4 ideas
10245 | One geometry cannot be more true than another [Poincaré] |
Full Idea: One geometry cannot be more true than another; it can only be more convenient. | |
From: Henri Poincaré (Science and Method [1908], p.65), quoted by Stewart Shapiro - Philosophy of Mathematics | |
A reaction: This is the culminating view after new geometries were developed by tinkering with Euclid's parallels postulate. |
22293 | Hilbert said (to block paradoxes) that mathematical existence is entailed by consistency [Hilbert, by Potter] |
Full Idea: Hilbert proposed to circuvent the paradoxes by means of the doctrine (already proposed by Poincaré) that in mathematics consistency entails existence. | |
From: report of David Hilbert (On the Concept of Number [1900], p.183) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 19 'Exist' | |
A reaction: Interesting. Hilbert's idea has struck me as weird, but it makes sense if its main motive is to block the paradoxes. Roughly, the idea is 'it exists if it isn't paradoxical'. A low bar for existence (but then it is only in mathematics!). |
13160 | To exist and be understood, a multitude must first be reduced to a unity [Leibniz] |
Full Idea: A plurality of things can neither be understood nor can exist unless one first understands the thing that is one, that to which the multitude necessarily reduces. | |
From: Gottfried Leibniz (Notes on Comments by Fardella [1690], Prop 3) | |
A reaction: Notice that it is our need to understand which imposes the unity on the multitude. It is not just some random fiction, or a meaningless mechanical act of thought. |
13161 | Substances are everywhere in matter, like points in a line [Leibniz] |
Full Idea: There are substances everywhere in matter, just as points are everywhere in a line. | |
From: Gottfried Leibniz (Notes on Comments by Fardella [1690], Clarif) | |
A reaction: Since Leibniz is unlikely to believe in the reality of the points, we must wonder whether he was really committed to this infinity of substances. The more traditional notion of substance is always called 'substantial form' by Leibniz. |