display all the ideas for this combination of texts
5 ideas
16901 | The equivalent algebra model of geometry loses some essential spatial meaning [Burge] |
Full Idea: Geometrical concepts appear to depend in some way on a spatial ability. Although one can translate geometrical propositions into algebraic ones and produce equivalent models, the meaning of the propositions seems to me to be thereby lost. | |
From: Tyler Burge (Frege on Apriority (with ps) [2000], 4) | |
A reaction: I think this is a widely held view nowadays. Giaquinto has a book on it. A successful model of something can't replace it. Set theory can't replace arithmetic. |
12451 | Scientific laws largely rest on the results of counting and measuring [Brouwer] |
Full Idea: A large part of the natural laws introduced by science treat only of the mutual relations between the results of counting and measuring. | |
From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.77) | |
A reaction: His point, I take it, is that the higher reaches of numbers have lost touch with the original point of the system. I now see the whole issue as just depending on conventions about the agreed extension of the word 'number'. |
16902 | Peano arithmetic requires grasping 0 as a primitive number [Burge] |
Full Idea: In the Peano axiomatisation, arithmetic seems primitively to involve the thought that 0 is a number. | |
From: Tyler Burge (Frege on Apriority (with ps) [2000], 5) | |
A reaction: Burge is pointing this out as a problem for Frege, for whom only the logic is primitive. |
12454 | Intuitionists only accept denumerable sets [Brouwer] |
Full Idea: The intuitionist recognises only the existence of denumerable sets. | |
From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.80) | |
A reaction: That takes you up to omega, but not beyond, presumably because it then loses sight of the original intuition of 'bare two-oneness' (Idea 12453). I sympathise, but the word 'number' has shifted its meaning a lot these days. |
12453 | Neo-intuitionism abstracts from the reuniting of moments, to intuit bare two-oneness [Brouwer] |
Full Idea: Neo-intuitionism sees the falling apart of moments, reunited while remaining separated in time, as the fundamental phenomenon of human intellect, passing by abstracting to mathematical thinking, the intuition of bare two-oneness. | |
From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.80) | |
A reaction: [compressed] A famous and somewhat obscure idea. He goes on to say that this creates one and two, and all the finite ordinals. |