10502
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We can rise by degrees through abstraction, with higher levels representing more things
[Arnauld,A/Nicole,P]
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Full Idea:
I can start with a triangle, and rise by degrees to all straight-lined figures and to extension itself. The lower degree will include the higher degree. Since the higher degree is less determinate, it can represent more things.
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From:
Arnauld / Nicole (Logic (Port-Royal Art of Thinking) [1662], I.5)
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A reaction:
[compressed] This attempts to explain the generalising ability of abstraction cited in Idea 10501. If you take a complex object and eliminate features one by one, it can only 'represent' more particulars; it could hardly represent fewer.
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9578
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If objects are just presentation, we get increasing abstraction by ignoring their properties
[Frege]
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Full Idea:
If an object is just presentation, we can pay less attention to a property and it disappears. By letting one characteristic after another disappear, we obtain concepts that are increasingly more abstract.
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From:
Gottlob Frege (Review of Husserl's 'Phil of Arithmetic' [1894], p.324)
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A reaction:
Frege despises this view. Note there is scope in the despised view for degrees or levels of abstraction, defined in terms of number of properties ignored. Part of Frege's criticism is realist. He retains the object, while Husserl imagines it different.
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9930
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Abstraction is on a scale, of sets, to attributes, to type-formulas, to token-formulas
[Burgess/Rosen]
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Full Idea:
There is a scale of abstractness that leads downwards from sets through attributes to formulas as abstract types and on to formulas as abstract tokens.
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From:
JP Burgess / G Rosen (A Subject with No Object [1997], III.B.2.c)
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A reaction:
Presumably the 'abstract tokens' at the bottom must have some interpretation, to support the system. Presumably one can keep going upwards, through sets of sets of sets.
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10524
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There is a hierarchy of abstraction, based on steps taken by equivalence relations
[Hale]
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Full Idea:
The domain of the abstract can be seen as exemplifying a hierarchical structure, with differences of level reflecting the number of steps of abstraction, via appropriate equivalence relations, required for recognition at different levels.
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From:
Bob Hale (Abstract Objects [1987], Ch.3.III)
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A reaction:
I think this is right, and so does almost everyone else, since people cheerfully talk of 'somewhat' abstract and 'highly' abstract. Don't dream of a neat picture though. You might reach a level by two steps from one direction, and four from another.
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