display all the ideas for this combination of philosophers
3 ideas
18883 | Any equivalence relation among similar things allows the creation of an abstractum [Simons] |
Full Idea: Whenever we have an equivalence relation among things - such as similarity in a certain respect - we can abstract under the equivalence and consider the abstractum. | |
From: Peter Simons (Modes of Extension: comment on Fine [2008], p.19) | |
A reaction: This strikes me as dressing up old-fashioned psychological abstractionism in the respectable clothing of Fregean equivalences (such as 'directions'). We can actually do what Simons wants without the precision of partitioned equivalence classes. |
18884 | Abstraction is usually seen as producing universals and numbers, but it can do more [Simons] |
Full Idea: Abstraction as a cognitive tool has been associated predominantly with the metaphysics of universals and of mathematical objects such as numbers. But it is more widely applicable beyond this standard range. I commend its judicious use. | |
From: Peter Simons (Modes of Extension: comment on Fine [2008], p.21) | |
A reaction: Personally I think our view of the world is founded on three psychological principles: abstraction, idealisation and generalisation. You can try to give them rigour, as 'equivalence classes', or 'universal quantifications', if it makes you feel better. |
9141 | Abstraction theories build mathematics out of second-order equivalence principles [Cook/Ebert] |
Full Idea: A theory of abstraction is any account that reconstructs mathematical theories using second-order abstraction principles of the form: §xFx = §xGx iff E(F,G). We ignore first-order abstraction principles such as Frege's direction abstraction. | |
From: R Cook / P Ebert (Notice of Fine's 'Limits of Abstraction' [2004], 1) | |
A reaction: Presumably part of the neo-logicist programme, which also uses such principles. The function § (extension operator) 'provides objects corresponding to the argument concepts'. The aim is to build mathematics, rather than the concept of a 'rabbit'. |