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Single Idea 9976
[filed under theme 18. Thought / E. Abstraction / 2. Abstracta by Selection
]
Full Idea
Frege's own conception of abstraction (although he disapproves of the term) is in agreement with the view that abstracting from the particular nature of the elements of M would yield the concept under which fall all sets equipollent to M.
Clarification
'Equipollent' means they map one-to-one onto each other
Gist of Idea
Frege accepts abstraction to the concept of all sets equipollent to a given one
Source
comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by William W. Tait - Frege versus Cantor and Dedekind III
Book Ref
'Philosophy of Mathematics: anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.44
A Reaction
Nice! This shows how difficult it is to slough off the concept of abstractionism and live with purely logical concepts of it. If we 'construct' a set, then there is a process of creation to be explained; we can't just think of platonic givens.
The
24 ideas
with the same theme
[mental acts which create abstract concepts]:
9789
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You can't abstract natural properties to make Forms - objects and attributes are defined together
[Aristotle]
|
9070
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We learn primitives and universals by induction from perceptions
[Aristotle]
|
10506
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Mathematics can be abstracted from sensible matter, and from individual intelligible matter
[Aquinas]
|
9104
|
A universal is the result of abstraction, which is only a kind of mental picturing
[William of Ockham]
|
23640
|
Only mature minds can distinguish the qualities of a body
[Reid]
|
23653
|
If you can't distinguish the features of a complex object, your notion of it would be a muddle
[Reid]
|
19577
|
Everything is a chaotic unity, then we abstract, then we reunify the world into a free alliance
[Novalis]
|
9145
|
We form the image of a cardinal number by a double abstraction, from the elements and from their order
[Cantor]
|
13454
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Cantor says (vaguely) that we abstract numbers from equal sized sets
[Hart,WD on Cantor]
|
9976
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Frege accepts abstraction to the concept of all sets equipollent to a given one
[Tait on Frege]
|
9361
|
We have to separate the mathematical from physical phenomena by abstraction
[Lewis,CI]
|
10644
|
A 'felt familiarity' with universals is more primitive than abstraction
[Price,HH]
|
10646
|
Our understanding of 'dog' or 'house' arises from a repeated experience of concomitances
[Price,HH]
|
9031
|
The basic concepts of conceptual cognition are acquired by direct abstraction from instances
[Price,HH]
|
10557
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Abstract objects are captured by second-order modal logic, plus 'encoding' formulas
[Zalta]
|
12657
|
Abstractionism claims that instances provide criteria for what is shared
[Fodor]
|
9149
|
To obtain the number 2 by abstraction, we only want to abstract the distinctness of a pair of objects
[Fine,K]
|
9150
|
We should define abstraction in general, with number abstraction taken as a special case
[Fine,K]
|
9982
|
Cantor and Dedekind use abstraction to fix grammar and objects, not to carry out proofs
[Tait]
|
9981
|
Abstraction is 'logical' if the sense and truth of the abstraction depend on the concrete
[Tait]
|
10141
|
Many different kinds of mathematical objects can be regarded as forms of abstraction
[Fine,K]
|
10229
|
Simple types can be apprehended through their tokens, via abstraction
[Shapiro]
|
9919
|
The old debate classified representations as abstract, not entities
[Burgess/Rosen]
|
8917
|
The Way of Abstraction used to say an abstraction is an idea that was formed by abstracting
[Rosen]
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