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Full Idea
On Frege's suggestion, functional terms that pick out abstract expressions (such as 'direction' or 'equinumeral') have a typical form of f(a) = f(b) iff aRb, where R is an equivalence relation, a relation which is reflexive, symmetric and transitive.
Gist of Idea
Functional terms can pick out abstractions by asserting an equivalence relation
Source
Gideon Rosen (Abstract Objects [2001], 'Way of Abs')
Book Ref
'Stanford Online Encyclopaedia of Philosophy', ed/tr. Stanford University [plato.stanford.edu], p.7
A Reaction
[Wright and Hale are credited with the details] This has become the modern orthodoxy among the logically-minded. Examples of R are 'parallel' or 'just as many as'. It picks out an 'aspect', which isn't far from the old view.
| 8912 | Nowadays abstractions are defined as non-spatial, causally inert things [Rosen] |
| 8913 | Chess may be abstract, but it has existed in specific space and time [Rosen] |
| 8914 | Sets are said to be abstract and non-spatial, but a set of books can be on a shelf [Rosen] |
| 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] |
| 8917 | The Way of Abstraction used to say an abstraction is an idea that was formed by abstracting [Rosen] |
| 8916 | Conflating abstractions with either sets or universals is a big claim, needing a big defence [Rosen] |
| 8915 | How we refer to abstractions is much less clear than how we refer to other things [Rosen] |