6 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. |
16776 | Substance is an intrinsic thing, so parts of substances can't also be intrinsic things [Duns Scotus] |
Full Idea: Substance ...is an ens per se. No part of a substance is an ens per se when it is part of a substance, because then it would be a particular thing, and one substance would be a particular thing from many things, which does not seem to be true. | |
From: John Duns Scotus (In Praed. [1300], 15.1), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 26.1 | |
A reaction: The tricky bit is 'when it is a part of a substance', meaning a substance must cease to be a substance when it is subsumed into some greater substance. Maybe. Drops of water? Molecules? Bricks? Cells? |
12709 | Motion is not absolute, but consists in relation [Leibniz] |
Full Idea: In reality motion is not something absolute, but consists in relation. | |
From: Gottfried Leibniz (On Motion [1677], A6.4.1968), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 3 | |
A reaction: It is often thought that motion being relative was invented by Einstein, but Leibniz wholeheartedly embraced 'Galilean relativity', and refused to even consider any absolute concept of motion. Acceleration is a bit trickier than velocity. |