barry Smith (Truth-maker Realism: response to Gregory [2000], p.20), quoted by Fraser MacBride - Truthmakers 1.5 | |
| A reaction: This is the axiomatic approach to such problems - treat the target concept as an undefinable, unanalysable primitive, and then give rules for its connections. Maybe all metaphysics should work like that, with a small bunch of primitives. |
| 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. |
| 12583 | Belief truth-conditions are normal circumstances where the belief is supposed to occur [Papineau] |
| Full Idea: The truth condition of the belief is the 'normal' circumstances in which, given the learning process, it is biologically supposed to be present. | |
| From: David Papineau (Reality and Representation [1987], p.67), quoted by Christopher Peacocke - A Study of Concepts 5.2 | |
| A reaction: How do we account for a belief in ghosts in this story? The notion of 'normal' circumstances and what is 'biologically supposed' to happen don't seem very appropriate. This is the 'teleological' view of belief. |