display all the ideas for this combination of philosophers
3 ideas
21444 | Modern geoemtry is either 'pure' (and formal), or 'applied' (and a posteriori) [Gardner] |
Full Idea: There is now 'pure' geometry, consisting of formal systems based on axioms for which truth is not claimed, and which are consequently not synthetic; and 'applied', a branch of physics, the truth of which is empirical, and therefore not a priori. | |
From: Sebastian Gardner (Kant and the Critique of Pure Reason [1999], 03 'Maths') | |
A reaction: His point is that there is no longer any room for a priori geometry. Might the same division be asserted of arithmetic, or analysis, or set theory? |
10480 | First-order logic can't discriminate between one infinite cardinal and another [Hodges,W] |
Full Idea: First-order logic is hopeless for discriminating between one infinite cardinal and another. | |
From: Wilfrid Hodges (Model Theory [2005], 4) | |
A reaction: This seems rather significant, since mathematics largely relies on first-order logic for its metatheory. Personally I'm tempted to Ockham's Razor out all these super-infinities, but mathematicians seem to make use of them. |
10286 | A 'set' is a mathematically well-behaved class [Hodges,W] |
Full Idea: A 'set' is a mathematically well-behaved class. | |
From: Wilfrid Hodges (First-Order Logic [2001], 1.6) |