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Single Idea 17785

[filed under theme 9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta ]

Full Idea

The abstractness of the old fashioned real numbers has been replaced by generality in the modern theory of complete ordered fields.

Gist of Idea

Real numbers as abstracted objects are now treated as complete ordered fields

Source

John Mayberry (What Required for Foundation for Maths? [1994], p.408-2)

Book Ref

'Philosophy of Mathematics: anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.408

A Reaction

In philosophy, I'm increasingly thinking that we should talk much more of 'generality', and a great deal less about 'universals'. (By which I don't mean that redness is just the set of red things).

The 16 ideas with the same theme [what we should take abstract object to be]:

9607 The greatest discovery in human thought is Plato's discovery of abstract objects [Brown,JR on Plato]
16086 Objects lacking matter are intrinsic unities [Aristotle]
12990 Real (non-logical) abstract terms are either essences or accidents [Leibniz]
8647 Not all objects are spatial; 4 can still be an object, despite lacking spatial co-ordinates [Frege]
10545 Abstract objects may not cause changes, but they can be the subject of change [Dummett]
9885 The existence of abstract objects is a pseudo-problem [Dummett]
10487 I am a fan of abstract objects, and confident of their existence [Boolos]
10414 Abstract objects are constituted by encoded collections of properties [Zalta, by Swoyer]
10558 Abstract objects are actually constituted by the properties by which we conceive them [Zalta]
17785 Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry]
10403 If properties are abstract objects, then their being abstract exemplifies being abstract [Swoyer]
10513 Many abstract objects, such as chess, seem non-spatial, but are not atemporal [Hale]
10514 If the mental is non-spatial but temporal, then it must be classified as abstract [Hale]
10518 Shapes and directions are of something, but games and musical compositions are not [Hale]
10523 Being abstract is based on a relation between things which are spatially separated [Hale]
8704 Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend]