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Single Idea 9981
[filed under theme 18. Thought / E. Abstraction / 2. Abstracta by Selection
]
Full Idea
If the sense of a proposition about the abstract domain is given in terms of the corresponding proposition about the (relatively) concrete domain, ..and the truth of the former is founded upon the truth of the latter, then this is 'logical abstraction'.
Gist of Idea
Abstraction is 'logical' if the sense and truth of the abstraction depend on the concrete
Source
William W. Tait (Frege versus Cantor and Dedekind [1996], V)
Book Ref
'Philosophy of Mathematics: anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.47
A Reaction
The 'relatively' in parentheses allows us to apply his idea to levels of abstraction, and not just to the simple jump up from the concrete. I think Tait's proposal is excellent, rather than purloining 'abstraction' for an internal concept within logic.
The
24 ideas
with the same theme
[mental acts which create abstract concepts]:
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9789
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You can't abstract natural properties to make Forms - objects and attributes are defined together
[Aristotle]
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9070
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We learn primitives and universals by induction from perceptions
[Aristotle]
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10506
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Mathematics can be abstracted from sensible matter, and from individual intelligible matter
[Aquinas]
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9104
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A universal is the result of abstraction, which is only a kind of mental picturing
[William of Ockham]
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23640
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Only mature minds can distinguish the qualities of a body
[Reid]
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23653
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If you can't distinguish the features of a complex object, your notion of it would be a muddle
[Reid]
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19577
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Everything is a chaotic unity, then we abstract, then we reunify the world into a free alliance
[Novalis]
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9145
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We form the image of a cardinal number by a double abstraction, from the elements and from their order
[Cantor]
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13454
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Cantor says (vaguely) that we abstract numbers from equal sized sets
[Hart,WD on Cantor]
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9976
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Frege accepts abstraction to the concept of all sets equipollent to a given one
[Tait on Frege]
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9361
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We have to separate the mathematical from physical phenomena by abstraction
[Lewis,CI]
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10644
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A 'felt familiarity' with universals is more primitive than abstraction
[Price,HH]
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10646
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Our understanding of 'dog' or 'house' arises from a repeated experience of concomitances
[Price,HH]
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9031
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The basic concepts of conceptual cognition are acquired by direct abstraction from instances
[Price,HH]
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10557
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Abstract objects are captured by second-order modal logic, plus 'encoding' formulas
[Zalta]
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12657
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Abstractionism claims that instances provide criteria for what is shared
[Fodor]
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9149
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To obtain the number 2 by abstraction, we only want to abstract the distinctness of a pair of objects
[Fine,K]
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9150
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We should define abstraction in general, with number abstraction taken as a special case
[Fine,K]
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9981
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Abstraction is 'logical' if the sense and truth of the abstraction depend on the concrete
[Tait]
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9982
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Cantor and Dedekind use abstraction to fix grammar and objects, not to carry out proofs
[Tait]
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10141
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Many different kinds of mathematical objects can be regarded as forms of abstraction
[Fine,K]
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10229
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Simple types can be apprehended through their tokens, via abstraction
[Shapiro]
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9919
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The old debate classified representations as abstract, not entities
[Burgess/Rosen]
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8917
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The Way of Abstraction used to say an abstraction is an idea that was formed by abstracting
[Rosen]
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