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18. Thought / E. Abstraction / 7. Abstracta by Equivalence

[defining abstraction by the principle of equivalence]

40 ideas
Frege's logical abstaction identifies a common feature as the maximal set of equivalent objects [Frege, by Dummett]
Frege's 'parallel' and 'direction' don't have the same content, as we grasp 'parallel' first [Yablo on Frege]
Fregean abstraction creates concepts which are equivalences between initial items [Frege, by Fine,K]
Frege put the idea of abstraction on a rigorous footing [Frege, by Fine,K]
We create new abstract concepts by carving up the content in a different way [Frege]
From basing 'parallel' on identity of direction, Frege got all abstractions from identity statements [Frege, by Dummett]
You can't simultaneously fix the truth-conditions of a sentence and the domain of its variables [Dummett on Frege]
Abstraction principles identify a common property, which is some third term with the right relation [Russell]
The principle of Abstraction says a symmetrical, transitive relation analyses into an identity [Russell]
A certain type of property occurs if and only if there is an equivalence relation [Russell]
Since abstract objects cannot be picked out, we must rely on identity statements [Dummett]
There is no reason why abstraction by equivalence classes should be called 'logical' [Dummett, by Tait]
We arrive at the concept 'suicide' by comparing 'Cato killed Cato' with 'Brutus killed Brutus' [Dummett]
An 'abstraction principle' says two things are identical if they are 'equivalent' in some respect [Boolos]
For most sets, the concept of equivalence is too artificial to explain abstraction [Lewis]
The abstract direction of a line is the equivalence class of it and all lines parallel to it [Lewis]
Mathematicians abstract by equivalence classes, but that doesn't turn a many into one [Lewis]
If we can establish directions from lines and parallelism, we were already committed to directions [Wright,C]
Abstracted objects are not mental creations, but depend on equivalence between given entities [Hale/Wright]
One first-order abstraction principle is Frege's definition of 'direction' in terms of parallel lines [Hale/Wright]
Abstractionism needs existential commitment and uniform truth-conditions [Hale/Wright]
Equivalence abstraction refers to objects otherwise beyond our grasp [Hale/Wright]
A sentence should be recarved to reveal its content or implication relations [Yablo]
Abstraction may concern the individuation of the set itself, not its elements [Tait]
Fine considers abstraction as reconceptualization, to produce new senses by analysing given senses [Fine,K, by Cook/Ebert]
We can abstract from concepts (e.g. to number) and from objects (e.g. to direction) [Fine,K]
Abstractionism can be regarded as an alternative to set theory [Fine,K]
An object is the abstract of a concept with respect to a relation on concepts [Fine,K]
An abstraction principle should not 'inflate', producing more abstractions than objects [Fine,K]
Abstraction-theoretic imperialists think Fregean abstracts can represent every mathematical object [Fine,K]
We can combine ZF sets with abstracts as urelements [Fine,K]
We can create objects from conditions, rather than from concepts [Fine,K]
Abstract objects might come by abstraction over an equivalence class of base entities [Shapiro]
I prefer the open sentences of a Constructibility Theory, to Platonist ideas of 'equivalence classes' [Chihara]
You can think of a direction without a line, but a direction existing with no lines is inconceivable [Lowe]
Functional terms can pick out abstractions by asserting an equivalence relation [Rosen]
Abstraction by equivalence relationships might prove that a train is an abstract entity [Rosen]
Any equivalence relation among similar things allows the creation of an abstractum [Simons]
Abstraction is usually seen as producing universals and numbers, but it can do more [Simons]
Abstraction theories build mathematics out of second-order equivalence principles [Cook/Ebert]