more from this thinker | more from this text
Full Idea
We can introduce a new type of object from the obtaining of some equivalence relation between objects of some already known kind, by identifying the new objects as equivalence classes of the old ones under that equivalence relation.
Gist of Idea
We can introduce new objects, as equivalence classes of objects already known
Source
report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Dummett - Frege philosophy of mathematics Ch.14
Book Ref
Dummett,Michael: 'Frege: philosophy of mathematics' [Duckworth 1991], p.167
A Reaction
Some accounts of abstraction merely describe the concept, but this is a rival to the traditional pyschological abstractionism that Frege attacked so vigorously. Should we take a platonist or constructivist view of the new objects?
| 9854 | We can introduce new objects, as equivalence classes of objects already known [Frege, by Dummett] |
| 9883 | Frege introduced the standard device, of defining logical objects with equivalence classes [Frege, by Dummett] |
| 9715 | An 'equivalence relation' is a reflexive, symmetric and transitive binary relation [Enderton] |
| 9716 | We 'partition' a set into distinct subsets, according to each relation on its objects [Enderton] |
| 8920 | Equivalence relations are reflexive, symmetric and transitive, and classify similar objects [Lipschutz] |
| 10104 | 'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words [George/Velleman] |
| 10103 | Grouping by property is common in mathematics, usually using equivalence [George/Velleman] |