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Single Idea 9881

[filed under theme 18. Thought / E. Abstraction / 7. Abstracta by Equivalence ]

Full Idea

Having rightly perceived that the fundamental class here was statements of identity between directions, Frege leapt to the conclusion that the basis for introducing new abstract terms consisted of determining the truth-conditions of identity-statements.

Gist of Idea

From basing 'parallel' on identity of direction, Frege got all abstractions from identity statements

Source

report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §64-68) by Michael Dummett - Frege philosophy of mathematics Ch.18

Book Ref

Dummett,Michael: 'Frege: philosophy of mathematics' [Duckworth 1991], p.232

A Reaction

This seems to be the modern view - that abstraction consists of the assertion of an equivalence principle. Dummett criticises Frege here (see Idea 9882). There always seems to be a chicken/egg problem. Why would the identity be asserted?

Related Idea

Idea 9882 You can't simultaneously fix the truth-conditions of a sentence and the domain of its variables [Dummett on Frege]

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]
9882 You can't simultaneously fix the truth-conditions of a sentence and the domain of its variables [Dummett on Frege]
9881 From basing 'parallel' on identity of direction, Frege got all abstractions from identity statements [Frege, by Dummett]
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]
8908 For most sets, the concept of equivalence is too artificial to explain abstraction [Lewis]
8907 The abstract direction of a line is the equivalence class of it and all lines parallel to it [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]
10562 We can combine ZF sets with abstracts as urelements [Fine,K]
10567 We can create objects from conditions, rather than from concepts [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]