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Single Idea 10135
[filed under theme 18. Thought / E. Abstraction / 7. Abstracta by Equivalence
]
Full Idea
A principle of abstraction is 'conceptual' when the items upon which it abstracts are concepts (e.g. a one-one correspondence associated with a number), and 'objectual' if they are objects (parallel lines associated with a direction).
Gist of Idea
We can abstract from concepts (e.g. to number) and from objects (e.g. to direction)
Source
Kit Fine (The Limits of Abstraction [2002], I)
Book Ref
Fine,Kit: 'The Limits of Abstraction' [OUP 2008], p.1
The
40 ideas
with the same theme
[defining abstraction by the principle of equivalence]:
9855
|
Frege's logical abstaction identifies a common feature as the maximal set of equivalent objects
[Frege, by Dummett]
|
10802
|
Frege's 'parallel' and 'direction' don't have the same content, as we grasp 'parallel' first
[Yablo on Frege]
|
10526
|
Fregean abstraction creates concepts which are equivalences between initial items
[Frege, by Fine,K]
|
10525
|
Frege put the idea of abstraction on a rigorous footing
[Frege, by Fine,K]
|
10556
|
We create new abstract concepts by carving up the content in a different way
[Frege]
|
9881
|
From basing 'parallel' on identity of direction, Frege got all abstractions from identity statements
[Frege, by Dummett]
|
9882
|
You can't simultaneously fix the truth-conditions of a sentence and the domain of its variables
[Dummett on Frege]
|
10583
|
Abstraction principles identify a common property, which is some third term with the right relation
[Russell]
|
10582
|
The principle of Abstraction says a symmetrical, transitive relation analyses into an identity
[Russell]
|
10584
|
A certain type of property occurs if and only if there is an equivalence relation
[Russell]
|
10549
|
Since abstract objects cannot be picked out, we must rely on identity statements
[Dummett]
|
9993
|
There is no reason why abstraction by equivalence classes should be called 'logical'
[Dummett, by Tait]
|
9857
|
We arrive at the concept 'suicide' by comparing 'Cato killed Cato' with 'Brutus killed Brutus'
[Dummett]
|
8693
|
An 'abstraction principle' says two things are identical if they are 'equivalent' in some respect
[Boolos]
|
8907
|
The abstract direction of a line is the equivalence class of it and all lines parallel to it
[Lewis]
|
8908
|
For most sets, the concept of equivalence is too artificial to explain abstraction
[Lewis]
|
15443
|
Mathematicians abstract by equivalence classes, but that doesn't turn a many into one
[Lewis]
|
13898
|
If we can establish directions from lines and parallelism, we were already committed to directions
[Wright,C]
|
10630
|
Abstracted objects are not mental creations, but depend on equivalence between given entities
[Hale/Wright]
|
8786
|
One first-order abstraction principle is Frege's definition of 'direction' in terms of parallel lines
[Hale/Wright]
|
12227
|
Abstractionism needs existential commitment and uniform truth-conditions
[Hale/Wright]
|
12228
|
Equivalence abstraction refers to objects otherwise beyond our grasp
[Hale/Wright]
|
10805
|
A sentence should be recarved to reveal its content or implication relations
[Yablo]
|
9985
|
Abstraction may concern the individuation of the set itself, not its elements
[Tait]
|
9142
|
Fine considers abstraction as reconceptualization, to produce new senses by analysing given senses
[Fine,K, by Cook/Ebert]
|
10135
|
We can abstract from concepts (e.g. to number) and from objects (e.g. to direction)
[Fine,K]
|
10137
|
Abstractionism can be regarded as an alternative to set theory
[Fine,K]
|
10138
|
An object is the abstract of a concept with respect to a relation on concepts
[Fine,K]
|
10527
|
An abstraction principle should not 'inflate', producing more abstractions than objects
[Fine,K]
|
10561
|
Abstraction-theoretic imperialists think Fregean abstracts can represent every mathematical object
[Fine,K]
|
10567
|
We can create objects from conditions, rather than from concepts
[Fine,K]
|
10562
|
We can combine ZF sets with abstracts as urelements
[Fine,K]
|
10231
|
Abstract objects might come by abstraction over an equivalence class of base entities
[Shapiro]
|
9568
|
I prefer the open sentences of a Constructibility Theory, to Platonist ideas of 'equivalence classes'
[Chihara]
|
8306
|
You can think of a direction without a line, but a direction existing with no lines is inconceivable
[Lowe]
|
8918
|
Functional terms can pick out abstractions by asserting an equivalence relation
[Rosen]
|
8919
|
Abstraction by equivalence relationships might prove that a train is an abstract entity
[Rosen]
|
18883
|
Any equivalence relation among similar things allows the creation of an abstractum
[Simons]
|
18884
|
Abstraction is usually seen as producing universals and numbers, but it can do more
[Simons]
|
9141
|
Abstraction theories build mathematics out of second-order equivalence principles
[Cook/Ebert]
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